Emmanuele DiBenedetto
Degenerate Parabolic Equations
,(
i
• Springer-Verlag
Emmanuele DiBenedetto
Degenerate Parabolic Equations With 12 Figures
Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest
Emmanuele DiBenedetto Northwestern University USA and University of Rome II Italy
Editorial Board (North America): I.H. Ewing Department of Mathematics Indiana University Bloomington, IN 47405 USA
F. W. Gebring Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA
P.R. Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA AMS Subject Classifications (1991): 35K65 Ubrary of Consress Catalo&ing-in-Publication Data DiBenedetto, Emmanue1e. Degenerate parabolic equationslEmmanue1e DiBenedetto. p. em. - (Universitcxt) Includes bibliographical references. ISBN 0-387-9402C).() (New York: acid-free). - ISBN 3-540-9402C).() (Berlin: acid-free) 1. Differential equations, Parabolic. I. Tide. QA377.062 1993 5W.353-dc20 93-285 @ 1993 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sip that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. This reprint has been authorized by Springer-Verlag (Berlin/Heidelberg/New York) for sale in the Mainland China only and not for export therefrom
987654321 ISBN 0-387-9402C).() Springer-Verlag New York Berlin Heidelberg ISBN 3-540-9402C).() Springer-Verlag Berlin Heidelberg New York
Preface
1. Elliptic equations: Harnack estimates and HOlder continuity Considerable progress was made in the early 1950s and mid-l960s in the theory of elliptic equations, due to the discoveries of DeGiorgi [33] and Moser [81,82]. Consider local weak solutions of {
(l.l)
u E
Wl~;(n),
(aijUz;)x., = 0
n
a domain in RN
in
n,
where the coefficients x --+ aij(x), i,j = 1,2, ... ,N are assumed to be only bounded and measurable and satisfying the ellipticity condition (1.2)
ajieiej ~ 'xleI 2 ,
a.e.
n, Ve E R N ,
for some ,X >
o.
DeGiorgi established that local solutions are HOlder continuous and Moser proved that non-negative solutions satisfy the Harnack inequality. Such inequality can be used, in turn, to prove the HOlder continuity of solutions. Both authors worked with linear p.d.e. 's. However the linearity has no bearing in the proofs. This pennits an extension of these results to quasilinear equations of the type (1.3)
{
wl!:(n),
p> 1 div a(x, u, Du) + b(x, u, Du)
U
E
= 0,
in
n,
with structure conditions (1.4)
a(x,u,Du)· Du ~ 'xIDuI P - tp(x), { la(x, u, Du)1 :5 AIDul p - 1 + tp(x), Ib(x, u, Du)1 :5 AIDul p -
1
+ tp(x) .
a.e. nT, p> 1
vi
Preface
Here 0 < A ~ A are two given constants and cp E Ll::(n) is non-negative. As a prototype we may take div IDuIP-2 Du = 0,
(1.5)
in
n,
p> 1.
The modulus of ellipticity of(1.5) is IDuI P- 2. Therefore at points where IDul =0, the p.d.e. is degenerate if p > 2 and it is singular if 1 < p < 2. By using the methods of DeGiorgi, Ladyzbenskaja and Ural'tzeva [66] established that weak solutions of (1.4) are finder continuous, whereas Serrin [92] and Trudinger [96], following the methods of Moser, proved that non-negative solutions satisfy a Harnack principle. The generalisation is twofold, i.e., the principal part a(x, u, Du) is pennitted to have a non-linear dependence with respect to UZi , i = 1,2, ... , N, and a non-linear growth with respect to IDul. The latter is of particular interest since the equation in (1.5) might be either degenerate or ·singular.
2. Parabolic equations: Harnack estimates and Holder continuity The first parabolic version of the Harnack inequality is due to Hadamard [50] and Pini [86] and applies to non-negative solutions of the heat equation. It takes the following fonn. Let u be a non-negative solution of the heat equation in the cylindrical domain n T == n x (0, T), 0 < T < 00, and for (xo, to) E nT consider the cylinder (2.1)
Qp == Bp(xo) x (to - p2, tol,
Bp(xo) == {Ix - xol < p} .
There exists a constant "y depending only upon N, such that if Q2p c nT, then
u(xo, to) ~
(2.2)
"y
sup u(x, to _ p2) . 8 p (zo)
The proof is based on local representations by means of heat potentials. A striking result of Moser [83] is that (2.2) continues to hold for non-negative weak solutions of (2.3)
{
u E V 1 ,2(nT)
== Loo (O,T;L2(n»)nL2 (o,T;Wl,2(n» ,
Ut-(ai;(x,t)uZi)zoJ =0,
in nT
where ai; E L 00 (nT ) satisfy the analog of the ellipticity condition (1.2). As before, it can be used to prove that weak solutions are locally II)lder continuous in T • Since the linearity of (2.3) is immaterial to the proof, one might expect, as in the elliptic case, an extension of these results to quasilinear equations of the type
n
(2.4)
{
u E V~,p(nT) Ut -
== Loo (0, T; L2(n»nV (O,~; Wl,p(n»),
dlva(x,t,u,Du)
= b(x, t,u, Du),
m nT,
where the structure condition is as in (1.4). Surprisingly however, Moser's proof could be extended only for the case p = 2, i.e., for equations whose principal
3. Parabolic equations and systems vii
IDul. This appears in the work of Aronson and Serrin [7] and Trudinger [97]. The methods of DeGiorgi also could not be extended. Ladyzenskaja et al. [67] proved that solutions of (2.4) are R>Ider continuous, provided the principal part has exactly a linear growth with respect to IDul. Analogous results were established by Kruzkov [60,61,62] and by Nash [84] by entirely different methods. Thus it appears that unlike the elliptic case, the degeneracy or singularity of the principal part plays a peculiar role, and for example, for the non-linear equation
part has a linear growth with respect to
(2.5)
Ut -
div
IDu IP-2 Du = 0,
one could not establish whether non-negative weak solutions satisfy the Harnack estimate or whether a solution is locally HOlder continuous.
3. Parabolic equations and systems These issues have remained open since the mid-1960s. They were revived however with the contributions ofN.N. Ural'tzeva [100] in 1968 and K. Uhlenbeck [99] in 1977. Consider the system (3.1)
Ui
E W,~:(n), p> 1, i=I,2, ... n,
in
n.
When p > 2, Ural'tzeva and Uhlenbeck prove that local solutions of (3.1) are of class c,t~:(n), for some aE (0, 1). The parabolic version of (3.1) is (3.2)
{
u
== (Ulo U2, ••. , un),
Ut -
div IDul
p - 2 DUi
Ui
= 0,
E V1,p(nT), i=l, 2, ... n, in aT.
Besides their intrinsic mathematical interest, this kind of system arises from geometry [99], quasiregularmappings [2,17,55,89] and fluid dynamics [5,8,56,57,74,75]. In particular Ladyzenskaja [65] suggests systems of the type of (3.2) as a model of motion of non-newtonian fluids. In such a case u is the velocity vector. Nonnewtonian here means that the stress tensor at each point of the fluid is not linearly proportional to the matrix of the space-gradient of the velocity. The function w = IDul 2 is formally a subsolution of (3.3) where at,k
_ {fJt,k + (P -
=
2)Ui,Zt Ui ,zlo }
IDul 2
•
This is a parabolic version of a similar finding observed in [99,I00J for elliptic systems. Therefore a parabolic version of the Ural'tzeva and Uhlenbeck result requires some understanding of the local behaviour of solutions of the porous media equation (3.4)
U~O,
m>O,
viii
Preface
and its quasilinear versions. Such an equation is degenerate at those points of OT where u=O ifm> 1 and singular ifO<m< 1. The porous medium equation has a life of its own. We only mention that questions of regularity were first studied by Caffarelli and Friedman. It was shown in [21] that non-negative solutions of the Cauchy problem associated with (3.4) are HOlder continuous. The result is not local. A more local point of view was adopted in [20,35,90]. However these contributions could only establish that the solution is continuous with a logarithmic modulus of continuity. In the mid-1980s, some progress was made in the theory of degenerate p.d.e. 's of the type of (2.5), for p > 2. It was shown that the solutions are locally HOlder continuous (see [39]). Surprisingly, the same techniques can be suitably modified to establish the local HOlder continuity of any local solution of quasilinear porous medium-type equations. These modified methods, in tum, are crucial in proving that weak solutions of the systems (3.2) are of class cl~; (OT). Therefore understanding the local structure of the solutions of (2.5) has implications to the theory of systems and the theory of equations with degeneracies quite different than (2.5).
4. Main results In these notes we will discuss these issues and present results obtained during the past five years or so. These results follow, one way or another, from a single unifying idea which we call intrinsic rescaling. The diffusion process in (2.5)
evolves in a time scale determined instant by iQstant by the solution. itself, so that, loosely speaking, it can be regarded as the heat equation in its own intrinsic timeconfiguration. A precise description of this fact as well as its effectiveness is linked to its technical implementations. We collect in Chap. I notation and standard material to be used as we proceed. Degenerate or singular p.d.e. of the type of (2.4) are introduced in Chap. II. We make precise their functional setting and the meaning of solutions and we derive truncated energy estimates for them. In Chaps. III and VI, we state and prove theorems regarding the local and global HOlder continuity of weak solutions of (2.4) both for p > 2 and 1 < p < 2 and discuss some open problems. In the singular case 1 < p < 2, we introduce in Chap. IV a novel iteration technique quite different than that of DeGiorgi [33J or Moser [83]. These theorems assume the solutions to be locally or globally bounded. A theory of boundedness of solutions is developed in Chap. V and it includes equations with lower order terms exhibiting the Hadamard natural growth condition. The sup-estimates we prove appear to be dramatically different than those in the linear theory. Solutions are locally bounded only if they belong to L ,oc ({}T) for some r ~l satisfying (4.1)
Ar == N(p - 2) + rp > 0
and such a condition is sharp. In Chap. XII we give a counterexample that shows that if (4.1) is violated, then (2.5) has unbounded solutions. The HOlder estimates and the Loo-bounds are the basis for an organic theory of local and global behaviour of solutions of such degenerate and/or singular equations.
4. Main results
ix
In Chaps. VI and VII we present an intrinsic version of the Harnack estimate and attempt to trace their connection with HOlder continuity. The natural parabolic cylinders associated with (2.5) are (4.2) We show by counterexamples that the Harnack estimate (2.2) cannot hold for nonnegative solutions of (2.5), in the geometry of (4.2). It does hold however in a time-scale intrinsic to the solution itself. These Harnack inequalities reduce to (2.2) when p = 2. In the degenerate case p > 2 we establish a global Harnack type estimate for non-negative solutions of (1.5) in the whole strip ET == RN X (0, T). We show that such an estimate is equivalent to a growth condition on the solution as Ixl - 00. If max{l; J~l} < p < 2, a surprising result is that the Harnack estimate holds in an elliptic form, i.e., holds over a ball Bp at a given time level. This is in contrast to the behaviour of non-negative solutions of the heat equation as pointed out by Moser [83] by a counterexample. These Harnack estimates in either the degenerate or singular case have been established only for non-negative solutions of the homogeneous equation (2.5). The proofs rely on some sort of nonlinear versions of 'fundamental solutions'. It is natural to ask whether they hold for quasilinear equations. This is a challenging open problem and parallels the Hadamard [50] and Pini [86] approach viafundamental solutions, versus the 'nonlinear' approach of Moser [83]. The number p is required to be larger than 2Nj (N + 1) and such a condition is sharp for a Harnack estimate to hold. The case 1 < P ~ 2Nj (N + 1) is not fully understood and it seems to suggest questions similar to those of the limiting Sobolev exponent for elliptic equations (see Brezis [19]) and questions in differen1, (2.5) tendsformally to a p.d.e. tial geometry. Here we only mention that as of the type of motion by mean curvature. HOlder and Harnack estimates as well as precise sup-bounds coalesce in the theory of the Cauchy problem associated with (2.4). This is presented in Chap. XI for the degenerate case p > 2 and in Chap. XII for the singular case 1 < p < 2. When p> 2, we identify the optimal growth of the initial datum as Ix I- 00 for a solution, local or global in time, to exist. This is the analog of the theory of Tychonov [98], Tacklind [94] and Widder [105] for the heat equation. When 1 < p < 2 it turns out that any non-negative initial datum U o E Lfoc(RN) yields a unique solution global in time. In general
p'"
2N I
Therefore the main difficulty of the theory is to make precise the meaning of solution. We introduce in Chap. XII a new notion of non-negative weak solutions and establish the existence and uniqueness of such solutions. We show by a counterexample that these might be discontinuous. Thus, in view of the possible singularities, the notion of solution is dramatically different than the notion of 'viscosity' solution. Issues of solutions of variable sign as well as their local and global behaviour are open. In Chaps. VIII-X, we tum to systems of the type (3.2) and prove that (4.3)
u~? E C/:'c(ilT),
i
= 1, 2, ... ,n,
j
= 1, 2, ... , N,
provided p > 2Nj(N + 1). Analogous estimates are derived for all p > 1 for solutions in L[oc(ilT), where r~1 satisfies (4.1). Again such a condition is sharp
x
Preface
for (4.3) to hold. Near the lateral boundary of ilT we establish C a estimates/or
all a E (0, 1), provided p > max {I; ~~2}' Estimates in the class boundary are still lacking even in the elliptic case.
c1,a near the
A similar spectrum of results could be developed for equations of the type (3.4). We have avoided doing this to keep the theory as organic and unified as possible. We have chosen not to present existence theorems for boundary value problems associated with (2.4) or (3.2). Theorems of this kind are mostly based on Galerkin approximations and appear in the literature in a variety of forms. We refer, for example, to [67] or [73]. Given the a priori estimates presented here these can be obtained alternatively by a limiting process in a family of approximating problems and an application of Minty's Lemma. These notes can be ideally divided in four parts:
1. 2. 3. 4.
HOlder continuity and boundedness of solutions (Chapters I-V) Harnack type estimates (Chapters VI-VII) Systems (Chapters VIII-X) Non-negative solutions in a strip ET (Chapters XI-XII).
These parts are technically linked but they are conceptually independent, in the sense that they deal with issues that have developed in independent directions. We have attempted to present them in such a way that they can be approached independently. The motivation in writing these notes, beyond the specific degenerate and singular p.d.e., is to present a body of ideas and techniques that are surprisingly flexible and adaptable to a variety of parabolic equations bearing, in one way or another, a degeneracy or singularity.
Acknowledgments The book is an outgrowth of my notes for the Lipschitz Vorlesungen that I delivered in the summer of 1990 at the Institut fUr Angewandte Math. of the University of Bonn, Germany. I would like to thank the Reinische Friedrich Wilhelm Universilit and the grantees of the Sonderforschungsbereich 256 for their kind hospitality and support. I have used preliminary drafts and portions of the manuscript as a basis for lecture series delivered in the Spring of 1989 at 1st. Naz. Alta Matematica, Rome Italy, in July 1992 at the Summer course of the Universidad Complutense de Madrid Spain and in the Winter 1992 at the Korean National Univ. Seoul Korea. My thanks to all the participants for their critical input and to these institutions for their support. I like to thank Y.C. Kwong for a critical reading of a good portion of the manuscript and for valuable suggestions. I have also benefited from the input of M. Porzio who read carefully the first draft of the first four Chapters, V. Vespri and Chen Ya-Zhe who have read various portions of the script and my students J. Park and M. O'Leary for their input.
Contents
Preface §1. Elliptic equations: Harnack estimates and HiUder continuity ....... v §2. Parabolic equations: Harnack estimates and Hi>lder continuity . . . . .. vi §3. Parabolic equations and systems. . . . . . . . . . . . . . . . . . . . . . . . .. vii §4. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. viii
I. Notation and function spaces §1. Some notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §2. Basic facts about W1,P(n) and w:,p(n). . . . . . . . . . . . . . . . . . . . §3. Parabolic spaces and embeddings . . . . . . . . . . . . . . . . . . . . . . . . . §4. Auxiliary lemmas .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §5. Bibliographical notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
I 3 7 12 15
II. Weak solutions and local energy estimates §1. Quasilinear degenerate or singular equations ................. §2. Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §3. Local integral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §4. Energy estimates near the boundary ....................... §5. Restricted structures: the levels k and the constant 'Y •••••••••••• §6. Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
16 20 22 31 38 40
xii
Contents
III. HOlder continuity of solutions of degenerate parabolic equations §1. The regularity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §2. Preliminaries ...................................... §3. The main proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §4. The first alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §5. The fllSt alternative continued . . . . . . . . . . . . . . . . . . . . . . . . . .. §6. The first alternative concluded .......................'. . .. §7. The second alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §8. The second alternative continued . . . . . . . . . . . . . . . . . . . . . . . .. §9. The second alternative concluded. . . . . . . . . . . . . . . . . . . . . . . .. §lO. Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §II. Regularity up to t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §12. Regularity up to ST. Dirichlet data ....................... §13. Regularity at ST. Variational data . . . . . . . . . . . . . . . . . . . . . . .. §14. Remarks on stability ................................. §15. Bibliographical notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
41 43 44 49 52 55 58 62 64 68 69 72 74 74 75
IV. HOlder continuity of solutions of singular parabolic equations §1. Singular equations and the regularity theorems . . . . . . . . . . . . . . .. §2. The main proposition .............................:.. §3. Preliminaries ...................................... §4. Rescaled iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §5. The first alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §6. Proof of Lemma 5.1. Integral inequalities. . . . . . . . . . . . . . . . . . .. §7. An auxiliary proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §8. Proof of Proposition 7.1 when (7.6) holds ................... §9. Removing the assumption (6.1) .......................... §10. The second alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §11. The second alternative concluded . . . . . . . . . . . . . . . . . . . . . . . .. §12. Proof of the main proposition ........................... §13. Boundary regularity .................................. §14. Miscellaneous remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §15. Bibliographical notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
77 79 81 84 88 92 95 97 101 102 106 109
no
114 116
v. Boundedness of weak solutions §1. §2. §3. §4. §5. §6.
Introduction....................................... Quasilinear parabolic equations. . . . . . . . . . . . . . . . . . . . . . . . .. Sup-bounds ....................................... Homogeneous structures. The degenerate case p > 2 ........... Homogeneous structures. The singular case 1 < p < 2 .......... Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
117 118 120 122 125 128
Contents xiii §7. Local iterative inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
(P>
............. §1O. Homogeneous structures and 1
137
Proof of Theorems 3.1 and 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorems 5.1 and 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . Natural growth conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138 140 142 143 144 147 149 155
max { 1; J~2}) 134 §8. Local iterative inequalities §9. Global iterative inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135 §11. §12. §13. §14. §15. §16. §17. §18.
VI. Harnack estimates: the case p>2 §1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §2. The intrinsic Harnack inequality . . . . . . . . . . . . . . . . . . . . . . . . . . §3. Local comparison functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §4. Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §5. Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §6. Global versus local estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . §7. Global Harnack estimates ....... '. . . . . . . . . . . . . . . . . . . . . .. §8. Compactly supported initial data . . . . . . . . . . . . . . . . . . . . . . . . . §9. Proof of Proposition 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. § 10. Proof of Proposition 8.1 continued . . . . . . . . . . . . . . . . . . . . . . .. § 11. Proof of Proposition 8.1 concluded . . . . . . . . . . . . . . . . . . . . . . .. § 12. The Cauchy problem with compactly supported initial data . . . . . . .. §13. Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
156 157 159 163 167 169 171 172 174 177 179 180 183
VII. Harnack estimates and extinction profile for singular equations § 1. §2. §3. §4. §5. §6. §7. §8. §9. § 10. §11.
The Harnack inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extinction in finite time (bounded domains) . . . . . . . . . . . . . . . . . . Extinction in finite time (in RN) ......................... An integral Harnack inequality for all 1 < p < 2 . . . . . . . . . . . . . . . Sup-estimates for J~l
184 188 191 193 198 199 203 204 206 211 214
xiv
Contents
VIII. Degenerate and singular parabolic systems §1. §2. §3. §4. §5. §6. §7.
Introduction....................................... Boundedness of weak solutions . . . . . . . . . . . . . . . . . . . . . . . . .. Weak differentiability of IDul2j1 Du and energy estimates for IDul Boundedness of IDul. Qualitative estimates .. , . . . . . . . . . . . . . . Quantitative sup-bounds of IDul . . . . . . . . . . . . . . . . . . . . . . . . . General structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
215 218 223 231 238 243 244
IX. Parabolic p-systems: HOlder continuity of Du §1. The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §2. Estimating the oscillation of Du . . . . . . . . . . . . . . . . . . . . . . . . . §3. HOlder continuity of Du (the case p > 2) . . . . . . . . . . . . . . . . . . §4. HOlder continuity of Du (the case 1 < p < 2 ) . . . . . . . . . . . . . . . . §S. Some algebraic Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §6. Linear parabolic systems with constant coefficients . . . . . . . . . . . . . §7. The perturbation lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. §S. Proof of Proposition l.l-(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §9. Proof of Proposition l.l-(ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . itO. Proof of Proposition l.l-(iii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 11. Proof of Proposition 1.1 concluded . . . . . . . . . . . . . . . . . . . . . . . . §12. Proof of Proposition 1..2-(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §13. Proof of Proposition 1.2 concluded. . . . . . . . . . . . . . . . . . . . . . . . §14. General structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §15. Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
245 248 251 256 258 263 268 275 278 282 284 286 288 291 291
X. Parabolic p-systems: boundary regularity §1. Introduction....................................... 292 §2. Flattening the boundary .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 §3. An iteration lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 §4. Comparing w and v (the case p > 2) . . . . . . . . . . . . . . . . . . . . 299 §5. Estimating the local average of IDwl (the case p > 2) .......... 304 §6. Estimating the local averages of w (the case p > 2) . . . . . . . . . . . 305 §7.
comparingwandv(thecasemax{1;;~2}
......... 309
§8. Estimating the local average of IDwl . . . . . . . . . . . . . . . . . . . . . 313 §9. Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
XI. Non-negative solutions in ET • The case p> 2 § 1. §2. §3. §4.
Introduction....................................... Behaviour of non-negative solutions as Ixl -+ 00 and as t "" 0 .... Proof of (2.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Initial traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
316 317 319 322
Contents xv
§5. §6. §7. §8.
Estimating IDulp - 1 in ET ............................ Uniqueness for data in LtoARN) ........................ Solving the Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . , Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
323 326 330 333
XII. Non-negative solutions in E T . The case 1
337 340 344 346 350 362 362 363 366 370 371 376 379
Bibliography ......................................
381
I Notation and function spaces
1. Some notation Let fl be a bounded domain in RN of boundary afl and for 0 denote the cylindrical domain flx (0, TJ. Also let,
8T == afl x [0, TJ,
< T < 00 let flT
r == 8 T u(flx {O})
denote the lateral boundary and the parabolic boundary of flT respectively. If fl is a sphere of radius P > 0 centered at some Xo E R N , we denote it by Bp(xo) == {Ix - xol < p}, and if Xo coincides with the origin, we let Bp(O) == Bp. The boundary afl will be assumed to satisfy the property of positive geometric density, i.e.,
(1.1)
there exists 0:* E (0,1) and Po> 0 such that 'r/xo E afl, { for every ball Bp(x o) centered at Xo and radius p::; Po
ISliBp(xo)I ::; (1 - o:*)IBp(xo)l, where 1171 denotes the Lebesgue measure of a measurable set E. At times it will be necessary to assume that afl is of class C1,A for some A E (0, 1). That is, there exist a positive number Po such that for all Xo E afl, the portion of afl within the ball BpJx o) can be represented, in a local system of coordinates, as the graph of a C1,A function ¢(x o ) such that ¢(xo)(x o ) =0. We set (1.2)
2
I. Notation and function spaces
Here for a smooth function fjJ defined on a compact subset IC of Rft. for some positive integer n [fjJh IC == sup IfjJ(x)1 + sup IfjJ(x) - fjJ~Y)I. 'zEIC (Z,II)EIC Ix - YI
(1.3)
The boundary all is piecewise smooth if it satisfies (1.1) and is the union of finitely many portions of (N-1) -dimensional hypersurfaces of class Cl,.x. If II is piecewise smooth, we say that a certain quantity, say C or 'Y, depends upon the structure ofall if it can be calculated apriori only in terms of the numbers o· ,Po in the definition (l.1), the number of the components making up all and the III . 11I1H norm of each of the components making up all. If 1 E Lq(ll), 1 ~ q ~ 00, denote by II/lIq,n the Lq(ll)-norm of I. The function 1 is in Lfoc(ll) if II/lIq,lC is finite for all compact subsets IC of ll. Let q, r ~ 1. A function 1 defined and measurable in llT belongs to
a
if
11/11 ••.,,,, '"
(l ([ IfI'dz) dT) f
1 < 00.
Also 1 E Lf::C(llT), iffor every compact subset IC of II and every subinterval
[t1' t2J C (0, TJ
J(/ I/lqdx) tl
i dr < 00.
IC
Whenever q = r we set Lq,r(llT) == Lq(llT), Lf::C(llT) == Lfoc(llT) and II/lIq,q;nT == II/lIq,nT' These definitions are extended in the obvious way when either q or r are infinity. If 1 E CI(llT), we denote by DI == (f"'11 1"'2"" ,I"'N) the gradient of 1 with respect to the space variables only. The spaces WI,p(n) and W!,P(ll). W l,p (ll)
p~ 1, are
defined by
is the completion of Coo (ll) under the norm IIvllwl,p(n) == livllp,n + IiDvllp,n,
W~'P(ll)
11
E COO(ll)nLP(ll).
is the completion ofC~(ll) under the norm IIvllw~,p(n)
== IIDvlip,n.
11
E C~(ll).
Equivalently WI,P(ll) is the Banach space of functions v E LP(ll) whose generalised derivatives v"" belong to LP(n) for all i= 1, 2, ... , N. A function v E Lfoc (ll) is in W,!;: (ll) if for every compact subset IC ell, v E Wl,P(K).
2. Basic facts about WI.pCo) and W~·P(O)
3
We let WI,oo(n) denote the space of functions v E LOO(n) whose disttibutional derivatives vz • are in LOO(n), for i = 1,2, ... , N. The space W,~:o(n) is defined analogously.
2. Basic facts about W 1,P(il) and W~'P(il) We collect here a few facts that will be of frequent use in what follows. The first is about the Gagliardo-Nirenberg multiplicative embedding inequality. THEOREM 2.1. Let v E w;,"(n), p ~ 1. For every fixed number s ~ I there exists a constant C depending only upon N, p and s such that
(2.1)
where Q E [0, 1], p, q ~ 1, are linked by (2.2)
and their admissible range is: (2.2-i)
q
N=I,
qE[S,OO],
(2.2-ii)
QE
if 1 ~ p < N, qE
[s, :!p]
qE[:!p'S] (2.2-iii)
if oE
p~N
> 1,
[O'P+S~_I)]; a E [0, I]
if if
and
s~ :!p' S~:!p;
q E [s, 00) and
[0, NP+~:-N»).
COROLLARY 2.1. LetvE W;,"(n), and assume pE [I,N). There exists a constant 'Y depending only upon N, and p, such that
(2.1)'
where
Np N-p
q=--.
We may take Q = 1 and S = 1 in (2.2-ii). H 8n is piecewise smooth, functions v in WI,"(n) are defined up to 8n via their traces. We will denote by vlan the trace on 8n of a function v E Wl,"(n) . PROOF:
4
I. Notation and function spaces
THEOREM 2.2. Assume that an is piecewise smooth. There exists a constant C depending only upon N,P and the structure of an such that
(2.3)
where (2.3-i)
1 (N - I)P] qE [ ' N -P '
if
1
(2.3-ii)
q E [1,00),
if
p= N.
< p < N,
If an is piecewise smooth. the space W;,"(n) can be defined equivalently as the set of functions v E WI,,, (n) whose trace on an is zero.
Remark 2.1. The embedding inequalities of Theorem 2.1 and Corollary 2.1 continue to hold for functions v in WI,"(n). not necessarily vanishing on an in the sense of the traces. provided we assume further that an is piecewise smooth and that
f
(2.4)
v(x)dx =
o.
n In such a case the constant C depends upon s, p, q, a, N and the structure of an. However it does not depend on the size of n. i.e., it does not change under dilations ofn. Let k be any real number and for a function v E WI,"(n) consider the truncations of v given by (2.5)
(v - k)+ == max{(v - k) ; O},
(v - k)_ == max{ -(v - k) ; O}.
LEMMA 2.1. LetvEW1,"(n). Thenforall kER, (v=Fk)± E Wl'''(n). Assume in addition that the trace of v on an is essentially bounded and
Ilvlloo,an :5 ko,
for some ko >
o.
Then for all k~ko, (v - k)± EW;,"(n). COROLLARY 2.2. Let Vi EW1,"(n),i=I,2, ... ,nEN. Then
w == min {Vl,V2, ... ,vn } E W1,"(n). PROOF: Assume first n=2. Then '.
mm
{
.} Vb V2
=
VI -
(V2 2
VI)+
The general case is proved by induction. If v is a continuous function defined in we set
+
V2 -
(VI -
2
V2)+
.
n and k < l is a pair of real numbers.
2. Basic facts about W1;P({}) and W.,';p({})
r>~
- {x [v < k] - {x [k < v < lj - {x
(2.6)
E
5
a Ivex) > l},
< k}, E n Ik < vex) < l} . E alv(x)
LEMMA 2.2. Let v E WI,I(Bp(X o)) nC(Bp(xo)) for some p > 0 and some Xo E RN and let k and I be any pair of real numbers such that k < I. There exists a constant "1 depending only upon N, p and independent of k, l, v, x o , p. such that
(2.7)
(l- k)1 [v
I
> 1]1 ~ "11 [~< kjl N+1
IDvldx.
[k
Remark 2.2. The conclusion of the lemma continues to hold for functions v E WI,I (n)nC( n) provided n is convex. We will use it in the case a is a hemisphere or a cube.
Remark 2.3. The continuity is not necessary to the conclusion of Lemma 2.2. The function v has been assumed to be continuous to give an unambiguous meaning to the definitions (2.6). If v is only in Wl,l (a). one could fix an arbitrary representative out of the equivalence class v say v and define (2.6) accordingly. The conclusion of the Lemma is independent of the choice of v.
2-(i). Poincare-type inequalities Inequality (2.7) is due to De Giorgi [33] and it is a particular case of a more general Poincare-type inequality. The embedding (2.1)' of Corollary 2.1 gives a majorisation of the Lq (n) -norm of u solely in terms of the V (n) -norm of its gradient. This is possible because one knows that u vanishes on aa in the sense of the traces. A Poincare-type inequality bounds some integral norm of a function u E Wl,p (n) in terms only of some integral norm of its gradient, provided some information is available on the set where u vanishes. PROPOSITION 2.1.
Let n be a bounded convex set in RN and let I.{) E C (1i)
satisfy
(2.8)
{
Let vE WI,p(a),
0
~ I.{) ~
the sets
"Ix E
1, [I.{)
n,
> k] are convex,
p~ 1, and assume
Vk E (0,1).
that the set
e == [v = Ojn[l.{) =
1]
has positive measure. There exists a constant C depending only upon Nand p and independent of v and I.{), such that (2.9)
6
I. Notation and fwiction spaces
PROOF: We fust prove (2.9) for p= 1. For every zEe and XE 0,
Iz-zl
Iv(x)1 = Iv(x) - v(z)1 = / J :p v(z+j:::jP) dp/ o Iz-zl
J /VV(Z+RP)/ dp.
$
o Multiply this inequality by cp(x) and integrate in dx over 0 and in dz over e. This gives Iz-zi
lei J cplvldx $ J dz J dx J CP(x)/DV(z+j:::jp)/ dp.
(2.10)
n
n
E
0
By virtue of the assumption (2.8)
cp(x)/Vv(z+j::=j"p)/
cpJDvl(z+j:::jP).
$
We put this inequality in (2.10) and compute the integral in dx on the right hand side, in polar coordinates with pole at z and radial variable r = Ix - zl. We let w be the angular variables and denote by 'R.(w) the polar representation of ao with pole at z. diam
lei Jcplvldx $ JdZ n
E
(
n
J rN- 1dr
)
1l(... )
J
dw
1... 1=1
0
$ -y(diamO)N
N1 J IxcpIDul(x) - z1N-1 r - dr 0
Je dz nJI:~~11~1 dx
$ -y(diamOt suPJ zEn
e
IX - d~N_1 JcplDvldx. Z n
Next, for all x E O. and for all 6> 0,
dz
Ix - zlN-l +
J
En{lz-zl~6}
Minimising with respect to the parameter 6 gives (2.11)
Jcplvl dx n
$ -y
(~;:N
JcplVvl dx, n
dz
Ix - z1N-1
3. Parabolic spaces and embeddings
7
for a constant 1'=1'(N,p). By replacing v with Ivl P in (2.11), we obtain
J
cplvl P dx
n
<
-
<
P'Y
(diamfl)N
1t:11--k
J
cplvlp-1lDvl dx
n
~jcplvlPdx + 1'(P) [(diam~N1P jCPIDvIPdx. 1t:1 1-
- 2
n
n
Remark 2.4. Inequality (2.7) follows by applying (2.9) with the function if v > k w= k if v ~ k.
cp == 1 and p = 1 to
{:in{V,l} -
By Corollary 2.2 such a function is in W1,1(fl).
3. Parabolic spaces and embeddings We introduce spaces of functions, depending on (x, t) E flT, that exhibit different regularity in the space and time variables. These are spaces where typically solutions of parabolic equations in divergence fonn are found. Let m, p ~ 1 and consider the Banach spaces
and
Vom,P(flT) == L oo (0, Tj Lm(fl))nLP (0, Tj W~,p(n)), both equipped with the nonn. v E Vm,P( flT ).
When m=p. we set V!,P(flT ) == V!(flT) and VP,P(flT ) ==VP(flT). Both spaces are embedded in Lq (flT ) for some q > p. In a precise way we have PROPOSITION 3.1. There exists a constant l' depending only upon N, p, m such that for every v E Vom,p( fl T )
(3.1)
jj1v(x,t)lqdxdt nT
",' ([/IDV(X,t)IPdxdt ) where
( esssupj Iv(x, t)lmdx) O
n
." ,
8
I. Notation and function spaces
N+m N
q=p--.
Moreover (3.2) The multiplicative inequality (3.l ) and the embedding (3.2) continue to hold for functions v E V m ,,,( T ) such that
n
J
v(x, t)dx
= 0,
fora.e.
tE(O,T),
n
provided an is piecewise smooth. In such a case the constant "f depends also on the structure of an. PROPOSITION 3.2. Assume that an is piecewise smooth. There exists a constant "f depending only upon N,p, m and the structure of an, such that for every v E
Vm,"(ilT ) ,
N+m N
(3.3)
q=p--.
PROOF OF PROPOSITION 3.1: Assume first that N(p-m)+mp > O. Write the embedding inequality (2.1) for the function x -+ v(x, t) for a.e. t E (0, T) and for the choice of the parameters s =m and
N+m
a=?!.;
N(P - m) + mp>O.
q=p--- ;
N
q
Taking the qthpower in the resulting inequality and then integrating over (0, T) proves (3.1). If N (p - m) + mp ::5 0, we must have p < N. Therefore applying Corollary 2.1, T
JJ
Ivl q dx dt =
nT
JJ
Ivl"lvl mN dx dt
0
n
~
! (j Ivl~dz (j T
$
$
)
N
IV1mdz)
~
([f IDV1Pdzdt) (~~'f [IV(X,tl1mdz) ~
To prove (3.2), we rewrite (3.1) as
3. Parabolic spaces and embeddings
9
and apply Young's inequality. PROOF OF PROPOSITION 3.2: IfvEVm,p(nT ), consider the function w(·, t)
= v(·, t) - I~I
J
v(x, t)dx,
a.e.
t E (0, T),
n
which has zero integral average over n for a.e. t E (0, T). By Remark 2.1, x-+ w(x, t) satisfies the embedding inequality (2.1) for a.e. t E (0, T] and with constant C depending also upon the structure of an. Proceeding as before, we arrive at (3.2) for w. For a.e. t E (0, T) ,
IIDwllp,nT = IIDvllp,nT'
IIw(" t)lIm,n :::; 21Iv(·, t)lIm,n.
Moreover
IIwll •.a. " II-II •. ",. - 1111 hI.
(I U
I_1 m dx)
1
;1;
Therefore
. The last term is majorised by
( Inl N(p~ml+mp) Nm
1 9
esssup IIv("
t)lIm,n,
O
and the proposition follows. We will use the following Corollaries obtained from the previous Propositions by taking m = p and by applying the HOlder inequality. COROLLARY 3.1. Let p > 1. There exists a constant 'Y depending only upon N and p, such that for every v E Vo"( T ).
n
IIvll:,nT :::; 'Yllvl > olmllvll~p(nT)'
(3.4) COROLLARY
3.2. Letp> 1. There exists a constant'Y depending only upon N,p
an. such that for every v E V P ( nT ). IIvll:,nT :5 'Y (1 + InlT)'lIIh Ilvl > 01 mpIIvll~p(nT)' N
and the structure of (3.5)
The next two Propositions hold in the case m = p.
10
I. Notation and function spaces
PROPOSITION 3.3. There exists a constant 'Y depending only upon N and p such that for every v E VJ'( (h ),
(3.6)
where the numbers q, r
~ 1 are
linked by N
1
N
-+-=r pq p'
(3.7)
and their admissible range is q E (p,ool,
(3.8)
{
r E [P2,00) ;
qE [p, :!p], q E [P, 00),
PROOF: Let v E follows that
r E [P,ool ;
r E (~, 00] ;
V!(nT ) and let r
if if
N
= 1,
1Sp
< N,
if 1 < N S p.
~ 1 to be chosen. From (2.1) with s
= pit
Choose ar=p. Then conditions (2.2)-(2.2-iii) imply (3.7)-(3.8), and the Proposition follows. The next Proposition holds for functions v E VP( nT ) not necessarily vanishing on the lateral boundary of nT. PROPOSITION 3.4. There exists a constant 'Y depending only upon N, p, m and the structure of 8 n, such that for every v E VP ( nT ),
(3.9)
Iivliq,r;iJT S 'Y
(1 + 1;11 )t
IivlivP(iJT)
where q and r satisfy (3.7) and (3.8). PROOF: Apply (2.1) to the function
w(·, t) = v(·, t) -
I~I
J
v(x, t)dx,
a.e.
t E (0, TI,
n
n
which has zero average in for a.e. t E (0, T). Proceeding as in the proof of Proposition 3.3 we arrive at (3.6) for w where 'Y now also depends upon the structure of 8n. From this
3. Parabolic spaces and embeddings
IIvll •.•,,,.
~ ~lIvllv.(".) + ~Inlt-l (1 :$1'lI v llvP(I1T ) +1'
U T)ldx) ·dT) Iv(x.
II
1
C!1~/N); ~:~~lIv(·,'T)lIp,I1'
We conclude this section by stating a parabolic version of Lemma 2.1 and Corollary 2.2 concerning the truncated functions (v - k)±.
Let v E V m ,P(IlT)' Then/or all k E R, (v - k)± E V m ,P(!1T)' Assume in addition that the trace 0/ x - v( x, t) on all is essentially bounded and LEMMA 3.1.
esssupllv(·,t)lIoo,al1:$ k o, /orsome ko > 0. O
3.3. Let Vi ELP (0, Tj W 1,P(Il)), i=1,2, ... ,nEN. Then
w == min{v1,v2, ... ,vn }
E
LP (0,TjW 1,P(Il)).
3-(i). Steklovaverages
°
Let v be a function in L1 (!1T) and for < h < T introduce the Steklov averages t) defined for all 0< t < T by
Vh(-,
_ *J v(·,'T)d'T, = t+h
Vh
{
t>T-hj
0,
vii.
t E (O,T - h},
t
_{* j
=
v(','T)d'T,
t E (h,T],
t-h
0,
t < h.
LEMMA 3.2. LetvE Lq,r(IlT ). Then, as h-O, Vh converges to v in Lq,r(IlT _t:) for every e E (0, T).lfv E C (0, Tj Lq(f1)). then as h - 0, Vh(-, t) converges to v(·, t) in Lq(Il)/or every tE (0, T - e), 'VeE (0, T).
A similar statement holds for vii.. The proof of the lemma is straightforward from the theory of LP spaces.
12
I. Notation and function spaces
4. Auxiliary lemmas 4-(;). Fast geometric convergence We state and prove two lemmas concerning the geometric convergence of sequences of numbers. LEMMA 4.1. Let {Yn } , n=O, 1,2, ... , be a sequence of positive numbers, satisfying the recursive inequalities
(4.1)
where C, b,> 1 and a > 0 are given numbers. If (4.2)
then {Yn } converges to zero as n-oo. The proof is by induction. LEMMA 4.2. Let {Yn } and {Zn} , n=O, 1,2, ... , be sequences ofpositive numbers, satisfying the recursive inequalities
{
(4.3)
Yn+l :5 Cbn (Y,t+a + Z~+"Yna) , Zn+1 :5 Cbn (Yn + Z~+,,)
where C, b> 1 and It, a > 0 are given numbers. If (4.4)
Y.o
1+ .. + Z 1+" < (2C)- " b--;;Z~
0
_
then {Yn } and {Zn} tend to zero as n PROOF:
,
where
u = min{lt; a},
00.
Set Mn =Yn + Z~+" and rewrite the second of (4.3) as
(4.5) Consider the tetm in braces in the first of (4.3). If Z!+" :5 Yn• such a term is majorised by 2M~+a. If Z!+" ~ Yn• then the same term can be majorised by
y~+a
+ (Z~+,,)l+a :5 M~+a.
Combining this with (4.5) we deduce that in either case
min {K.,a} 2C 1+K.b(1+K.)nMl+ M n+l < _ n ' The proof is concluded by induction as in Lemma 4.1.
4. Auxiliary lemmas
13
4-(ii). An interpolation lemma LEMMA 4.3. Let {Yn } , n = 0,1,2 ... , be a sequence of equibounded positive numbers satisfying the recursive inequalities
(4.6)
where C, b> 1 and a E (0, 1) are given constants. Then
2C Yo ~ ( bl-~
(4.7)
)±
Remark 4.1. The Lemma turns the qualitative infonnation of equiboundedness of the sequence {Yn } into a quantitative apriori estimate for Yo. PROOF OF LEMMA
4.3: From (4.6), by Young's inequality VeE (0,1),
n=O,1,2, ....
By iteration
C )~
Yo
b±
~ enYn + ( e1- a
Q
n-l
.
~ (b±e)'.
!
Choose e = so that the sum on the right hand side can be majorised with a series convergent to 2. Letting n -+ 00 proves the Lemma.
4-(iii). An algebraic lemma We conclude this section by recording two algebraic inequalities needed in what follows. LEMMA
4.4.
Letp~2.ThenVa,bERm,mEN
(4.8)
where 'Yo depends only upon p, m. Let 1
(4.9) where 'Yl depends only upon p, m.
14
I. Notation and function spaces
PROOF:
J(p)
= (laI P - 2 a - Iblp - 2b, a - b) 1
~ (/ ! I" + (1 -
!
8)bl'-'(80 + (1 - 8)b)da, 0 -
b)
1
= Isa + (1 -
s)bI P- 2 Ia - bl 2 ds
o
!
1
+ (p -
2)
Isa + (1 - s)bI P- 4 1(sa + (1 - s)b, a - b}1 2 ds.
o Ifp~2. J(p) ~ la -
1
bl 2 J Isa + (1 o
Isa + (1 - s)bl ~ and (4.8) follows. If lal < Ib -
!
s)bI P- 2ds. If lal ~ Ib -
lial- (1 - s)la - bll
~
al. we have
sla - bl
al.
1
la - bl 2
Isa + (1 - s)bI P- 2dx
o
>
la _ b12
-
!
1
o
~~
2
(Isa + (1- s)bl )P (2 - s)la - bl 2
ds
)P/2
1 (
/2
/ Isa + (1-
s)bl 2 ds
= ~ ~/2 (la1 2 + Ibl2 + (a, b) )P/2 ~
'iola - bjP.
Remark 4.2. The reverse inequality to (4.8) is. in general. false. Indeed. if a E R + • b=a+l J(p) = (p - 1)~P-2; for some ~ E (a, a + 1). Letting a -+ 00 shows that the inequality J(p) :5 independent of a and ~. Next, if 1 < p < 2,
!
"rIa -
bl P cannot hold with "r
1
J(p) :5 (p - 1)la -
bl 2
o If lal ~ la -
bl. since p < 2,
Isa + (1 - s)bI P- 2 ds.
5. Bibliographical notes
Isa + (1- s)bI P- 2 ~
15
Ilal- (1- s)la - bIlP-2~ sp- 2 la - bl p - 2
and (4.9) follows since sp-2 is integrable. If lal < Ib -
ai, let s. be defined by
bl = lal.
(1 - s.)la Then estimate 1
/(P)
~ (p -I)la - bl /lIa l 2
(1- s)la - b!lP-2ds
o 8.
~ la - bl
:s
P /
(lal -
(1 - s)la - bDp-1ds
o 1
+ la - bl /
! (Ial -
(1 - s)la - bl)p-1ds
s.
Remark 4.3. The reverse inequality is false, in general, with 'Y independent of
a,b.
5. Bibliographical notes For the theory of Sobolev spaces we refer to the monographs of Adams [I] and Mazja [76]. The embedding theorems 2.1 and 2.2 are special cases of more general embedding theorems. No attempt has been made to state them in the most general setting and under the best assumptions on the regularity of afJ. For a proof of Lemma 2.1 we refer to Mazja [76] and Stampacchia [93]. Lemma 2.2 is due to DeGiorgi [33]. Also the statement in Remark 2.2 follows from the proof in [33]. The parabolic spaces Vm,P(fJT ) and Vom,P(fJT ) are standard in the theory of parabolic partial differential equations and we refer for example to [67,73]. The embedding theorems 3.2 and 3.3 are a modification of similar statements and proofs in [67]. The lemmas on rapid geometric convergence are stated in [67]; we have given a different proof. The interpolation inequality of Lemma 4.3 is taken from Campanato [22,23]. Lemma 4.4 is taken from [27].
II Weak solutions and local energy estimates
1. Quasilinear degenerate or singular equations We introduce a class of quasilinear parabolic equations with the principal part in divergence fonn, that are either degenerate or singular due to the vanishing of the gradient IDul of their solutions. (1.1)
Ut - diva(x,t,u,Du) = b(x,t,u,Du)
n
in'D'(nT)'
n
The functions a : T x R N + 1 -+ RN and b : T x RN +1 -+ R are only assumed to be measurable and satisfying the structure conditions
a(x, t, u, Du)· Du ~ ColDul P - f{)o(x, t), la(x, t, u, Du)1 S C11Dul p - 1 + f{)l (x, t), Ib(x, t, u, Du)1 C21Dui P + f{)2(X, t)
s
for p > 1 and a.e. (x, t) E nT • Here Ci , i = 0, 1,2, are given positive constants and f{)i, i =0, 1, 2, are given non-negative functions, defined in (IT and subject to the condition -'!.-
f{)o,
• •
f{)1p-I ,f{)2 E Lq,r(rl UT )
where q, r ~ 1 satisfy (As)
and
1
N
-=r + -::: pq
= 1-
. 1\:1.
1. Quasilinear degenerate or singular equations
17
(As-i)
(As-ii)
(A5 -iii)
A measurable function u is a local weak sub(super)-solution of (1.1) in nT if (1.2)
u
E
Gloc
(0, T; L~oc(n»nLfoc (0, T; wl~:(n)) ,
and for every compact subset JC of n and for every subinterval [h, t2J of (0, TJ t
(1.3)
t2
fUrpdXlt2 + f f{ -urpt + a(x, T, u, Du)·Drp} dxdT /C
1
tl/C t2
:5
(~) f
fb(X, T, u, Du)rp dxdT,
tl/C
for all locally bounded testing functions (1.4)
rp
~
o.
The local boundedness of the testing functions rp is required to guarantee the convergence of the integral on the right hand side of (1.3). A function u that is both a local subsolution and a local supersolution of (1.1) is a local solution. Remark 1.1. If p = 2. then (1.1) is non-degenerate. In such a case it is known that locally bounded weak: solutions are locally mlder continuous; moreover the assumptions (AI )-( As) are optimal for a mlder modulus to hold. It would be technically convenient to have a formulation of weak solution that involves Ut. Unfortunately solutions of (1.1). whenever they exist. possess a modest degree of regularity in the time variable and. in general, Ut has a meaning only in the sense of distributions. The following notion of local weak: sub(super)-solution involves the discrete time derivative of u and is equivalent to (1.3). Fix t E (0, T) and let h be a small positive number such that 0 < t < t + h < T. In (1.3) take tl = t, t2 = t + h and choose a testing function rp independent of
18
ll. Weak solutions and local energy estimates
the variable T E (t, t + h). Dividing by h and recalling the definition of Steklov averages we obtain (1.5)
J
{Uh,tCP + la(x, T, u, Du)Jh ·Dcp -Ib(x, T, U, DU)]h cp} dx ::;;
(~)O,
K:x{t}
for all 0 < t < T - h and for all cp E WJ,p(,qnL~c(.fl),
cp ~ O.
To recover (1.3), fix a subinterval O
1-(i). Subsolutions and parabolic equations Tbe structure conditions (AI)-(As) are not sufficient to characterize parabolic
p.d.e. 's. For example the 'principal part'
a(x, t, u, Du}
= Du -
Du/IDul
satisfies (Ad-(As) with p= 2. However its 'modulus of ellipticity' changes type at IDul = 1. In what follows we assume that (1.1) is weakly parabolic in the sense that it satisfies (AI )-( As) and in addition, whenever u is a weak solution of ( 1.1), (A6)
for all k E R the truncated functions (u - k)± are weak subsolutions of (1.1) in the sense of (1.3).
with a(x, t, u, Du) replaced by
±a(x, t, k ± (u - k)±, ±D(u - k)±) and b(x, t, u, Du} replaced by
±b(x, t, k ± (u - kh, ±D(u - k)±). To clarify the connection between subsolutions and parabolic structures, we derive some sufficient conditions on a(x, t, u, Du) for (A6) to be verified. Let u be a local weak solution of (1.1), and in (1.5) take the testing function e>O
and cp satisfying (1.4).
We integrate in dt over It I , t2J C (0, T) and let first h -+ 0 and then e -+ 0 to obtain
1. Quasilinear degenerate or singular equations
19
t2
ju+C{)(x,t)I:: n
+ fj{-u+C{)t+a(x,t,u+,DU+).DCP} dxdr tIn
t2
=f f
(1.6)
b(x, t, u+, Du+) cpdxdr
tIn
LEMMA 1.1. Assume that
a(x,t,u,11)·11
~
0,
Then (1.1) is weakly parabolic. PROOF: It suffices to verify (A6) for (u - k)+ and k = 0. This is the content of (1.6).
A more general condition for (As) to hold can be given by the notion of monotonicity. We say that 11-+a(X, t, u, 11) is monotone i£
(a(x,t,u,11d - a(x, t,u, 112),111 - "12) ~ 0,
V11i ERN, i=I,2.
LEMMA 1.2. Assume that 11-+a(x, t, u, 11) is monotone and (1.8)
Then (A6) holds. PROOF: Write the last integral on the right hand side of (1.6) as t2
e fy tl n
[(a(X,t,u,DU+) -a(x,t,u,O)).Du+l d d ()2 cp X r U++e
t2
-f}diV a(x, t, u, 0) ~ cp dxdr U++e tl n t2
-
°
f"fa(x,t,u,O).Dcp~dxdr.
J
U++e
tIn
We let e '\. and discard the non - negative contribution of the first integral. The sum of the last two terms tends to zero since (1) The monotonicity assumption (1.7) is natural in the existence theory. It permits one to apply Minty's Lemma [78] to identify the weak limit of the principal part of the p.d.e. when (1.1) is approximated by a sequence of regularised problems.
20
n. Weak solutions and local energy estimates
(1.9) t2
=j j
div a(x, t, u+, O)
tin t,
=-
jja(x,t,u+,O)'D
One checks that the assumptions of the lemma are verified for example by equations with principal part
where 1/Jo is bounded, non-negative and .p;,x; E Ll(nT) and the matrix (ai;) is only measurable and positive definite.
Remark 1.1. The 'regularity' assumption (1.8) is only needed to justify the limit in (1.9). It can be dispensed with when working with a sequence of approximating solutions.
2. Boundary value problems We will give regularity results for weak solutions of (1.1) up to the lateral boundary ST, provided u satisfies appropriate Dirichlet or Neumann boundary conditions. We also prove that weak solutions are HBlder continuous up to t = if the initial datum is HBlder continuous. Since the arguments are local in nature, for these results to hold, the prescribed boundary data have to be taken only locally. However, for simplicity of presentation we will state them globally, in tenns of boundary value problems.
°
2-(i). The Dirichlet problem Consider fonnally the Dirichlet problem
(2.1)
ut .- div a~, t,.u, Du) { u( ,t)18n - g( ,t), u(',O)
= b(x, t, u, Du),
in nT, a.e.
t E (0, T),
= uoO,
where the structure conditions (Al)-(As) are retained. On the Dirichlet data 9 and U o we assume
2. Boundary value problems 21 (D)
9 is continuous onST with modulus of continuity, say wg (·),
(Uo )
Uo
is continuous in
n with modulus of continuity, say woO.
A weak: sub(super)-solution of the Dirichlet problem (2.1) is a measurable function u, satisfying (2.2)
and for all te (0, T] I uY'(x, t) dx I I {-uY't
(2.3)·
n
+ a(x, T, u, Du)·DY'} dxdT
nc
~ (~) luoY'(X, 0) dx + Ilb(X, T,U, DU)Y'dxdT, n
nc
for all bounded testing functions
In addition the second of (2.1) holds in the sense that u ~ ~ 9 on an in the sense of the traces of functions in W1,p(n) for a.e. t e (0, T). A function u that is both a sub-solution and a super-solution of (2.1) is a solution of the Dirichlet problem. The formulation can be rephrased in terms of Steldov averages as in the previous section, namely
(2.5)
l{uh,tY' + [a(x, T, u, Du)lh ·DY' - [b(x, T, u,Du)lh Y'} dx
~ (~) 0,
,Ux{t}
for all 0 < t < T - h and for all Y'
e w!,p(n)nLOO(n),
Y' ~
o.
Moreover the initial datum is taken in the sense of L 2 (n), i.e., (2.6)
2-(ii). Variational boundary data Assume that an is piecewise smooth, so that the outward unit normal, which we denote with n, is defined a.e. on an and consider formally the Neumann problem
(2.7)
Ut - div a(x, t, u, Du) = b(x, t, u, Du), { a(x, t, u, Du)·n = ",,(x, t, u), u(·,O) = u o (-).
in nT, on ST,
22
ll. Weak solutions and local energy estimates
We retain the structure conditions (A1HAs) and the assumption (U o ) on the initial datum. We assume that ",,(., t, u(·, t» admits, fora.e. tE (0, T),anextension into n which we denote by 1fi(., t, u( ,t», such that
11fi1 :5 ""ou + ""10 a.e. nT, { l1fiul :5""0' l1fiz.1 :5 ""1. i =1, 2, ... ,N,
(N) where (N-i)
""1, ""0 are given non-negative functions satisfying
1/Jr,1/Jrr
Ltl,r(nT), whereq and f satisfy (As).
E
To give a notion of weak sub(super)-solution we let /C be an arbitrary compact subset of RN and consider testing functions
cp
(2.8)
~
0.
A function u, (2.9)
is a weak sub(super)-solution of the Neumann problem (2.7) if for every compact
subset /C of RN and for every subinterval [t1' t2J of [0, T] t2
t
(2.10)
j UCPdXlt2 ICnn
+j
:5
j{-uCPt + a(x,T,U,Du)·Dcp}dxdT
tllCnn
1
(~) j
~
~
jb(x,T, U, Du)cpdxdT + j j 1/J(x, t, u)cpdudT,
tllCnn
tl ICnan
where du denotes the surface measure on an. We remark that the testing functions cp vanish, in the sense on the traces, on the boundary of /C and not on the boundary of n. The variational datum is reflected in the boundary integral on the right hand side of (2.10). The formulation in terms of Steklov averages is (2.11)
j {Uh,tCP + [a(x, T, u, Du)J h ·Dcp - Ibex, T, u, Du)J h cp} dx (ICnn]x{t}
:5
(~)
[1/J(x, t, u)Jh cp du,
j
(ICnan]x{ t}
for all
°<
t
< T - h and for all cp E W~·P(/C), cp ~ 0,
and the initial datum is taken in the sense of (2.6).
3. Local integral inequalities We will derive some integral inequalities in the interior of nT. which will be the main tools in establishing local HOlder estimates for the solutions. Analogous es-
3. Local integral inequalities 23
timates near the lateral boundary ST as well as near t = 0 will be derived in the next section. Let K p denote the N-dimensional cube centered at the origin and wedge 2p. i.e.•
If Xo E RN. we let [xo congruent to Kp. i.e.•
+ Kp] denote the cube of centre Xo and wedge 2p which is
[xo + Kp] == {x E RNll~~N IXi - xo.il < p}. Let 6 be a given positive number and consider the cylinder
Q (6, p) == Kp x {-6, O} , and if (xo, to) ERN+!. we let [(xo, to) + Q (6, p)] denote the cylinder with 'vertex' at (xo, to) congruent to Q (6, p). i.e .•
[(xo, to) + Q (6,p)] == {x E RNll~~N IXi - xo,il < p} x {to - 6, to}. We will refer to these as cubes and cylinders of 'radius' p and height 6. Fix (xo, to) E nT and let p and 6 be so small that [(xo, to) + Q (6, p)] E nT. Let (denote a piecewise smooth cutoff function in [(xo, to) + Q (6, p)] such that (3.1)
(E [0,1], ID(I
< 00,
and ((x, t) = 0, for x outside [xo
+ Kp].
Assume that (3.2)
and construct the truncated functions (u - k)±. We will choose levels k satisfying (3.3)
esssup
I(u - k)±1 == Ht ~ 0,
[(z",t,,)+Q(II,p»)
where 0 is a positive parameter to be chosen later.
Remark 3.1. Suppose (3.3) is written for (u - k)+ and assume the number
o is
small. Then the levels k are forced to be near the essential sup of u in
[(xo, to) + Q (6, p)]. Likewise if (3.3) is written for (u - k)_. then k has to be close to the essential inf of u within [(xo, to) + Q (6, p)]. Roughly speaking. the function u is HOlder continuous if. within [(xo, to)+ Q(6, p)]. it is close in some integral norm to its integral average. Accordingly the sets where the function is near its supremum or near its infimum. within [(xo, to) + Q (6, p)]. have relatively small measure. Our energy inequalities reflect this through the sets
24
II. Weak solutions and local energy estimates
In estimating the contribution of the lower order terms !.pi, i = 0, 1, 2, it is convenient to introduce the numbers q, r,,,, constructed starting from q, f''''1 as follows: q=
(3.5)
qp(1 + ",) q- 1 ;
r
A
=
fp(1
+ ",) 1
rA
;
It is seen from (As-i)-(As-iii) that they satisfy
1
(3.6)
N
N
;+pq=p2'
and their admissible range is
qE(P'OO)' {
(3.7)
qE (p, :!p), qE(P,OO),
if N
rE(p2,oo);
if1
r E (p,oo);
rE(~'OO);
= 1, < p < N,
if! < N
~
p.
1be statement that a constant "'I depends only upon the data means that it can be determined a priori only in terms of the numbers N, p, q, r, "', the constants Ci , i = 0,1,2, and the norms
3-(i). Local energy estimates PROPOSITION 3.1. Let u be a locally bounded weak solution 0/(1.1) in nT . There exist constants "'I and 00 that can be determined a priori only in terms 0/ the data such that/or every cylinder [(x o, to) + Q (0, p)] c nT and/or every level k satisfying (3.3) for 0 ~ 00
(3.8)
sup to-8
$
I(u -k)~(P(x,
t)dx + "'1- 1
[xo+Kp]
I(U - k)~(P(x,to
flD(u - k)±(IPdxdT
[(X o,to)+Q(8,p)] - O)dx + "'I
[xo+Kp]
II(u -k)~(P-1(tdxdT +
+"'1 [(x o,to)+Q(8,p)]
frJ
II(U - k)~ID(IPdxdT
[(Xo,to)+Q(8,p)]
"'I
{J IAt,(T)l 0-
~
i dT}
•
8
PROOF: After a translation we may assume that (x o, to) coincides with the origin and it will suffice to prove (3.8) for the cube Q (0, p). In the weak formulation (1.5) take the testing functions
3. Local integral inequalities 25
"P = ±(Uh - k)::I::(P
(-e, t), t
and integrate over we have flfSt
E
(-e, 0). Estimating the various terms separately
Therefore integrating by parts and letting h - 0 with the aid of Lemma 3.2 of Chap. I,
~ j(u - k)~(P(x,t)dx - ~ j(u - k)~(P(x, -e)dx Kp
Kp
t
-
~I
fiu -
k)~(p-l(t dxd-r.
-8Kp
In estimating the remaining parts we first let h _ 0 and then use the structure conditions (AI)-(As). To simplify the notation, set
tE(-e,O). Then (3.9)
± j j[a(x, t, u, DU)]h·D ((Uh - k)::I::(P) dxd-r
~
Q'
II a(x, t, u, Du)· [±D(u - k)::I::(P ± p(u - kh(P-1 D(] dxd-r Q'
~ Co IIID(U Q'
khlP(PdxdT - II "Po(P X [(u - k)::I:: > 0] dxd-r Q'
- pCI IIID(U - k)::I::lp-l(U - k)::I::(P-IID(ldxd-r Q'
- p I j "PI (u - k)::I::(P-IID(ldxdT, Q'
where x( E) denotes the characteristic function of the set E. By Young's inequality
26
ll. Weak solutions and local energy estimates
pCI /
(i)
/ID(U - k)±IP-I(U -
k)±(P-IID(ldxdr
Q'
~ ~o //ID(U Q'
(ii)
k)~ID(IPdxdr,
k)±IP(Pdxdr + 'Y(Co) //(U Q'
k)~ID(IPdxdr
p / / CPI(U - k)±(,,-IID(ldxdr:5: //(U Q'
Q'
+1' //cprX[(U-k)± >Ojdxdr. Q'
Combining this in (3.9) we arrive at
/ /a(x, r, u, Du)·D ((u - k)±(P) dxdr Q'
~ ~o / /ID(U -
k)±(IPdxdr - l' / /(U -
Q'
k)~ID(IPdxdr
Q'
-1' //(CPo+cp(=r)X[(U-k)± >Ojdxdr. Q'
Finally
(3.10)
/ /lb(X, r, u, Du)(u - k)±(Pldxdr Q'
:5: C2 //ID(U
-
k)±IP(u - k)±(Pdxdr + / / CP2(U - k)±(Pdxdr.
Q'
Q'
Now if we impose on the levels k the restriction
(3.11)
esssup I(u Q(9,p)
we deduce from (3.10)
k)±1 :5: 6 == 0
Co 4C ' 2
3. Local integral inequalities 27
f flb(x, r, u, Du)(u - k)±(PI dxdr
(3.10')
Qt
:5
~o fflD(u -
k)±IP(Pdxdr + II C{)2(U - k)±(Pdxdr
Qt
:5
Qt
~o !!ID(U -
k)±(IPdxdr + Do
Qt
+')'
!!
C{)2X [(u - k)± > 0] dxdr
Qt
f I(u -
k)~ID(IPdxdr.
Qt
Combining these estimates and recalling that t E ( -6, 0) is arbitrary, we obtain
sup f(u - k);(P(x, t) dx + C2° jr flD(u - k)±(IPdxdr
J
-9
Q(9,p)
:5 I (u - k);(P(x, -6)dx Kp
+ ')' f I(u - k)~ID(IPdxdr + ')' If(u Q(9,p)
k);(p-l(t dxdr
Q(9,p)
+')' fl(C{)o + C{)r +C{)2)x[(u-k)± >Ojdxdr. Q(9,p)
By HOlder's inequality
11(C{)o+C{)r +C{)2)x[(u-k)± >Ojdxdr Q(9,p)
:5
II". +"r + '1'211.,.,,,, {} A~p
1'-1
(T
>I'" '" dT } . .
Therefore recalling the definition (3.5) of the numbers q, r, It, inequality (3.8) follows.
Remark 3.2. The proof shows that the number Do in (3.11) has to be chosen small according to the constant C2 • If in (1.1) b(x, t, u, Du) =0, then Do can be taken to be infinite and no restriction is imposed on the levels k. Remark 3.3. If the lower order terms C{)i are all zero, then in (3.8) the last term can be discarded.
28
II. Weak solutions and local energy estimates
3-(ii). Local logarithmic estimates Introduce the logarithmic function (3.12)
1/1 (Hr, (u - k)±,c) == In+
{± Hie - (u - k)± + c Hr
}, 0
< c < Hr,
where Hr is defined in (3.3) via the levels k, and for s > 0 In+ s == max{lns; OJ.
In the cylinder [( x o, to)
+ Q «(J, p)I we take a cutoff function satisfying (3.1) and
( is independent of t E (to - (J, tol.
(3.13)
PROPOSITION 3.2. Let u be a locally bounded weak solution of (1.1) in th. There exist constants "( and 60 that can be determined a priori only in terms of the data. such that for every cylinder (Xo, to) + Q«(J,p) E {h andforevery level k satisfying (3 J) for 6 S 60
(3.14)
!
sup to-9
1/12 (Hr, (u - k)±, c) (x, tKP(x) dx
[zo+K,,]
<
f !!
1/12 (Hr,(u-k)±,c) (x,to-(JKP(x)dx
[zo+K,,]
+ 'Y
1/111/11£ (Hr, (u - k)±, c) 12 -
P
IV(I P dxdr
[(zo,to)+Q(9,p)] ~
+; (1 +In
~t) ll.IAUT)li dT } ·
PROOF: As before, we may take (x o, to) == (0, 0) and will work within the cylinder Qt introduced earlier. Also, to simplify the symbolism let us set
(3.15)
In (1.5) take the testing function I{)
= ~h
[1/I2(Uh)] (P
= [1/12(Uh)]' (P.
By direct calculation
[1/I2(Uh)]" = 2(1 + 1/1)1/112 E Lioc(lh) which implies that such a I{) is an admissible testing function in (1.5). Since 1/1 (Uh) vanishes on the set where (Uh - k)±=O,
3. Local integral inequalities 29
//!
=
1£h [1/J2]' (PdxdT
Qt
//!
1/J2(PdxdT
Qt
=/
1/J2(1£hKPdx -
Kpx{t}
/1/J2(1£hKP dX. Kpx{-9}
Therefore letting h -+ 0
//!
1£h [1/J2]' (PdxdT
1/1 2
----+ /
(Ht, (1£ - k)±,c) (Pdx
Kpx{t}
Qt
-
/ 1/1 2 (Ht, (1£ - k)±, c) (Pdx. Kpx{ -9}
To estimate the remaining terms we let h structure conditions (At}-(As).
/ / a(x,T,1£, D1£)·DIPdxdT Qt
-+
0 first and then make use of the
//(1 + -2//(1 + ~ 2Co
1/J) 1/J,2ID1£I P(PdxdT
Qt
1/J) 1/Jf2IPo (x, TKPdxdT
Qt
- 2pCl / /ID1£ IP- 11/J1/J'(P- 1ID(ldxdT Qt
- 2pCl / / 1/J1/J'IPl(X, TKP-1ID(ldxdT. Qt
From this, by repeated application of Young's inequality
/ /a(x,T,1£, D1£)·DIPdxdT Qt
//(1 + -2//(1 + ~ Co
1/J)1/J,2ID1£I P(PdxdT
Qt
1/J) 1/J,2 IPo (x, TKPdxdT
Qt
- -y(P) //1/J (1/J,)2- p ID(IPdxdT Qt
- -y(P) / / 1/J (1/J,)2 IPr (PdxdT. Qt
For the lower order terms we have
30
n. Weak solutions and local energy estimates / /lb(X, T, U, DU)1/J'I/l(PldxdT :5 C2 //IDuIP (1 + 1/J) ,t/P1/J,-l(Pdxdr Qt
Qt
+ //'P21/J1/J'(Pdxdr. Qt
Next we observe that by virtue of (3.3) and the definition (3.12) and (3.15) of 1/J
1/J'-1
= Ht -
(u - k)± + c < 26
and
Therefore by virtue of the choice (3.11) of the levels k we have
/ /lb(x,T, U, Du)1/J1/J'(PldxdT :5 Qt
~o //IDUIP (1 + 1/J) 1/J12(PdxdT Qt
+ ~ In ( ~t)
//1'P2IX [(u - k)± > OJ dxdr. Qt
Collecting these estimates we arrive at
/ !lI2 (Ht, (u -
k)±, c) (Pdx
Kpx{t}
:5
/!lI
2
(Ht, (u - k)±,c) (Pdx + "Y //1/JI1/J'1 2 - PID(I PdxdT
Kpx{ -9}
Q(9,p)
+; (1+1n~t) //('Po +'Pr +'P2)X[(u-k)±>OJdxdT Q(9,p)
where we have used the fact that c < 1. Treating the last integral as before proves the Proposition.
Remark 3.4. If the constant C2 in (A3) is zero, then we may take 6 =00 and there is no restriction on the levels k. Also if 'Pi =0, i=O, 1,2, then the last term on the right hand side of (3.14) can be discarded. Remark 3.5. In any case, whence the constant 60 has been chosen according to (3.11). the constant "Y on the right hand side of either (3.8) or (3.14), is independent o/u.lt is only the levels k that might depend upon the solution u via (3.11).
4. Energy estimates near the boundary 31
4. Energy estimates near the boundary We assume u is a weak solution of either the Dirichlet problem (2.1) or the Neumann problem (2.7), satisfying in addition (4.1)
uEV>C'J({h)
and
UEL 2
(0,T;W 1 ,PUh»).
The assumptions (D), (Vo), (N), (N - (i» on the boundary data will be retained. We will derive energy and logarithmic estimates, similar to those of Propositions 3.1 and 3.2, near the lateral boundary ST as well as at t = O. Fix a point (x o, to) on ST, and construct the box [(x o, to) + Q (8, p)], where 8 is so small that to - 8>0. In [(xo, to) + Q (8, p)] introduce a piecewise smooth cutoff function (x, t) - «x, t) satisfying (3.1). We observe that for all t E (t o 8, to), x-«x, t) vanishes on the boundary of[xo + Kp] and not on the boundary of [x o + Kp] n n. Here the interior quantities introduced in the previous section are modified as follows
esssup I(u - k)±1 == D~ $ 6, [(xo,t o)+Q(9,p)]nnT
(4.2)
where 6 $ 60 and 60 is a parameter chosen according to (3.11). Analogously we define the logarithmic function (4.3)
!Ii(D~,(U-k)±,C)==ln+{ Dk± - (uD~) },C
and introduce the sets (4.4)
B~/T) == {x E [xo + Kp] n
nl
(U(X,T) - k)± >
O}.
4-(i). Variational boundary data Let u be a weak solution of (2.7) satisfying (4.1) and assume in addition that
on is of class Cl+ A forsome AE (0,1).
(4.5)
PROPOSITION 4.1. There exist constants "( and 60 that can be determined a priori only in terms of the data and the quantities lIulloo,nT and IIlonllh+).,suchthat for every (xo, to) E ST./or every cylinder [(xo, to) + Q (8, p)] such that to-O>O and for every level k satisfying (4.2) for 6 $ min {60 ; I}
(4.6)
sup j(u - k)~(P(x, t) dx + ,,(-1 jr f ID(u - k)±(IPdxdT to-9
+"(
f
ff
(u - k)'i(p(x, to - 8) dx + "( (u - k)~ID(IPdxdT [xo+Kp]nn [(xo,t o)+Q(9,p)]nnT
jj(U-k)~(P-l(tdxdT+"({lIB~p(T)lidT}
[(x o ,to)+Q(9,p)]nnT
0-
9
1!.ll±.cl r
32
II. Weak solutions and local energy estimates
Moreover if ( is independent oft e (to - fJ, to), (4.7)
sup to-9
J!li2 (D~, (u -
k)±, c) (x, t)(P(x) dx
[zo+Kp)nn
~
J!li 2 (D~, (u -
k)±, c) (x, to - fJ)(P(x) dx
[zo+Kp)nn
+ ')'
JJ !lil!li
u
(D~, (u -
k)±,c)
1
2 P -
ID(I P dxdT
[(Zo,to )+Q(9,p»)nnT ~
+;
(I +In D!) Ll.IB~p(T>I:dT} ·
where the numbers q, r, It satisfy (3.6)-(3.7).
=
Remark 4.1. If the Neumann data are homogeneous, i.e., .,pi 0, i = 0, 1, and 'Pi =0, i=O, 1,2, and b(x, t,u,Du) =0. then we may take 6=00 and the levels k are not restricted. Remark 4.2. The proof below is local in nature and it shows that (4.6)-(4.7) hold
true for weak solutions that satisfy the Neumann data on a portion of ST. Accordingly, only such a portion is required to be of class CHA. Also. no reference to initial data is necessary. PROOF OF PROPOSITION 4.1: Fix (xo, to) e ST, assume that (xo, to) coincides . with the origin and work with cubes Kp and cylinders Q(fJ,p). Since an is of class CHA, for sufficently small p, the portion of an within the cube K p. can be represented in a local system of coordinates as a portion of the hyperplane x N = 0 and KpnnC {XN >O}. Set
(4.8)
K:
=Kpn{XN > O} and Q+ (fJ, p) =Q (fJ, p)n{XN > O}.
Without loss of generality we may assume that (2.11) is written in such a coordinate system. To derive (4.6), in (2.11) we take the testing functions
and let h -+ O. All the terms in (2.11) are treated as in the proof of Proposition 3.1, except for the boundary integral. We arrive at
4. Energy estimates near the boundary 33
sup
(4.9)
-8
I
k)~(P(x, t)dx + C2°
(u -
K:
~
jr f ID(u - k)±(IPdxdT
J
Q+(8,p)
I
(U -
k)~(P(x, -9)dx
Kp
+1'
II k)~ID(IPdxdT II II (<,00 + <,Or + <,02) (u -
+ l'
Q+(8,p)
+1'
(u -
k)~(p-l(tdXdT
Q+(8.p)
X [(U - k)±
> OJ dxdT
Q+(8.p)
o
+1'
II -8
where
¢(X, T, U)(U - k)±(PdxdT.
Kp
Kp =Kpnan is the (N-l)-dimensional cube
We estimate such a boundary integral by transforming it into an interior integral as follows
IJI
.,(x, T, u)(u - k)±C"dXdTI
-8K p
~
11(/a:N
(,i(x, T, u)(u -
k)±(p) dx
N) dXdT
p
~1'
II (I~ZN II
I(u - k)±(P + 1~IID(u - k)±(I) dxdT
Q+(8,p)
+1'
(1,z,I(u - k)±(P-1ID(1 + l~ullD(u - k)±I(u - k)±(p) dxdT.
Q+(9,p)
By virtue of assumption (N),
II I~ZNI(uQ+(9,p)
k)±(PdxdT
~ Dt
II
¢lX[(u-k)±
Q+(9.p)
where Dt is defined in (4.2). Also by Young's inequality
> OjdxdT,
34
n. Weak solutions and local energy estimates
I~I {ID(u -
II
k)±(1 + (u - k)±(P-1ID(I} dxdr
Q+(9.p)
~ ~o
II ID(u - k)±(IPdxdr + 'Y(P) II (u - k)'±ID(IPdxdr
Q+(9.p)
Q+(9.p)
+ 'Y (p, lIulloo.UT) II (?/Io + ?/Il)~ X [(U - k)± > OJ dxdr. Q+(9.p)
Finally
II
l~uIlD(u -
k)±I(u - k)±(Pdxdr
Q+(9.p)
~ 'Y I I I~ul {ID(u -
k)±(I(u - k}±(p-l + (U -
k)~(P-IID(I} dxdr
Q+(9.p)
~ ~o
II ID(u-k)±(IPdxdr+'Y(p,60 ) II(u-k)'±ID(IPdxdr
Q+(9.p)
Q+(9.p)
+ 'Y (p, lIulloo.uT) II ?/1ft X [(u - k)± > OJ dxdr. Q+(9.p)
Combining these estimates implies that the boundary integral on the right hand side of (4.9) can be estimated by
~o
IIID(U - k)±(IPdxdr + 'Y(P, lIulloo.uT) II (u - k)'±ID
Q+(9.p)
Q+(9.p)
+'Y(p,60 ) II
(1+?/Io+?/Id~x[(u-k)± >Ojdxdr.
Q+(9.p)
We put this in (4.9) and, to conclude the proof, estimate the integral involving the functions !Pi, i=O, 1,2, and ?/Ii, i=O, I, as in the proof of Proposition 3.1. The proof of the logarithmic estimate (4.7) near the lateral boundary ST is similar to the proof of the interior logarithmic estimate (3.14), modulo the modifications indicated above and we omit the details.
4-(;;). Dirichlet boundary data Let u be a weak solution of the Dirichlet problem (2.1), which in addition satisfies (4.1). The assumption (D) on the boundary datum 9 is retained. Fix (xo,to) EST and consider the cylinder [(xo,t o) + Q(8,p)j, where 8 is so small that to - 8 > O. Local energy estimates for u near (x o, to) are obtained by taking, in the weak formulation (2.5), the testing functions
4. Energy estimates near the boundary
35
integrating over [(xo, to) + Q (8, p)) and letting h -+ O. Such a choice of testing functions is admissible if for a.e. t E ( to - 8, to),
(u(·, t) - k)± (P(x, t) E W~'P ([xo
(4.10)
+ Kp) n 0).
Since x-+((x, t) vanishes on the boundary of [x o + Kp) and not on the boundary of [x o + Kp) n 0, condition (4.10) will be verified iffor a.e. tE (to -8, to) (u - k)± = 0 in the sense of the traces on
a[xo + Kp) n O.
In view of Lemma 3.1 of Chap. I, this can be realised for the function (u - k)+ if
k is chosen to satisfy k
(4.11)
~
sup
[(zo ,to )+Q(lI,p »)nST
g.
Analogously the functions -(Uh - k)_(P can be taken as testing functions in (2.5) if (4.12)
With these choices of k we may repeat calculations in all analogous to those of Proposition 3.1 and derive energy inequalities for u near ST. Analogous considerations hold for a version of the logarithmic estimates along the lines of Proposition 3.2. We summarise 4.2. There exist constants "( and 60 that can be determined a priori only in terms ofthe data and such that for every (xo, to) E ST .for every cylinder [(x o, to) + Q (8, p)) such that to-8 > 0 andfor every level k satisfying (4.2)for 6 ~ 60 and in addition (4.11)for the functions (u - k)+ and (4.12) for (u - k)_ , the following inequalities hold: PROPOSITION
(4.13)
Jr
sup I(u - k)~(P(x, t)dx + ,,(-1 f ID(u - k)±(IPdxdT to-lI
~ !(U-k)~(P(x,to-(})dX+"( !!(u-k)~ID(IPdxdT [zo+Kp)nn
+"( I I (u -
[(zo,to)+Q(II,p»)nnT
k)~(P-l(tdxdT + "(
[(zo,to)+Q(II,p»)nnT
{J
~
IBt,p(T)lidT}
0-6
Moreover if the cutofffunction ( is independent of t E (to - 8, to),
r
36
II. Weak solutions and local energy estimates
1!li2 (nt, (u - k)±, c) (x, t)(P(x)dx
sup
(4.14)
to-9
[zo+Kp]nn
:5
!
!li 2 (nt,(u-k)±,c) (x,to-O)(P(x)dx
[zo+Kp]nn +')'
II
!lil!liu (nt,(u-k)±,c) 12 - Pln(I Pdxdr
[(Zo,to)+Q(fJ,p)]nnT ~
+
~ (1+ In Dn LlIBUT) It dT } ·
where the numbers q, r, It satisfy (3.6)-(3.7). Local considerations as those in §3 apply to the present case. In particular the Proposition continues to hold for weak solutions that satisfy the Dirichlet data on a portion of ST.
4-(iii). Initial data Consider a weak solution of (1.1) that takes the initial datum U o in the sense that
k! h
(4.15)
u(·, r)dr
-+ Uo
in L~oc(fl) as h-+O.
o Thus u could be a solution of either the Dirichlet problem (2.1) or the Neumann problem (2.1). In either case the assumption (Uo ) is in force. Fix (x o, to) E flT and consider the cylinder [(xo, to) + Q (0, p)] where 0 is such that to-O=O. Therefore [(x o, to) + Q (0, p)]lies on the bottom of the cylindrical domain flT • Consider a cutoff function ( satisfying (3.1) and in addition (
is independent of E (0, to).
°
Local energy estimates for u near t = are derived by taking in the weak formulation (1.5) testing functions
"': = ±(Uh - k)±(P, integrating over (0, t), t E (0, to)' and letting h-+O. The fllSt term in (1.5) gives
~
!(uh - k)l(x,t)(Pdx ~+K~
~
!(Uh - k)l(x,O)(Pdx. ~+K~
If k is chosen so that k~sUP[zo+Kp] u o , then in view of (4.15) we have
4. Energy estimates near the boundary
37
o.
j (Uh(X, 0) - k)!
Also from the definition (3.12) of the function lli(·). it follows that lli (Dt, (u - k)±, c)
=0
whenever (u - k)± = o.
j lli 2 (Dt, (Uh - k)+, c) (x,O)
-+
as
0
h - O.
[xo+Kp]
Analogous considerations hold for (Uh - k) _
PROPOSITION 4.3. There exist constants 'Y and 60 that can be determined a priori only in terms of the data such that for every (xo, to) E {h,for every cylinder [(xo, to) + Q (8, p)] such that t o-8 = 0 andfor every level k satisfying (4.2)for 6 $ 60 and in addition
{k ~
(4.16)
k$
sUP[xo+Kp] Uo
for the function (u - k)+
inf[xo+K p] U o
for the function (u - k) _,
the following inequalities hold:
j(u -
sup
(4.17)
to-9
k)~(x, t)
[xo+Kp]
jr flD(u - k)±
J
[(xo,t o )+Q(9,p)]
$ 'Y j j (u - k)'±ID
{JIB~/T)lidT}
~
r
0
Moreover
(4.18)
sup
jlli2 (Dt,(u- k)±,c) (x,t)
to-9
+ 'Y
j j llillliu (Dt, (u - k)±, c) 12 -
P
ID
[(X o ,to )+Q(9,p)]
li!±.!tl
~ J(1+ In D!) tl.IB~)T)lidT} · where the numbers q, r, K, satisfy (3.6)-(3.7).
Remark 4.3. Local considerations apply to this case along the lines of similar remarks in the previous sections.
38
D. Weak solutions and local energy estimates
Remark 4.4. The constant 'Yon the right hand sides of either (4.13)-(4.14) or (4.17)-(4.18) is independent ofu. It is only the levels k that might depend upon the solution u via (3.11). Moreover if "Pi == 0, i = 0, 1, 2, and C 2 = 0, the levels k are independent of u. Remark 4.5. We conclude this section by observing that all the energy estimates as well as logarithmic estimates for (u - k)+ hold true if merely u is a subsolution of (1.1) and for (u - k)_ if u is a supersolution of (1.1).
5. Restricted structures: the levels k and the constant 'Y We will make a few remarks on the dependence of the constant 'Y in the energy and logarithmic estimates and on the restrictions to be placed on the levels k.
5-(i). About the constant 'Y For the interior estimates of Propositions 3.1 and 3.2, the constant 'Y depends only upon the data and it is independent of the apriori knowledge of lIulloo,nT. It can be calculated apriori only in terms of the numbers N, p, r, It, the constants Ci, i = 0,1,2, and the norms
lI"Po, "Pr , "P2114,r; nT· 1be same dependence holds for estimates near the parabolic boundary of nT in the case of Dirichlet data (see §4-(ii) and §4-(iii». In the case of variational data, 'Y depends also upon the structure of an (see §1, Chap. I), and the norms
5-(;;). Restricted structures The choices (3.11) and (4.2) of 60 impose a restriction on the levels k. Such a restriction is needed to handle the lower order terms b( x, t, u, Du) in (1.1). It follows from (3.1 0) and (3.10)' that the choice (3.11) of 60 permits the absorption of the term
C2
jjlD(U ~
k):l:I"(u -
k):l:C"dxdr :5 6 C2 j jID(U 0
k):l:I"C"dxdr
q
into the tenns generated by the principal part of the operator in (1.1). Also. the coefficient of the integral involving "P2 depends only upon the data (i.e., Co, C2 ), if the levels k are chosen according to (3.11).
5. Restricted structures: the levels k and the constant 'Y 39
Such a choice of ~o impose~ on k to be close to either the supremum or the infimum ofuin Q(6, p). Thus, in particular, the apriori knowledge of li u lioo,Q(8,p), is required. We will introduce structure conditions on (1.1) that yield energy and logarithmic estimates analogous to (3.8) and (3.14) for the truncated functions (u - k)±. with no restriction on the levels k. We will limit ourselves to the interior estimates of Propositions 3.1 and 3.2. First, it is obvious from the remarks above that Propositions 3.1 and 3.2 continue to hold for all the levels k if b(x, t, u, Du) == O. A more general condition is (A
3)
where (A~)
and q, r satisfy (As) and (As-i)-(As-iii). The structure condition (A3) implies (A3)' The source term '{J2 is required to be more integrable than the corresponding source tenn in (A3). Let us consider local weak solutions u of (1.1) with the structure conditions (Ad, (A2), (A3), (A4), (A~), (As) and (As-i)-(As-iii). We do not require that u be locally bounded. To derive local energy and logarithmic estimates for u we proceed as in the proof of Propositions 3.1 and 3.2. The lower order tenns in (3.10) are now estimated by repeated use of the Young's inequality as follows.
!
!lb(X, r, u, Du)(u - k)±("ldxdr
Q&
~ ~o + 'Y
!
!!
Q&
Q&
!ID(U - k)±(I"dxdr + 'Y
!
!(u -
'{J2(U - k)±("dxdr
k)~ max {ID(I" ; ("} dxdr.
Qt
We conclude that, for such solutions, inequalities (3.8)-(3.14) hold true for every level k, with a constant 'Y independent of u, provided the integral
!
j(U -
k)~ID(I"dxdr
[(x o ,t o )+Q(8.p)]
in (3.8) is replaced by
j j(U -
k)~ max {ID(I"j ("} dxdr
[(xo ,t o )+Q(8,p)]
and the integral
40
II. Weak solutions and local energy estimates
JJ!lil!li (Ht, (u - k)±, c) 1-"ID(I"dxdr 2
u
[(xo,to )+Q(9,p))
in (3.14) is replaced by
JJ!lil!li (Ht, (u - k)±, c) 1 u
2 -"
max
{ID(I"; ("} dxdr.
[(x o ,to )+Q(9,p))
6. Bibliographical notes When p = 2, assumptions (A d-( As) are optimal to obtain a HOlder modulus of continuity for the solutions (see [67]). The weak fonnulation of local and global weak sub(super)-solutions is standard and we refer for example to [67,73]. When 1 < p < 2, it seems more suitable to work with cubes of the type of K p rather than balls. For this reason we have introduced a unified geometry. The notation [xo + Kp] to denote a cube about Xo is introduced in Krylov-Safonov (64).
The idea of deriving energy inequalities for the truncated functions (u - k) ± seems to appear fust in Bernstein (12), in a global way, i.e., with the integrals extended to the whole nT • A local version of such estimates by use of local cutoff functions was introduced in the celebrated paper of DeGiorgi (33). Since then they have been widely used eSJ)C':ially in the russian literature (see for example (67) and references therein). Logarithmic estimates seem to be crucial in the study of the local behaviour of solutions of elliptic and parabolic equations in divergence fonn. For this we refer to Kruzkov [60,61,62), Moser [81,82,83] and Senin (92). The logarithmic function in (3.12) has been introduced in (35) and is now a standard tool in studying the local behaviour of degenerate and singular p.d.e. 's.
III Holder continuity of solutions of degenerate parabolic equations
1. The regularity theorem Consider solutions u of (1.1) or of Chap. II for the case P > 2. The equation is degenerate since the modulus of ellipticity vanishes when IDul =O. We will prove that if u E L~(nT), then it is HOlder continuous within its domain of definition. It will shown in Chap. V that local weak: solutions of such degenerate equations are indeed locally bounded. To simplify the presentation we will assume that u E L 00 (nT ). If u is only locally bounded, it will suffice to work: within a fixed compact subset of nT. In the theorems below, the statement that a constant 'Y depends upon the data means that it can be determined a priori only in terms of the norm II ull oo,nT , the constants Ci , i=O, 1,2, and the norms liepa, cpr ,cp2I1ti,T;nT appearing in the structure conditions (A 1 )-( A3)' We let /C denote a compact subset of nT and let p - dist (/C j r) be the intrinsic parabolic distance from /C to the parabolic boundary of nT, i.e., (1.1)
p - dist (/Cj rjp)
(
)
~n It - sll/p . == (.,.t)elC inf Ix - yl + lIull~ ' T (1I •• )er
l-(i). Interior HOlder continuity 1.1. Let u be a bounded local weak solution of (1.1) of Chap. 1/ in n T . Then (x, t) - u(x, t) is locally Holder continuous in nT. Moreover there
THEOREM
42
lll. HOlder continuity of solutions of degenerate parabolic equations
/or every pair o/points (x}, h), (X2' t2) EK-.I/thelowerordertermsb(x, t, u, Du) satisfy (A~) 0/§5 o/Chap. II. then 'Y and a are independent o/liulloo.nT.
1-(ii). Boundary regularity (Dirichlet data) THEOREM 1.2. Let u be a bounded weak solution 0/ the Dirichlet problem (2.1 ) o/Chap. II and let (D) and (Uo ) hold. The boundary an is assumed to satisfy the propertyo/positive geometric density (1./) o/Chap.l. Then uEC caT)' and there exists a continuous positive non-decreasing/unction s -. w( s) : R + -. R + • such that IU(XI' tt)
- U(X2' t2)j :5 W
(ixi -
x21 + It I
-
t21 i ) ,
/orevery pair o/points (Xl, tt), (X2' t2) E nT.lnparticular if the boundary datum 9 is Holder continuous in ST with exponent say a g • and if the initial datum U o is Holder continuous in a with exponent say a uo ' then u is Holder continuous in aT aiad there exist constants 'Y> 1 and a E (0, 1) such that
lu(xt, tt} - U(X2' t2)1 :5 'Yllulloo.nT~XI
-
x21 + lIulI:f.nT It I
-
t211/Pr '
for every pairo/points (Xl, tt}, (X2' t2) E aT. The constants 'Y and a depend only upon the data. Moreover the constant a depends also upon the HOlder exponents a g , Quo 0/ 9 and U o respectively. I/the lower order terms b(x, t, u, Du) satisfy (A~) 0/§5 o/Chap.lI. then 'Y and a are independent 0/ lIulloo.nT. Even though we have stated the Theorem in a global way the proof has a local thrust. For example the boundary datum 9 could be continuous or HOlder continuous only on a open portion of ST (open in the relative topology of ST), say E. Then the solution u of the Dirichlet problem would be continuous (respectively HOlder continuous) up to every compact subset of E. Analogous remarks hold in the case U o is only locally continuous or locally HOlder continuous. In particular to establish the continuity (HOlder continuity respectively) of u up to nx {O}, no reference is needed to the Dirichlet problem or any boundary value problem.
2. Preliminaries 43
1-0;;). Boundary regularity (Variational data) To stress such a locality we state our next theorem as if no information were available on the initial datum uO • THEOREM 1.3. Let u be a weak solution of the Neumann problem (2.7) of Chap. II, satisfying
u E LaO
(fi X [E, TJ) ,
EE
(0, T).
Assume that an is of class Cl,~ and let (N) and (N - i) hold. Then u is Holder continuous in n x [Eb T],for all E < El < T, and there exist constants "y and Q such that
IU(Xb td
- U(X2, t2) I
~ 'Yllulloo,iix[€,T] (IXl -
x21
+ lIull.:rox(€,T)ltl
-
t211/P)
Q
,
for every pair of points (Xl, tl), (X2, t2) E n x [El, T]. The constants 'Y > 1 and Q depend only upon E, lIulloo,iix(€,T] and the data, including the structure of an and the norms IItPl, tPrr IIq,r;UT appearing in (N - i). In addition the constant 'Y depends upon the distance (El - E). If the Neumann data are homogeneous, i.e., if tPo == tPl == 0, and if in addition the lower order terms b(x, t, u, Du) satisfy (Aa) of§5 of Chap. II, then 'Y and Q are independentofllulloo,iix(€,T]'
Remark 1.1. The continuity of u can be claimed up to t = 0 provided (Uo ) of Chap. II holds. Also, if U o is HOlder continuous in n then u is HOlder continuous innT.
2. Preliminaries The HOlder continuity of u, either in the interior of nT or at the parabolic boundary, will be, heuristically, a consequence of the following fact. The function (x, t) -+ u( x, t) can be modified in a set of measure zero to yield a continuous representative out of the equivalence class u E Vj!:(nT ), if for every (xo, to) E nT there exist a family of nested and shrinking cylinders [(xo, to) + Q (6n , Pn)] with the same vertex such that the essential oscillation Wn of u in [(xo, to) + Q (6n, Pn)] tends to zero as n -+ 00 in a way quantitatively determined by the structure conditions (AD!-(A6). The key idea of the proof is to work with cylinders whose dimensions are suitably rescaled to reflect the degeneracy exhibited by the equation. To make this precise, fix (xo, to) E nT and construct the cylinder
m. ltilder continuity of solutions of degenerate parabolic equations
44
where E is a small positive number to be determined later. After a translation we may assume that (xo, to) (0, 0). Set
=
p.+
=
esssup u,
p.- =
Q(RP-',2R)
ess inf
Q(RP-',2R)
u,
W=
essosc
Q(RP-e,2R)
=p. + -p. -
and construct the cylinder
2. = (~)"-2
(2.1)
ao
A
where A is a constant to be determined later only in terms of the data. We will assume that
(W)"-2 A > [lE.
(2.2)
This implies the inclusion (2.3)
and the inequality essosc u < w.
Q(aoRP,R)
-
By cylinders rescaled to rejlectthe degeneracy, we mean boxes of the type (2.1) where the length has been suitably stretched to accommodate the degeneracy. If p = 2, these are the standard parabolic boxes reflecting the natural homogeneity
of the space and time variables.
3. The main proposition PROPOSITION 3.1. There exist constants Eo, " E (0, 1) and C, A > 1, that can be determined a priori depending only upon the data, satisfying the following. Construct the sequences
Ro=R,
wo=w
andfor n= 1, 2, ... ,
R". = C-nR, Construct also the family of cylinders ~ an
__ (Wn ),,-2,
Thenfor all n=O, 1,2, ... Q(n+1)
c
Q(n)
and
A consequence of this Proposition is:
A
n=0,1,2, ....
3. The main proposition 45 LEMMA 3.1. There exist constants 'Y > 1 and 0 E (0, 1) that can be determined a priori only in terms of the data. such that for all the cylinders
o < p 5: R,
Q (aopp, p) ,
~ = (~)P-2,
A
ao
essosc u 5: 'Y (w + ,neo)
Q(aop",p)
(-RP)Q .
PROOF: From the iterative construction of Wn it follows that Wn+1 5: C R~o and by iteration
We may assume without loss of generality that
7JWn
+
eo is so small that" 5: C-Eo. Then
Let now 0 < P 5: R be fixed. There exists a non-negative integer n such that c-(n+1) R
5: P 5: C- n R.
This implies the inequalities
Therefore
. { ;"2eo} .
o=mm
0 1
To conclude the proof we observe that since Wn 5: w. the cylinder Q (aopp , p) is included in Q(n) =Q (an~' Rn). so that essosc u5:wn.
Q(aop",p)
Statements of HOlder continuity over a compact set now follow by a standard covering argument.
46
ill. HOlder continuity of solutions of degenerate parabolic equations
Remark 3.1. The proof of Proposition 3.1 will show that indeed it is sufficient to work with the number w and the cylinder Q (aoRP, R) linked by (3.1)
essosc u < w.
Q(aoRp,R)
-
This fact is in general not verifiable, for a given box, since its dimensions would have to be intrinsically defined in terms of the essential oscillation of u within it. Therefore the role of having introduced the cylinder Q (RP-E, 2R) and having assumed (2.2) is that (3.1) holds true for the constructed box Q (aoRP, R). It will be part of the proof of Pf<)position 3.1 to show that at each step the cylinders Q(n) and the essential oscillation of u within them satisfy the intrinsic geometry dictated by (3.1). To begin the proof, inside Q (aoRP , R) consider subcylinders of smaller size constructed as follows. The number w being fixed, let So be the smallest positive integer such that w
(3.2)
-28 0 <6 0, _
where the number 60 is introduced in (3.11) of Chap. II in the derivation of the local energy estimates. Then construct cylinders (3.3)
[(0, t) + Q (dRP, R)] ,
R
Figure 3.1
3. The main proposition 47
These are contained inside Q (aoRP, R) if the number A is chosen larger that
280 and if t ranges over
_ {AP-2 _ (280 }P-2} RP < t < wp - 2
o.
The structure of the proof is based on studying separately two cases. Either we can find a cylinder of the type of [(0, l) + Q (dRP, R)] where u is mostly large, or such a cylinder cannot be found. In either case the conclusion is that the essential oscillation of u in a smaller cylinder about (xo, to) decreases in a way that can be quantitatively measured. In the arguments to follow we assume (2.2) is in force and determine later the numbers A, e and eo.
Remark 3.2. For later use we estimate the quantity G(w, R}
w ) -2 1!1!±.cl == 'YRNIt ( 28 dr, 0
where 'Y is a constant depending only upon the data and K. is defined in (3.5) of Chap. II. From the defu\ition of din (3.3) it follows that (3.4)
G(w , R}
< _ A I RNltw- b,
and
Al
where b = 2 + (p _ 2) p(1 + K.} r
= A2+(P-2)P(1;,,) •
Along the proof we will encounter quantities of the type AiRNltw-b, i = 1,2, ... ,t, where Ai are constants that can be determined a priori only in terms of the data and are independent of w and R. We may assume without loss of generality that they satisfy (3.5)
Indeed if not, we would have w:5 C Reo for the choices C = max A~/b l~i9
and
t
and the first iterative step of the Proposition would be trivial.
Remark 3.3. The proof below and (2.2) show that the numbers e and eo can be taken as
eo
NK.
= b'
e = (p- 2}eo.
In the estimates to follow we denote with 'Y a generic positive constant that can be calculated a priori depending only upon the data and that may be different in different contexts.
48
ill. HOlder continuity of solutions of degenerate parabolic equations
3-(i). About the dependence on lIulloo.nT We will use the energy and logarithmic estimates of Propositions 3.1 and 3.2 of Chap. II for the truncated functions (u - k)± over cylinders contained in Q (aoRP , R). When working with (u - k) _ we will use the levels for some i
~
o.
These levels are admissible since
lI(u - k)-lIoo.Q(IJoRP.R) ::;; 60 • When working with (u - k) + we will take levels for some i
~
o.
These are also admissible since
II (u -
k)+ lloo.Q(lJoRP.R) ::;; 60 •
Let us fix 60 as in (3.11) of Chap. II. Then, sincew::;;2I1ull oo .nT' (3.2) holds true if we choose So so large that
280 =
8~2I1ulloo.nT.
Having chosen So this way. (3.2) is verified when working within any subdomain of DT • The a priori knowledge of the norm lIulloo.nT is required through the number So. If the lower order terms b(x, t, u, Du) in (1.1) satisfy (A~) of §5 of Chap. II, then. as remarked there. the energy and logarithmic inequalities hold true for the truncated functions (u - k)± with no restriction on the levels k. Thus in such a case So can be taken to be one and no a priori knowledge of lIulloo.nT is needed. The numbers A and Ai introduced in (3.5) will be chosen to be larger than 280 • In the proof below we will choose them of the type and
.-
A . - 2;0+h, ,
i
= 0,1,2, ... ,
where hi ~ 0 will be independent of lIulloo.nT. We have just remarked that if the lower order terms b(x, t, u, Du) satisfy (A~) of§5 of Chap. II, then So can be taken to be one. We conclude that for equations with such a structure, the numbers Ai can be determined a priori only in terms of the data and independent of the norm
lIulloo.nT·
4. The first alternative 49
4. The first alternative LEMMA 4.1. There exists a number Vo E (0, 1) independent 0/ w, R, A such that if/or some cylinder of the type [(0, f) + Q (dRP, R)]
I(X, t) E [(0, f) + Q (dRP, R)] lu(x, t} < IL- + 2~0 15 volQ (dRP, R) I, then (4.1)
u(x,t) > IL-
+ 2S~+1
a.e. (x,t)
E
[(O,f) +Q(d(f)",f)]·
PROOF: Fix a cylinder for which the assumption of the lemma holds. Up to a translation we may assume that (0, f) (0, O), and we may work within cylinders Q(dPP,p) , O
=
Rn
R
R
="2 + 2n+1'
n = 0,1,2, ... ,
construct the family of nested cylinders Q (dR~, Rn) and let (n be a piecewise smooth cutoff function in Q (dR~, Rn) such that
(4.2)
We will use the energy inequalities of Proposition 3.1 of Chap II. written over the cylinders Q (d~, Rn), for the functions (u - kn )-. where forn=O, 1,2, ... ,
k = n IL
W +~ 280+1 + 280+1+n '
In this setting. (3.8) of Chap. II takes the form (4.3)
esssup
-dR~
J(u - k n ): (!(x, t)dx + jr flD (u - knL (nlPdxd1'
J ,
Q(dR~,Rn)
KRn
57: {JJ(U-kn)~dXd1'+~ Q(dR~,Rn)
!!(U-kn):dXd1'}
Q(dR~,Rn)
,,(1+,,)
+7 {
J
IAkn,Rn (1')1 i d1'}
-dR~
We will show that as n -+ 00
r
50
DI. HOlder continuity of solutions of degenerate parabolic equations
Ilx[(U-knL >O]dxdr-+O. Q(dR!:.Rn)
+ 2.::'+1' this would imply that
Since kn '\. koo = J,L-
thereby proving the lemma. We observe that
and estimate above the first two tenns on the right hand side of (4.3) by
7
~: (2: )2 (2: 0
0
r-
2
II X[(u - knL > 0] dxdr
Q(d~.Rn)
+7:
:57:
(2:J ~ II X [(u - knL > 0] dxdr (2:J" Ilx[(U-knL >O]dxdr, 2
Q(dR~,R,,)
Q(d~.R,,)
where we have used the defmition (3.3) of d. Combining these remarks in (4.3) and dividing through by d, we arrive at (4.4)
I(u - kn)~ (:(x, t) dx + -d1
esssup
-dR~
I liD (u - knL (nl"dxdr 11 Q(d~.R,,)
KRn
:57~: (2:J" ~ Ilx[(U-knL >O]dxdr Q(d~.Rn)
+7
( W
2 Bo
),,-2 d,. ~
{
}~ I ~dR~IAkn.R,,(r)l·dr 1
0
In (4.4) we introduce the change of time-variable z
1:
= tid which transfonns
Q (d~, Rn)into Qn Setting also
== Q(~,Rn) == KR" x{-~,O}.
v(·,z) = u(·,zd)
and
(n(-,z) = (n(·,zd),
the inequality (4.4) can be written more concisely as
4. The first alternative 51
where we have set
o
J
IAnl =
and
IAn(z)ldz.
-Rl:
Since (11 - knL
II (11 - knL 1I:,Q,,+1 $11 (11 $ II (v -
knL
$11 (v - knL
11(11 - knL
1I:,Qn+l ~ Ikn -
kn+1 IPIAn+1I
~ 2P(~+2)
(;.'J PIAn+1l·
Combining these estimates gives (4.6)
IA 11+~ IAn+d ~ -y4np nRP + 14-'
t
(2~' r' dol':"' lA_I'm LIA,,(
Ell±!!.l.
Divide by IQn+11 and introduce the quantities
Using also the fact that. by virtue of Remark 3.2 and (3.5) RNIC
(.!!!...) -2 2
dP(1:IC)
80
we obtain from (4.6) in dimensionless fonn
<1 -
,
Ii
z) dz }
•
52
m. H6lder continuity of solutions of degenerate parabolic equations y.n+ 1 < "'4np {y'l+~ _I n
+ Y.~Zl+"} n n ,
'tin
= 0,1,2, ....
Next by the .embedding of Proposition 3.3 of Chap. I
Therefore np {y. + Zl+"} Z n+l < _ ",4 I nn
,
'tin = 0, 1,2, ....
From Lemma 4.2 of Chap. I it follows that Yn and Zn tend to zero as n -+ provided
where 80
00,
=min { N;"p; It}.
5. The frrst alternative continued Suppose the assumptions of Lemma 4.1 are verified for some box of the type [(0, l) + Q (dRP, R)]. We will exploit the fact that at the time level
-8
(2BoW)2-P (R)P "2'
=t -
the functionx-+u(x, -8) is strictly above the level ",- + 2.::'+1' in the cube K R / 2 • To simplify the symbolism let us set p = ~ and construct the cylinder
Q(8,p) == Kpx(-8,0),
p= R/2.
The length 8 of such a cylinder satisfies (5.1)
(-2-W)2-P < -pP8 -< (W)2-P -A 2
P•
0
-
The next lemma asserts that, owing to (4.1), the set where u(·, t) is close to I.e , within the smaller cube KR/4' can be made arbitrarily small for all time levels -8~t~0.
S. The fllSt alternative continued S3 LEMMA 5.1. For every number Vi E (0,1), there exists a positive integer 81. depending only upon the data and independent of w, R, such that
'lit E (-9,0). PROOF: Consider the logarithmic estimates of Proposition 3.2 of Chap. II, written over the cylinder Q (0, p), for
(u-k)_, As a number c in the definition of W, we take W c-~"':"":'"":"
- 2s o+ i + n
n> 1,
'
where n is to be chosen. Thus we take
where
Hi;
= esssup (u - (JL- + 2 W+ i ) ) Q(fJ.p) So
For t= -0, by virtue of(4.1) we have
::; _
2 w+ i
.
80.
(u - (JL- + 2.~+dL =0, and therefore
W(x, -0) = 0, These remarks in (3.14) of Chap. II yield
Iw
II
W!Wu !2- Pdxdr
2(x, t)(P(x)dx ::; ;
(5.2)
Kp
+
Q(fJ.p)
~(2":'+' r' ~ + In H, (2<.:,+0>
n(lIA',p(T) dT) · , .ell±!!l
Ii
where A;.i·) is defined in (3.4) of Chap. II and x -+ (x) is a piecewise smooth cutoff function in Kp that equals one on Kp/2 and such that ID(I::; 4/ p. Next
W::;ln( and
!WU !2-P = !Hk_-
~
~
(u - k)_
) =nln2
W n !P-2 ::; (W + 2so +l+ 2
80
)P-2 •
Therefore, in also view of (5.1), the first term on the right hand side of (5.2) is estimated above by
54
m. Itilder continuity of solutions of degenerate parabolic equations ; ffl[lll[lul2-PdXdT S 'Yn;
(2~JP-2IKp/21 S 'YnAP-2IKp/21,
Q(9,p)
where l' and A are constants depending only upon the data. and A has to be determined later. The second tenn is estimated by using the conditions (3.6). (3.7) of Chap. II. linking the parameters r, q, K.. This gives
n
Ei!±cl
~(2'.:,+or'(l + In H; (2•.:'+0 (lIA'-"
)-2 (W)-Ei!±cl(P-2) A R r
NIt IKp / 2
1.
The number n will be detennined shortly. depending only upon the data and independent of w, p. Therefore by virtue of Remark 3.2 and (3.5) we may estimate n
(
w
2Bo +l+ n
)-2 (~)_P(1:")(P_2) Nit < A R _ 1.
Combining these remarks into (5.2) yields
I
f 1[12(x,t)dx S 'YnAP- 2 Kp/2
(5.3)
I,
Kp/2
where we have used the fact that (== 1 on K p/2. The integral in (5.3) is estimated below by extending the integration to the smaller set {x E K p/ 2 1 u(x, t)
< ,.,.- + 2Bo~l+n} ,
t E (-0,0).
On such a set
and since the right hand side of this inequality is a decreasing function of Hi; • we have 1[12 ~ In2( 20:+1 ) = (n _ 1)21n2 2. 2°o+ n
Putting this into (5.3) gives that for all t E (-0,0) (5.4)
Ix E K
p / 2 Iu (x,
t)
<,.,.- + 2Bo~l+n IS 'Y AP- (n ~ 1)2I Kp/21· 2
To prove the lemma we have only to choose n sufficiently large.
6. The fmt alternative concluded 55
6. The first alternative concluded The infonnation in Lemma 5.1 will imply that u is strictly bounded away from p.in a smaller cylinder. To make this precise. consider the box
=Kp/2X(-0,0),
Q(O,~)
P = R/2,
where (J satisfies the bounds in (5.1). LEMMA 6.1. The numbers V1 E (0,1) and 81 » 1 can be chosen a priori dependent only upon the data and independent of w and R. so that
u(x, t) > p.-
w
+2
a.e. (x, t) E Q ((J,~)
81 +1'
.
PROOF: We will use the local energy estimates of Proposition 3.1 of Chap. II in the following setting. Let
n = 0,1,2, ... ,
construct the cylinders Q ((J, Pn) and let x -+ (n (x) be a piecewise smooth cutoff function in KPn that equals one on KPn+1 and such that ID(n 1:$ 2n+3 / p. Write the inequalities (3.8) of Chap. II over Q (0, Pn) for the functions (u - kn )_. where kn
= p.- + 2
w 1
81 +
+ 2s
w l+ Hn
'
and observe that. owing to the conclusion of Lemma 4.1. (u - knL (x, -0) = 0,
IrIx E KPn,
n
= 0,1,2, ....
With these choices. the energy inequalities yield (6.1)
sup -8
::;
j(u - kn)~ '~dx + KPn
,),-1
r riD (u - knL (niP dxdr
JJ Q(8,Pn)
')'~P jj(u - kn)~ dxdr +"( {JIAkn,pJr)lidr} Q(8,Pn)
£lli!!.!. r
-8
The first tenn on the left hand side is estimated below. for all t E (-0,0). by
j(u - kn)~ '~dx ~ (2~1) 2-p j(U - kn)~ '~dx KPn > 2-P (2 )P-2 ~ j(u - k )P I"Pdx > ~ j(u - k )P I"Pdx A n - '>n n - '>n ,
KPn
81
pP
pP
KPn
KPn
56
DI. HOlder continuity of solutions of degenerate parabolic equations
if 81 is chosen so large as to satisfy the conclusion of Lemma 5.1 and the inequality 2- P (281 I A)p-2 ~ 1. We put this in (6.1), divide through by BI pP and introduce in the cylinders Q (B, Pn), the change of variable z = tPP lB. This maps Q (B, Pn) into the boxes Qn == KPn x(-PP,O). Let us also set v(x, z) = u(x, zBI PP) and
o
IAnl = jIAn(z)ldZ. -pP
By the embedding of Corollary 3.1 of Chap. I,
2-(n+2)1' (2~1 )"IAn+l1
~
kn)~ dxdz
jj(v -
Qn+1n[v
~
jj(v -
kn)~ (~dxdz
Qn
~ 111 (v - knL (nll~p(Q,,)IAnlwT,; < 1 2n1' (~)p IA Il+~ -
pP
28 1
n
+~(~t"¥" 1A.lm {}An(Z>li dz } I
Divide through by the coefficient of An+ 11 and set
Using also (5.1) and (3.5) to estimate
I!i!±!!l
•
6. The first alternative concluded 57
we arrive at
Yn+1 :5 -y4np {y~+m:;;
+ Yn;vT,; Z~+/( } .
Proceeding as in the proof of Lemma 4.1, we have
Zn+1 :5 -y4 np {Yn + Z~+/(}. By Lemma 4.2 of Chap. I it follows that Yn and Zn tend to zero as n -+ 00, provided (6.2)
{NT,; K}.
where 90 = min j To prove the lemma, we fix according to Lemma 5.2. We summarise the results obtained so far.
III as in (6.2) and pick
81
PROPOSITION 6.1. There exists numbers 110 , TJo E (0, 1) and Al » 1 depending only upon the data and indepeiuJent o/w, R, such that if/or some cylinder o/the type [(0, l) + Q (dRP, R»),
(6.3)
I(X, t) E [(0, l) + Q (dRP, R)llu(x, t) < jJ- + 2~o
I
:5 1I0 IQ(dRP,R) I, then either (6.4)
or
essosc
(6.5)
Q(d( I)".f)
u:5
TJo W
where b is introduced in (3.4). PROOF: Assume (6.4) is violated. By Lemma 6.1, we can determine a positive number 81 such that .f u > jJ- +-W essm Q( 8,-f) 281 +1
where 9 satisfies (5.1) with p = R/2. Change the sign of this inequality and add the quantity esssuPQ(8.i) u to the left hand side and jJ+ to the right hand side. This gives
ess OBC U :5 Q(8.f)
(1 - 21+1) w.
Therefore the proposition follows with TJo =
81
(1 - 2.:+1), since
58
m. IIUder continuity of solutions of degenerate parabolic equations
Remark 6.1. Let us trace the dependence of "10 and Al upon lIulloo,nT. The numbers 110 and III depend only upon the data and are independent of u. The number 81 is given by 81 = 8 0 + n where n is chosen from (5.4). Thus
depends upon lIulloo,nT via 8 0 through (3.2). Also A and Al are of the type Ai = 2so +h ,. where hi, i = 0, 1,2, ... , can be determined a priori only in terms of the data and are independent of II u II OO,nT. We conclude that if the lower order terms b(x, t, u, Du) satisfy the structure condition (Aa) of§5 of Chap. II. we have 8 0 = 1 and therefore "10' A, Ai can be determined a priori only in terms of the data and are independent of lIulloo,nT.
7. The second alternative We assume in this section that the assumptions of Lemma 4.1 are violated. i.e. for every subcylinder [(0, t) + Q (dRP, R)] cQ (aom', R)
I(X,t) E [(O,t) +Q(dm',R)]lu(x,t) < p.-
+ 2~o
I> 1I0lQ (dRP, R) I·
Since we rewrite this as (7.1)
I(X, t) E [(0, t) + Q (dRP, R)lIu(x, t) > p.+ -
2~o I
~ (1- 110) IQ(dRP,R) I,
valid/or all cylinders [(0, t) + Q (dRP, R)]
~
c Q (aoRP, R)
ao
=
(~)P-2. A
In view of (7.1) we will study the behaviour of u near its supremum p. + and will be working with the truncated functions (u - k) + for the levels i
~
0.
LEMMA 7.1. Let [(0, t) + Q (dRP, R)] cQ (aom', R) befixedand let (7.1) hold. There exists a time level
t·
such that
E
[f - dRP , f -
110 dRP]
2
'
7. Tho second alternative 59
PROOF: If not, for aU tE [i - dJlP, i - ~dJlP],
Ix E KR I u(x, t) > p.+ - 2~o I> (11_-v:i2) IKRI and I(X,t) E [(O,t)
~
+ Q(dJlP,R)] I u(x,t) > p.+ -
2~o I
t-!f-dRP
f
Ix E KR Iu(x, r) > p.+ - 2~o Idr
t-dRP
> (l-vo ) IQ(dR"',R) I, contradicting (7.1). The lemma asserts that at some time level t* the set where u is close to its supremum occupies only a portion of the cube K R. The next lemma claims that this indeed occurs for all time levels near the top of the cylinder [(0, t) + Q (dJlP, R)]. LEMMA
7.2. There exists a positive integer 82 > 80 such that
Ix fora/ltE
[i -
E
I
KR u(x,t) > p.+ -
2~21 :5 (1- (~f) IKRI,
~dJlP,fj.
PROOF: Consider the logarithmic inequalities (3.14) of Chap. II written over the box K R X (t* , t) for the function (u - k) + for the levels k =p. + - 2":0 . As for the number c in the definition of the function 1[1, we take
w c- -2Bo+n'
n
°
> to be chosen.
Thus we take
(7.2) where
Ht ==
ess sup [(O,f)+Q(dRP ,R)]
(u
-
( p.+
- -W ) ) . 280 +
The cutoff function x-+(x) is taken so that ( = 1 in the cube K(l-O')R, uE (0, I), and ID(I:5 (UR)-l. With these choices, inequality (3.14) of Chap. II yields for all tE(t*,t)
60
m. HiUder continuity of solutions of degenerate parabolic equations
I
(7.3)
t
j llf2(x,to<)dx+ «(1~)P j j llf lllful2- PdxdT
p 2(X,t)dx:5
K(l-,,)R
KR
tOKR
E1.!±.!!.l
(2'~+') -2 [1 + I. Hi C.~"") -1] {}At(Tl11dT} ·
+0
The various tenns in (7.3) are estimated as follows. First
llf:5nln2; Illfu
2P :5 2P (2"0W)P-2 ; [l+lnHt(2"0+n W )-1] l-
:5-ynln2.
Next, from (7.2) it follows that llf vanishes on the set [u<JL+ - 2":0]. Therefore, using Lemma 7.1, the first integral on the right hand side of (7.3) is estimated above by
The second integral is estimated by
«(1~)P
t
ff
llflllfuI2-PdxdT:5 ;pnIKRI,
tOKR
since f - to< :5dRP, and d is given by (3.3). Finally for the last tenn, we have E1.!±.!!.l
o
(2'~+.r' [1+ I.Ht (2.:'+.r'] {jIAt(Tl11dr} ·
:5 -ynA2w- bRNItIKRI,
where A2 = 2(B o +n)b and b is defmed in (3.4). By Remark 3.2 and (3.5) we may assume that nA2W-b RNIt $1. Combining these remarks in (7.3) we conclu(,ie that for all tE (to<, f) (7.4)
j
22
llf2(x, t) dx :5 n 1n 2 (11_-II:i2) IKRI
+ ;P nIKRI·
K(1-,,)R
The left hand side of (7.4) is estimated below by integrating over the smaller set {x E K(1-
1'+ -
2B~+n }
.
On such a set, since the function llf in (7.2) is a decreasing function of Ht, we estimate
7. The second alternative 61
After carrying this in (7.4) and dividing through by (n - 1)21n2 2 we obtain Ix E
K(l-u)R I u(x, t) > J1.+ - 211~+n I
~ (n:
lr (/--V:/2)
IKRI + u;n IKR1 ·
On the other hand
Ix E KR I u(x,t) > J1.+ - 28~+n I ~ Ix E K(l-u)R Iu(x, t) > J1.+ - 2s~+n 1+ IKR\K(l-u)RI ~ Ix E K(l-u)R I u(x, t) > J1.+ - 2B~+n 1+ NuIKRI· Therefore
Ix E KR Iu(x, t) > J1.+ - 2s~+n I
~ [(n: 1) (/--v:i2) + u;n + NU]IKRI, 2
for all t E (t* , f). Choose u so small that u N ~ ~ v~ and then n so large that
Then for such a choice of n the lemma follows with 82 =80
+ n.
Remark 7.1. Since the number Vo is independent of w and R, also 82 is independent of these parameters. The number A that determines the length of Q (aoRP, R) is still to be chosen. We will determine it later independent of w and R and subject to the condition A > 282 • Since (7.1) holds for all cylinders of the type [(0, f) + Q (dRP, R)], the conclusion of Lemma 7.2 holds true for all time levels satisfying
where ao and d are defined in (2.1) and (3.3) respectively. If the number A is chosen sufficiently large, we deduce
62
lli. HOlder continuity of solutions of degenerate panboIk equations
COROLLARY
7.1. ForalltE(-!fR",O),
From now on we will focus on the cylinder Q (If R", R) and to simplify the symbolism we set A.(t)
= {x E KRlu(x,t) > J.I.+ -
As = {(x,t) E Q
;.},
(~ RP,R) 11£(x,t) > J.I.+ -
;.}.
8. The second alternative continued The information of Corollary 7.1 will be employed to deduce that the set where u is close to its supremum J.I.+, within the cylinder Q (!fR", R), can be made arbitrarily small. In this section we will also detennine the length of the cylinder Q (aoR", R) by detennining the number A. LEMMA 8.1. For every II. E (0, 1) there exists a number S. >S2 itulepetulento! w and R, such that
Remark 8.1. Assume for the moment that the number B. has been chosen. Then we detennine the length of the cylinder Q (aoR", R) by choosing (8.1) PROOF OF LEMMA 8.1: Consider the local eneqy estimates (3.8) of Chap. II written over the box Q (aoRP, 2R), for the functions (1£ - k)+. The levels k are given by
w k=,,+--8 r-
2
'
where 82 :::; 8 :::; B. and B. is to be chosen. We take a cutoff function ( that equals one on Q (!f RP, R), vanishes on the parabolic boundary of Q (ooR", 2R) and such that
Neglecting the firsttenn on the left hand side of these energy estimates, and using the indicated choices, we obtain
8. The second alternative continued 63
(8.2)
IIID(U-k)+IPdxdT~;;' II(U-k)~dxdT Q(a RP,2R)
Q(.y.RP,R)
+
o
aO~ II(U-k)!dxdT+'Y{ Q(ao RP,2R)
~
JIAt.2R(T)lidT} -a';RP
r
The various tenn on the right hand side of (8.2) are estimated as follows. First (i)
II(U-k)~dxdT~ ;P (;rIQ(~RP,R) I·
;;,
Q(a Rp,2R) o
Next by virtue of the choice (8.1) of the parameter A, and the defmition (2.1) of ao , (ii)
ao~
If (u - k)! dxdT ~ ;" (; Q(a RP,2R)
r IQ C; R", R) I·
o
Finally making use of Remark 3.2 and (3.5) ~
(iii)
~ {j~At.2R(T)I\dT} · . ~ ;" (;r IQ (~ R",R) I(A 3w- b RNK )
~;" (;rIQ(~RP,R) I, where A3 = 2b8 • and b is defined in (3.4). These estimates in (8.2) give (8.3)
fflDulPdxdT
~ ;" (;r IQ (~ RP,R) I·
A.
Next we use Lemma 2.2 of Chap. I applied to the function u(·, t) for all times RP ~ t ~ 0, and for the levels
-!f
(I - k) Notice that by virtue of Corollary 7.1 we have
Applying Lemma 2.2 of Chap. I in this setting, gives
w
= 2 +1 . 8
64
III. RUder continuity of solutions of degenerate parabolic equations
JIDul
4-yR N H
w
2S+1IAs+1(t)1 :5 v~ IKRI
dx,
A.(t)\AO+l (t)
for all t E ( - ~ flP , 0). From this, integrating over such a time interval we get
2:1 lAsH I
:5
~ R JJ IDul dxdr A.\A'+1 1
,; Take the
~R ([fIDU1PdxdT) 'IA,\A.+d7'.
;tr power, estimate the integral on the right hand side by (8.3) and divide
through by
(2;+1 ) ~ . This gives
IAs+1l~ :5 -y(vo)-t!r IQ (~ RP,R) 1;;!-rIAs\As+1l. These inequalities are valid for all 82 :5 8:5 8 •• We add them for 8
= 82,82
+ 1,82 + 2, ...
8. -
1.
The right hand side can be majorized by a convergent series bounded above by (~RP, R) Therefore
IQ
I.
(8. - 82)
IAs.l~ :5 -y(vo)-t!r \Q (~ RP,R) \~.
To prove the lemma we divide by (8. - 82) and take 8. so large that -y ----.:..--E.::.!--..,-I
v~ (8. - 82)
:5 v•.
p
Remark 8.2. If v. is independent of wand R, also 8. and hence A are independent of these quantities. Remark 8.3. The process desCribed in Lemma 8.1 has a double scope. Given v., it determines a level p. + - 2":' and a cylinder so that the measure of the set where u is above such a level can be made smaller than v., on that particular cylinder.
9. The second alternative concluded Next we show that indeed u is strictly below its supremum p.+ in a smaller box coaxial with Q (~RP, R) and with the same vertex. To simplify the symbolism ~~~
a. = !ao = ! (~)P2
2
2
w
and write accordingly Q (~RP, R) =Q (a.flP, R).
,
9. The second alternative concluded 65 LEMMA
9.1. The number II. (and hence s. and A) can be chosen so that u(x, t) :5 JI.+ -
2S~+l '
PROOF: We will apply the local energy estimates of Proposition 3.1 of Chap. II over the boxes Q (a. R~, Rn) to the function (u - kn ) +, where for all n =
0,1,2, ... , and The cutoff functions (n are taken to satisfy
0< (n(X, t),
in Q (a.R~+l,Rn+d
(n == 1 (n = 0
a..
and
j
on the parabolic boundary of 0 < 8 r < 2 P (n+1) R' - 8T ..n a. RP ,
I< .. n -
IDr .!.
Vex, t) E Q (a.R~, Rn) ,
Q (a.~, Rn),
2"+1
= 2 (!:!L)P-2 A '
A = 2S ••
With these choices. inequalities (3.8) of Chap. II take the form (9.1)
esssup
I(u - k n )! (~(x, t)dx +
f f ID (u 11 Q(a.R:;,R,,)
-a.R!:
Rp2
:5 'Y
np
k n )+ (nlPdxdr
ff (u-kn)~dxdr+ :.RP 2np ff 11 11 (u-kn)!dxdr
Q(a.R:;,Rn)
Q(a.R!:,R,,) p(1+")
+~{j~At..''
I(U - kn )! (!:(x, t) dx KRn
~ (2~.) 2-pI
(u -
kn)~ (!:(x, t) dx
KRn
~ 2a.1I (u - kn )+ (nll:,KRn (t). Next. using again the definition of a., the first two terms on the right hand side of (9.1) are estimated above by
66
m. HOlder continuity of solutions of degenerate parabolic equations
Substituting this in (9.1) and dividing through by aoo gives
By (3.5) and Remark 3.2 we may estimate a
*:">-1 -< (~)P A w-b < (~)11 R-NIC 2". 2". '
•
4
-
where A4 = Ab. Next, in the cylinders Q (a .. ~, Rn) we introduce the change of variable z =tfa.. which maps Q (aoo~, Rn) into Qn =KR" X (-~, 0). Setting
v(·,z)
= u(·,aooz),
and
IAnl =
o
J
-R~
inequality (9.2) can be rewritten more concisely as
This inequality and Corollary 3.1 of Chap. I give
IAn(z)ldz,
9. The second alternative concluded 67 2-(n+2)p
(~)P 28 •
IAn+l I
= (kn+l - knt I(x, z) E Qn+!
I v(x, z) > kn+ll
:5 II (v - k n )+ lI~n+l :5 II (v - kn )+ (nll~n :5 'YIAnl ~ II (v - k n)+ (nllt"(Qn) np < '"I (~)P 2Rp IA n 11+~ 28 •
+~ (2~.r R-N!:dZ}
Ei!.±!!l
•
Thus setting
we have the recursive inequalities
Yn+! :5 'Y4np {Y~+~
+ Yn~ Z!+I< } ,
Zn+l :5 'Y 4np {Yn + Z!+I<} . It follows from these with the aid of Lemma 4.2 of Chap. I that Yn and Zn tend to zero as n - 00 provided
Yo where 00 = min
1 + " 1 + ..
+ Z~+I< :5 'Y -er 4- p er == II.,
{~; "'}.
The following proposition summarises the results of the second alternative and it is proved arguing as in the proof of Proposition 6.1 PROPOSITION 9.1. There exists numbers 110 ,111 E (0,1) and A2 » 1 depending only upon the data and independent of w and R, such that if for all cylinders of the type [(0, f) + Q (dRP, R)]
I(x, t)
E
[(0, f) + Q (dRP, R)] lu(x, t) > Jl.+ -
:5 (1 then either (9.3)
2~o I 110
)
IQ (dRP, R) I,
68
m. UUder continuity of solutions of degenerate parabolic equations
or (9.4)
essosc u < 111 W [Q(a.( ft.f)] -
where b is introduced in (3.4).
Remark 9.1. The constants 111 E (0,1) and A2 depend only upon the data and, in general, also upon the nonn lIulloo.nT via the number So. If the lower order tenn b(x, t, u, Du) satisfies the structure condition (A~) of §5 of Chap. II, we have So = 1 and therefore 111, A, A2 can be detennined a priori only in tenns of the data and are independent of lIulloo.nT'
10. Proof of Proposition 3.1 The two alternatives just discussed can be combined to prove the main Proposition 3.1. Let us recall that
~
=
ao
(~)P-2. A
The concluding statement of the first alternative is that, starting from the cylinder
and going down to the smaller cylinder
Q(d(ft,f), the essential oscillation w decreases by a factor 110 E (0,1), unless w :5 A1R~, where A 1 is a large constant that can be computed a priori only in tenns of the data and the number So is introduced in (3.2). Analogously, the conclusion of the second alternative is that starting from the same cylinder and going down to the smaller box
Q(!f(ft.f) , the number w decreases by a factor 111 E (0, 1), unless w:5 A2R~ , where A2 is a constant that can be computed a priori in tenns of the data. We combine these two facts into LEMMA
10.1. There exist constants
that can be determined a priori only in terms of the data, such that either
w:5.AR~
or
11. Regularity up to t = 0 69 We comment further on the content of Remark 3.1. The arguments presented do not require that the starting cylinder be Q (RP-E, 2R). It would have been sufficient to have started from the box
if we had known a priori that (l0.1)
essosc
Q(aoRP,R}
U
< w. -
Next we will construct a box for which information of the type of (l0.1) can be derived. Set
WI == max {Tlw;AR~}
and
~
= (WI)P-2,
al
A
and let us estimate from below the length of the cylinder Q ( d (
i)P ,i) for which
the conclusion of Lemma 10.1 holds. We have
where and
It follows that, for the cylinder Q (aIRf, R I ), inequality (l0.1) is verified and the process can now be repeated starting from such a box, thereby proving Proposition 3.1. As indicated in §3 this implies the interior mlder continuity stated in Theorem 1.1. The constant dependence indicated in the statement of the theorem follows from the arguments of §3-( I) and Remarks 6.1 and 9.1.
11. Regularity up to t = 0 Let u be a weak solution of (1.1) of Chap. II that takes initial data U o in n. We assume U o is continuous with modulus of continuity, say wo (')' The regularity of u up to t = 0 will follow from a proposition analogous to Proposition 3.1. Fix (xo, 0) E x {O}, and R> 0 so that [xo + K2RJ c After a translation we may assume Xo = 0 and construct the cylinder
n
n.
70
m. HOlder continuity of solutions of degenerate parabolic equations
where c is a positive number to be chosen. As before. set
p.+
=
esssup
Qo(RP-~ ,2R)
u, p.- =
essinf
Qo(RP-< ,2R)
u, w =
essosc
Qo(RP-~ ,2R)
u.
Let So be the smallest positive integer satisfying (3.2) and construct the box
For all R> 0, these boxes are lying on the bottom of ilT. PROPOSITION 11.1. There exist constants co, ij E (0, 1) and C, A > 1 that can be determined a priori depending only upon the data. satisfying the following. Construct the sequences Ro = R, Wo =W and
Rn = C- nR,
Wn+l
= max {iiWn; CR~o},
n= 1, 2, ... ,
and the family of boxes
~
__ (~)P-2,
n
A
an
= 0,1,2, ....
Thenfor all n=O, 1,2, ... and
{w
essoscu :5 max Q~")
n ;
2essoscuo }. Kiln
°
The proof of the continuity (or the HOlder continuity) of u up to t = follows from a simple variant of Lemma 3.1. Statement and proof of such a variant goes along the lines of similar arguments in §3. Here we indicate how to prove Proposition 11.1. Assume without loss of generality that p.+ ~ Ip.-I and that dRP < RP- e,
. I.e..
( 2':0 )P-2 > R£. Indeed otherwise we would have . C: o
E
= --.
p-2
This implies that Qo (dRP, R) is all contained in the box Qo (RP-e, 2R). and we may work within Qo (dRP, R). Also without loss of generality we may assume that So ~ 2. Set
and consider the two inequalities (11.1)
+
w
p. - 280
< P.o+ ,
_ p.
w
_
+ 2 > P.o • 80
If both hold, subtracting the second from the first we obtain
11. Regularity up to t =0 71
and there is nothing to prove. Let us assume. for example. that the second of (11.1) is violated. Then for all 8:::: 8 0 • the levels
_ k = Jl.
w
+2
8 '
satisfy the second of (4.16) of Chap. II. Therefore we may derive energy and logarithmic estimates for the truncated functions (u - k) _. These take the form (11.2)
sup
}(u - k):(x, t}("dx + JfJf ID(u - k)-(rdxdT
O
Qo(dRP,R)
II(U-k)"-'D('''dxdT+1'{jiBk.R(T)'~dT}
$1' Qo(dRP,R)
,.
,
0
1!li2(Dk",(u-k)-,c)("(x)dx
sup
(11.3)
.ei.!±.cl
O
$ l' II
!lil!li
u
2 -"ID(I"dxdT (Dk", (u - k)_,c) 1
Qo(dRP,R)
+; (Hln ~') {fBk.n(fll'df ('"' , Dt
where and Bt,R are defined as in (4.2) and (4.4) of Chap. II. The proof can now be completed as follows. First by using the logarithmic estimates (11.3) and proceeding as in Lemma 5.1. given any eo E (0, 1) we can find positive numbers to and Ao. depending only upon the data. such that either (b defined in (3.4»
(11.4)
or. for all t E (0, dJlP).
Ix
E
KR/21 u(x, t) < Jl.-
+ 2~0 I< eoIKR/21·
Second. using the energy inequalities (11.2) and the procedure of Lemma 6.1. we conclude that if (11.4) does not hold. then (11.5)
esSl'nf u > Jl.Qo(d(ft.f)
W + --. 1
2 0+1
Changing the sign of (11.5) and adding ess sUPQo( d( ft. f) u to the left hand side and Jl.+ to the right hand side we obtain
72
ill. ltilder continuity of solutions of degenerate parabolic equations
1
ij = 1 - 2to +1 .
If the frrst of (11.1) is violated, we write the energy and the logarithmic inequalities for (u - k)+, k=JI.+ - ~ for s~so and proceed as before. To summarise, going down from Qo (RP-E, 2R) to the smaller box
the essential oscillation decreases by a factor of ij, unless either W
< 2essoscuo KR
or
LEMMA 11.1. There exist constants Ao > 1 and ijE (0,1), that can be computed a priori only in terms of the data, such that either
or To prove Proposition 11.1 we iterate this process over a sequence of boxes all lying on the bottom of nT • This is done by arguments similar to those in the previous sections.
12. Regularity up to ST. Dirichlet data Let (x o, to) be fixed and consider the cylinder !(xo, to)
+ Q (RP-£, 2R)], where
e = to(P - 2), where the number b is defined in (3.5) and It is introduced in (3.2) of Chap. II. We let R> 0 be so small that to - RP-E ~ 0, and change variables so that (xo, to) == (0,0). The function u solves (1.1) of Chap. II and takes boundary data 9 on ST in the sense of the traces of functions in V 2,p(nT). The Dirichlet datum (x, t)--+ g(x, t) is continuous in ST with modulus of continuity wg (·). Set JI.+ =
u, JI.
esssup
=
essinf
Q(RP-< ,2R)nI1T
u,
W
=
Q(RP-£ ,2R)nI1T
essosc Q(RP-£ ,2R)nI1T
and construct the box
Q(dRP,R) , where the number So is introduced in (3.2). Let also JI.;
=
sup Q(dRP,R)nST
g,
JI.- = 9
inf
Q(dRP,R)nST
g.
u,
12. Regularity up to Sr. Dirichlet data 73
If the two inequalities (12.1) are both true, subtracting the second from the first gives
w
~
2
osc
Q(dRP,R)nST
g,
and the oscillation of u over Q (dJtP, R) n {h is comparable to the oscillation of 9 over Q (dRP, R) n ST. Let us assume, for example, that the first of (12.1) is violated. Then the levels
satisfy (4.11) of Chap. II. and we may derive energy estimates for (u - k) +. Since (u - k)+ vanishes on Q (dJtP, R)nST. wemayextendittothe wholeQ (dJtP, R) by setting it to be zero outside nT within the box Q (dRP, R). Also. in (4.13) of Chap. II we take a cutoff function vanishing on the parabolic boundary of Q (dRP, R). Taking into account these remarks. we obtain the energy estimates (12.2)
sup
f(U-k)!(P(x,t)dx+jrrID(u-k)+(IPdxdr
-dRp
J
Q(dRP,R)
~ "Y f f(u - k)~ ID(IPdxdr + "Y ff(u Q(dRP,R)
k)! (p-l(t dxdr
Q(dRP,R) ~
"'{1.IBt.R(T)11dT} . , where Bt,R(r) is defined in (4.4) of Chap. II. We observe that the conclusion of Lemma 7.2 is automatically verified for (u - k)+. Indeed the function x -+ (u(x, t) - k)+, vanishes outside nnKR. for all t E (-dJtP, 0) and an satisfies the property of positive geometric density of Chap. I. Therefore we may use Lemma 8.1 and its proof to deduce that for all el E (0,1). there exist positive numbers Al and £1 that can be detennined a priori only in tenns of the data such that either w < Al RlYf' or
I(X, t) E Q (dRP, R) 1 u(x, t) > p.+ - 2~1 An application of Lemma 9.1 now gives
I< ellQ (dRP, R) I·
74
m.
HOlder continuity of solutions of degenerate parabolic equations
LEMMA 12.1. There exist numbers Al > 1 and ijE (0,1) that can be computed a priori only in terms of the data such that either
w
< AIRl!f-
or
essosc
Q(d( ~)",~)
u:5 max
{ijwi
osc Q(dR",R)nST
g}.
The proof of the theorem can now be completed by stating a proposition similar to Proposition 11.1.
13. Regularity at ST. Variational data First we remark that the proof of interior regularity is only based on the energy and logarithmic estimates of §3 of Chap. II. In particular if such estimates were available for some locally bounded function U E vt:':( nT), then the conclusion of Theorem 1.1 would hold for u, irrespective of the differential equation u might satisfy. Keeping this in mind, one realises that the proof of Theorem 1.3 is the same as that of interior HOlder continuity, owing to the energy and logarithmic inequalities of Proposition 4.1 of Chap. II. If (xo, to) E ST is fixed, after a translation to (xo, to) == (0,0) and a local inequalities (4.6) and (4.7) of §4 of Chap. II, can be viewed as flattening of written over cylinders of the type Q+(8, p), defined in (4.8) of Chap. II. The cutoff function x -+ (x, t) vanishes on the boundary of Kp and not on the boundary of This affects the proof only in the application of the embedding Corollary 3.1 of Chap. I. Such an embedding was applied, after rescaling, to functions VE V"(Qn), where Qn==KR" x {-~,O} (see Lemmas 4.1,6.1,8.1). Now for these domains, the ratio T Ilnl,,/N is a constant depending only upon the dimension N. We also remark that the application of Lemma 2.2 of Chap. I, in the context of half cubes is possible since such a lemma holds for convex domains (see Remark 2.2 of Chap. I).
on,
K"t.
K"t,
14. Remarks on stability As p'\. 2 the equation becomes less degenerate. The proof presented in the previous sections shows that "Y and Q are stable in the sense that lim "Y(p) = "Y(2) < ",2
00
and
lim Q(p) = Q(2) E (0,1). ",2
Thus the classical results of HOlder continuity of weak solutions of quasilinear non-degenerate parabolic equations can be recovered from our results by letting p'\. 2 in the structure conditions of § I of Chap. II.
15. Bibliographical notes 75
14-(i). Continuous dependence on the operator A similar stability holds for the local behaviour of solutions of a family of equations of the type of (1.1) of Chap. II. To be specific, let us consider as an example the family of equations
8 8T UA UA E
.
- dlvaA
(x, t, UA, DUA) =
bA
(x, t, UA, DUA)
C'oe (0, Ti L~oc(n))nLfoe (0, Ti wl~:(n») ,
where A ranges over some subset I of the real numbers. Assume that for all AE I, the functions
satisfy the structure conditions (At}-(A5) uniformly in A, i.e., for constants Ci and functions "Pi, i =0, 1, 2, independent of A. Assume moreover that
uniformly in A. Then LEMMA 14.1. {UA} is a family of uniformly Holder continuous functions over compact subsets of nT.
Results of this kind are referred to in the literature as continuous dependence of the solution on the operator. Stability results also hold for a family of equations where also the parameter p ranges over a compact subset of [2, 00).
15. Bibliographical notes Questions regarding the local behaviour of solutions of equations of the type of the p-Iaplacian were raised by Ladyzenkaja-Solonnikov-Ural'tzeva [67], Aronson-Serrin [7] and Trudinger [97]. The first results for the elliptic case appear in Ural'tzeva [100] and Uhlenbeck [99] and, for the parabolic case, in [39]. These results hold also for systems and we will comment further on them in Chap. VIII. The proof presented here is taken from [36,37]. The structure conditions (Ad-(A5) are optimal for Theorems 1.1-1.3 to hold, as pointed out in [67] in the non-degenerate case p = 2. The iteration technique of Lemma 4.1 is a parabolic version of a similar elliptic technique due to DeGiorgi [33]. The new input regards the space-time geometry intrinsically defined by the solution itself. A first version of this technique appears in [38] in a simpler situation. It turns out that the same idea can be used to establish the local HOlder continuity of solutions of the porous medium equations and its generalisations. Here we mention the contributions of [37] and [24]. It can also be used to prove the local HOlder continuity
76
m. HOlder continuity of solutions of degenerate parabolic equations
of doubly degenerate equations. To be specific consider the p.d.e. (1.1) of Chap. II satisfying the structure conditions (AI)
a(x, t, u, Du)·Du ~ Co~(Iul)IDuIP - lPo(x, t),
(A2)
la(x,t,u,Du)1 $ C1~(luI)IDulp-l +~;(U)lPl(X,t), Ib(x, t, u, Du)1 $ C2~(lu1)1Du1P + lP2(X, t).
(A3)
The non-negative functions lPi, i=O, 1,2, satisfy (A4)-(A5) of§1 of Chap. II. The function ~( .) is degenerate near the origin in the sense that there exists a number lTo > such that
°
°
°
for given positive constants < 1'1 $ 1'2 and $ f32 $ /31. This behaviour has to hold only near the degeneracy, i.e., for s near zero. For s > lTo it will suffice that ~(s) be bounded above and below by given positive constants, i.e., for example,
We require that u E Cloc (O,TjL~oc(.a»),
~;6(u)IDul
E
Lfoc(!1r).
Let F(.) denote a primitive of ~;6 (.). Then the p.d.e. can be interpreted weakly by requiring that
F(u)
E
Lfoc (O,T;W,!::(il»).
If~(s) = 1, 'r:Is >0, then (1.1) is of the p-Iaplacian type. Ifp=2 and ~(s) =sm-1 for some m > 0, then 0.1) exhibits a degeneracy (m > 1), or singularity (0 < m < 1) of the type of porous medium equation. In the latter case a weak. solution is required to satisfy
lul m E
L~oc (o,TjW,!:(ilT»).
The mlder continuity of solutions of such doubly degenerate equations can be proved by methods similar to the ones presented here and has been established independently by Porzio-Vespri [88] and Ivanov [52]. The technique is also flexible enough to handle equations bearing a power-like degeneracy at two values of the solutions. These arise in the flow of immiscible fluids in a porous medium and have as a prototype Ut
= Llu(1 -
u)
= 0,
O:5u:51.
Results on continuous dependence appear in [9] in a different context.
IV Holder continuity of solutions of singular parabolic equations
1. Singular equations and the regularity theorems Evolution equations of the type of (1.1) of Chap. II for 1 < p < 2 are singular since their modulus of ellipticity becomes unbounded when Dul =0. We will lay out a theory of local and global HOlder continuity of solutions 11. of such singular p.d.e. 'so We assume that 11. E L 00 (nT ). If 11. is only locally bounded it will suffice to work within compact subsets K. of nT. The intrinsic p-distance dist (K.j rjp) from K. to the parabolic boundary of nT is defined as in (1.1) of Chap. ID. In the theorems below, the statement that a constant 'Y depends upon the data means that it can be determined a priori only in terms of lIulloo,nT • the constants Gi , i=O, 1, 2, and the
I
norms lI'i'o, 'i'r ,'i'2I1q,r';nT appearing in the structure conditions (At}-(A5). For pin the singular range 1
1)
Cz P n Ix - yl + It - sip == (".'lEIC inf ( lIuli oo . ' T (1I •• lEr
1-(i). HOlder continuity in the interior THEOREM 1.1. Let 11. be a bounded local weak solution of (1.1) of Chap. 11 and let (At}-(A5) hold. Then 11. is locally HOlder continuous in nT • and there exists constants 'Y> 1 and a E (0,1) depending only upon the data. such that VK. c nT.
78
IV. HBlder continuity of solutions of singular parabolic equations
/oreverypairo/points (Xl, td, (X2, t2) E IC. I/the lower order terms b(x, t, u, Du) satisfy (A~) 0/§5 o/Chap.lI, then 'Y and a are independento/llulloo.or .
1-(ii). Boundary regularity (Dirichlet data) THEOREM 1.2. Let u be a bounded weak solution o/the Dirichlet problem (2.1) o/Chap. II and let (D) and (Uo ) hold. Assume also that the boundary an has the property 0/ positive geometric density (1.1) 0/ Chap. I. Then u E C (liT) and there exists a continuous non-decreasingfunction s -+ w(s) : R+ -+ R+, such that w(O);:O and
lu(XI. tl) - U(X2' t2)1 :5 w (ixi
- x21 + It I
-
t21t;) ,
/orevery pairo/points (XI. tt), (X2, t2) E nT.lnparticu/ar, ifthe boundmydatum 9 is Holder continuous in ST with exponent say a g , and if the initial datum U o is Holder continuous in Ii with exponent say auo' then (x, t) -+ u(x, t) is Holder continuous in nT and there exist constants 'Y > 1 and a E (0, 1) such that
for every pair o/points (Xl, tl), (X2, t2) E liT, The constants 'Y and a depend only upon the data and the number a* 0/ (1.1) o/Chap.l. Moreover the constant a also depends upon the Holder exponents a g , auo 0/ 9 and U o respectively. I/the lower order terms b(x, t, u, Du) satisfy (A~) 0/§5 o/Chap.lI, then 'Y and a are independent o/llulloo.or .
1-(iii). Boundary regularity (variational data) THEOREM 1.3. Let u be a bounded weak solution 0/ the Neumann problem (2.7) o/Chap.1I and let (N) and (N - i) hold. Assume that the boundary an is 0/ class CI.~. Then u is Holder continuous in nT and there exist constants 'Y and a such that
for every pair o/points (XI. td, (X2, t2) E nT.
2. The main proposition 79
The constants 'Y > 1 antfo: only depend upon lIulloo.i'iT and the data, including the structure ofafl and the norms IItP1, tPf=r IIq.r;,aT appearing in the assumptions (N) - i. If the Neumann data are homogeneous. i.e., iftPo=tP1 =0, and ifin addition the lower order terms b(x, t, u, Du) satisfy (Aa) of§5 of Chap. II. then 'Y and 0: are independent ofllulloo.i'iT.
l-(iv). Some comments
The last two Theorems have been stated in a global way. The proof however uses only local arguments so that they could be stated within any compact portion, say /(, of fl. Accordingly, the hypotheses on the boundary data need only to hold within /(,. For example, in the case of Dirichlet data, the boundary datum 9 could be continuous or HOlder continuous only on a open portion of ST (open in the relative topology of ST), say E. Then the solution u of the Dirichlet problem would be continuous (respectively HOlder continuous) up to every compact subset of E. Analogous considerations can be made for Neumann data satisfying (N)-(N-i) on relatively open portions of ST. Similar remarks hold if U o is only locally continuous or locally HOlder continuous. In particular, to establish the continuity (lli)lder continuity respectively) of u up to fl x {O}, no reference is needed to any boundary data on ST. Finally we comment on the assumption that u be locally bounded. It will be shown in the next Chapter, that when p > 2, solutions of (1.1) are locally bounded. This is no longer true, in general, if 1
~
1 satisfying N(p - 2) + rp>O
and such a condition is sharp. Thus, unlike the degenerate case, when p is near one, the local boundedness is not implicit into the notion of weak solution and must be obtained by other information such as boundary data. We refer to Chap. V for a systematic study of local and global boundedness.
2. The main proposition The lli)lder continuity of u, either in the interior of flT or at the parabolic boundary, will be, heuristically, a consequence of the following fact. The function (x, t) -+ u( x, t) can be modified in a set of measure zero to yield a continuous representative out of the equivalence class uE Vi!;;(flT ), if for every (xo, to) E flT there exist a family of nested and shrinking cylinders [(xo, to) + Q (On' Pn)] with same vertex such that the essential oscillation Wn of u in [(xo, to) + Q (On, Pn)] tends to zero as
80
IV. HOlder continuity of solutions of singular parabolic equations
n - 00 in a way that can be quantitatively detennined by the structure conditions (AlHA 5). \ To begin the proof of Theorem 1.1 we introduce a space-time configuration that reflects the singularity exhibited by the p.d.e. Fix (xo, to) E T and construct the cylinder
n
[(X o, to)
+ Q (RP, R l - e)] C nT,
where e is a small positive number to be detennined later. After a translation one may assume that (xo, to) == (0, 0) and set 1'+
=Q(RP,RI-C) esssup u,
1'- =
ess inf u,
W=
Q(RP,Rl-<)
essosc u==J.L+ - 1'-.
Q(RP,Rl-<)
Consider the box
Q (RP, CoR) ,
(2.1)
where
Co
= (~)
E=! P
and where A is a constant to be detennined later only in tenns of the data. If we assume that (2.2) then we have and
essosc u
Q(RP,coR)
< w.
-
Cylinders of the type of (2.1) have the space variables stretched by a factor
(w / A) ~ , which is intrisically detennined by the solution. If p 2 these are the standard parabolic cylinders with the natural homogeneity of the space and time variables.
=
PROPOSITION 2.1. There exist constants eo," E (0,1) and C,A,A > 1, that can be determined a priori depending only upon the data, satisfying the following. Construct the sequences Ro = R, WO =w
Rn = c-nR, Wn+1
= max{1JWn;.A~t},
n=I,2, ... ,
and the boxes Q(n)
== Q (R", enR,,),
en = (:;)
Then for all n=O, 1,2, ...
A consequence of this proposition is
~,
n=O, 1,2, ....
3. Preliminaries 81 LEMMA 2.1. There exist constants 'Y> 1 and aE (0, 1) that can be determined a priori only in terms of the data, such that for all the cylinders
0< p
~
R,
essosc
Q(pP,cop)
U~'Y(w+JtEO)(RP)Q.
This is the analog of Lemma 3.1 of Chap. III. The proof is the same and it implies the HOlder continuity of u over compact subsets of {}T via a covering argument.
Remark 2.1. The proof of Proposition 2.1 will show that it would suffice to work with the number w and the cylinder Q (IV', CoR) linked by essosc u < w.
(2.3)
Q(RP,coR)
-
This fact. is in general not verifiable for a given box since its dimensions would have to be intrinsically dermed in terms of the essential oscillation of u within it. The reason for introducing the cylinder Q (RP, R1-E) and assuming (2.2) is that (2.3) holds true for the constructed box Q (RP, coR). It will be part of the proof of Proposition 2.1 to show that at each step the cylinders Q(n) and the essential oscillation of u within them satisfy the intrinsic geometry dictated by (2.3).
Remark 2.2. Such a geometry is not the only possible. For example, one could introduce a scaling with different parameters in the space and time variables. Examples of such mixed scalings will occur along the proof of Proposition 2.1. Here we mention that the proof could be structured by introducing the boxes Q (RP-E , 2R) and Q (aoIV', R) formally identical to those of §2 of Chap. III and rephrasing the Proposition 2.1 in terms of such a geometry.
3. Preliminaries Inside Q (RP, CoR) consider subcylinder: of smaller size constructed as follows. The number w being fixed, let So be the smallest positive integer such that
w
(3.1)
-280
where fJo is introduced in (3.11) of Chap. II. Then construct cylinders (3.2)
[(ii,O)
+ Q (RP, doR)1 ,
do
= (~) 2 80
E.::l P
82
IV. ltiider continuity of solutions of singular parabolic equations
.-
Figure 3.1 These are contained inside Q (RP , coR) if the number A is larger that 260 and if x ranges over the cube K'R.(w). where
(3.3) !=J!
= Lo (doR) • where
Lo
==
(:'0)" -
1.
One may view these as boxes moving inside Q (Rf', CoR) as the coordinates x of dleir vertices range over the cube K'R.(w)' The cylinders [(x. 0) + Q (Rf' , doR) I can also be viewed as the blocks of a partition of Q (Rf', coR). Indeed we may arrange that Lo be an integer and view the cube KeoR as the union, up to a set of measure zero, of L: disjoint cubes each congruent to KdoR. Analogously Q (Jl1', CoR) is the disjoint union, up to a set of measure zero of open boxes each congruent to Q (Jl1', doR). The proof of Theorem 1.1. is based on studying the following two cases. Let 110 be a small positive number. Then either
L:
the first alternative there exists a cylinder of the type of [(x, 0) + Q (Jl1', doR)]. making up the partilion of Q (Jl1' , coR), such that (3.4)
meas{ (x,t) E [(x,O) +Q(RP,doR)11 u(x,t) < #r + 2~0} < 1I0 IQ(RP,doR) I,
or
the second alternative for all cylinders [(x,O)
+ Q (RP, doR)] making up the partition of Q (Jl1', CoR),
3. Preliminaries 83
(3.5)
meas { (x, t) E [(x,O)
+ Q (R", doR)] 1u(x, t) < JJ- + 2~0 } ~
volQ (R", doR) I·
In either case the conclusion is that the oscillation of u in a smaller cylinder with vertex at the origin, decreases in a way that can be quantitatively measured. In the arguments to follow we assume (2.2) holds. Indeed if not, pE
eo
= (2 _ p)
and the fll'St iterative step of Proposition 2.1 would be trivial. Remark 3.1. Along the proof we will encounter quantities of the type
i=I,2, ... ,IEN, where Ai are constants that can be detenoined a priori only in tenos of the data and
(3.6)
bo
= 2 + N(p _ 2) (~ _
1:
K.) .
From the range of K. and q as defmed in (3.5)-(3.7) of Chap. II, one checks that bo ~ p. We may assume that (3.7) Indeed if not. we would have w $ A'wo for the choices
At.; l~i~1
A = max
NK. eo=-·
and
bo
I
Remark 3.2. The proof shows that the numbers e and eo can be taken as NK. eo=-, bo
e=
eo(2-p) p
.
3-(i). About the dependence on lIull oo ,I1T In the arguments below we will use the energy and logarithmic estimates ofPropositions 3.1 and 3.2 of Chap. II, for the truncated functions (u- k)±, over cylinders contained in Q (R", CoR). When working with (u - k)_ we will use the levels
_ k=JJ
w
+-2 8 +., I 0
for some i
~
0,
~
O.
and when working with (u - k) + we will take for some i
84
IV. Holder continuity of solutions of singular parabolic equations
These are admissible since
11(1.£ -
W
k)±lIoo,Q(RJ>,coR)
::5 2so +i ::5 Do·
Let us fix Do as in (3.11) of Chap. II. Then. since w::5 2l11.£ll oo ,nT' (3.1) holds true. within any subdomain of nT. if we choose So so large that (3.8)
280 =
4P+8 C 2 Co 1l1.£lloo,nT'
The a priori knowledge of the nonn 1I1.£lloo,nT is required through the number So. If the lower order tenns b(x, t, 1.£, D1.£) in (1.3) satisfy (A3) of§5 of Chap. II. then, as remarked there, the energy and logarithmic inequalities hold true for the truncated functions (1.£ - k)± with no restriction on the levels k. Thus in such a case. So can be taken to be one and no a priori knowledge of 1I1.£lloo,nT is needed. The numbers A and Ai introduced in (3.7) will be chosen to be larger than 280 • In the proof below we will choose them of the type
i=1,2, ... , where fi ~ 0 will be independent of 1I1.£lloo,nT' We have just remarked that if the lower order tenns b(x, t, 1.£, D1.£) satisfy (A3) of §5 of Chap. II. then So can be taken to be one. We conclude that for equations with such a structure the numbers Ai can be detennined a priori only in tenns of the data and independent of the nonn 1I1.£lloo,nT'
4. Rescaled iterations The following rescaled iteration technique applies to any subcylinder of fh and it is crucial in both alternatives. Let m > 0 be given by
and consider the cube Kd1R
== { I$;i$;N max IXil <
dIR} ,
and the box QR (mI' m2)
== Kd!R x
{_2
m2
(P-2) RP,
o}.
Fix (x, i) E nT, and let R> 0 be so small that
[(x, i) + QR (ml' m2)] C nT· Remark 4.1. If (x, i) == (0, 0) and 2m ! = A, m2 =0, then the cylinder [(x, i) + QR (mI' m2)] coincides with Q (RP, coR). Analogously. ifm2 =0, mi = So and l = 0, then. for a suitable choice of x the cylinder [(x, i) + QR (mIt m2)] coincides with one of the boxes making up the partition of Q (RP, CoR).
4. Rescaled iterations 85 LEMMA 4.1. There exists a number Vo that can be determined a priori only in terms o/the data and independent o/w, R and mt, m2 such that:
(I).I/u is a super-solution 0/(13) in [(x, l)
+ QR {m}, m2)] satisfying
essosc
[(f,l)H2R(mlom 2)]
u<w -
and
I
meas {(x, t) E [(x, l) + QR (mll m2)] u{x, t) < J.t-
+
2: }
:::; vo IQR{mllm2)
I,
then either or
u{x, t) ~ J.t
_
w
+ 2m+1 '
where bo is defined in (3.6) and Ao is a constant depending only upon the data and the numbers m}, m2. Analogously (ll).I/u is a sub-solution 0/(13) in
[(x, t) + QR (m}, m2)] satisfying
essosc
[(f,l)+QR(ml,m2)]
u <w -
and
I
meas { (x, t) E [(x, l) + QR (mb m2)] u(x, t) > J.t+ -
2: }
:::; volQR (mt, m2)
I,
then either or
u(x, t) :::; J.t+ -
2:::+
1'
We only prove the statement regarding super-solutions. Assume (x, t) == (0,0) and construct the decreasing sequences of numbers
PROOF:
kn
w
w
= J.t- + 2m+! + 2m+}+n'
and the families of nested cubes and cylinders
n=O, 1,2, ... ,
IV. mldcr continuity of solutions of singular parabolic equations
86
Consider (3.8) of Chap. n, written over the boxes Qn for (u choice of the cutoff functions Cn
kn) _ and with the
0 < Cn(X,t):5 1, V(x,t) E Qn, andCn:l in Qn+1i { Cn = 0 on the parabolic boundary of Qni ~ 0 < r < 2(2-p)m2 2,,("+2) . IDr I < 2,,+2 (...!L) m ..n -
R
2
, - ..n,t
l
-
R"
In this setting, (3.8) takes the fonn
sup
(4.1)
j(u -
_2(,,-2)m2 R"
kn)~ C:(x,t)dx
+j jlD (u - knL Cnl PdxdT Q"
:5 ..,:
(2:
1
f-Pjj(U - kn)~ dxdT Q"
+ ..,: 2(2- p)m2j j(u - kn)~ dxdT Q" ~
+.., { JIA;"
,dl R,. (T)I idT}
r
-2(,,-2)"'2 Rt
Since
sup(u _ k)
< ~-:w_ m m
n - -
Q"
2 l+ 2'
the first two tenns on the right hand side of (4.1) are estimated above by
..,: G:f2(2-p)m2j/X.[(u-knL >O]dxdT. Q"
To estimate below the two integrals on the left hand side, introduce the level 1 k n : 2 (kn + k n +1)
< kn.
Then for all tE (_2(p-2)m 2 Rl:, 0)
j(u -
kn)~ C:(x,t)dx ~
Kn
j(kn - kn )2-P (u -
kn)~ C:(x,t)dx
K"
~ G~a)2-P 2(p-2)(n+3)j(u - kn)~ C:(x,t)dx. K" Also we have
4. Rescaled iterations 87
!
!ID
(u -
knL (niP dxdr ~
Qn
!JID
(u - kn) _ (niPdxdr
Qn
- 'Y:
(2:)2 2(2-p)m !!X [(U 2
knL > 0] dxdr.
Qn
Put these estimates in (4.1), divide through by ( ; ) 2-1'
2(p-2)(n+3)
and in the resulting integrals introduce the change of variables y=
which maps
C:J
~ p
x,
Qn into
Setting also v(y,z)=u (d1y,2(p-2)m2t)
(y,z)=«d 1y,2(p-2)m2t) ,
I
and
o An(z)={y E KRnlv(y,z)
< kn }, IAnl=! IAn(z)ldz, -R~
we arrive at
II (v - knL (nll~p(Qn)
+
~ 'Y:;n
(;r IAnl
~ G~.r A.R'"K
W
-'{!
IA.(z) I~ dz }
~
•
By (3.7), AoRNICW-bo ~ I, and by Corollary 3.1 of Chap. I,
2-(n+3)p (.!:!-.)P 2m IAn+1 I
~ (kn - kn+t}P l(y,Z) E Qn+ll v(y,z) < kn+11 ~ II (v - kn )+ 1I:,Qn+l
~ II (v - kn )+
~ 'YIAnl~ II (v -
k n )+
"~ G~.r [:- 1A.I'+"r, + 1A.1"r,
U
1A.(z)I' dz }
"'P"] .
88
IV. HOlder continuity of solutions of singular parabolic equations
Thus setting
we have the recursive inequalities
Yn+l ~ -y4n " {y~+m:;
+ y~ Z!+K } ,
Zn+l ~ -y4n " {Yn + Z!+K}. It follows from these and Lemma 4.2 of Chap. I that Yn and Zn tend to zero as n -+ 00, provided
Yo
+ Z~+K
~ (2-y)-(l+K)/60 4-"(l+K)/6~
== vo,
where 80=min{~;K}.
Remark 4.2. The proof shows that the number Vo depends upon p but it is 'stable' asp/2, i.e., as p
--+
2.
Remark 4.3. The conclusion of Lemma 4.1 continues to hold for cylinders of the type
(4.2)
QR (m,{3)==Kr x (-{3R" , 0) ,
r=~:)
E=l P
{3 > 0,
R,
provided {3 is independent of w and R. In such a case we take m = depend also upon {3.
ml
and Vo will
5. The first alternative Suppose that there exists a cylinder of the type of [(x, 0) + Q (RJ', doR)] making up the partition of Q (R", CoR) for which (3.4) holds. Then we apply Lemma 4.1 with ml = 8 0 and m2 = 0 to conclude that (5.1)
u(x, t) ~ Il-
+ 28~+l '
'I(x, t) E [(x,O)
«
+ Q i)" ,dof)] .
We view the box [(x,O) + Q«i)" ,doi)] as a block inside Q (R", coR). Let R(w) be the 'radius' introduced in (3.3). The location of x within the cube K'R.(w) is only known qualitatively. We will show that the 'positivity' of (5.1) 'spreads' over the full cube KcoR, for all times
(f)" ~ t ~ In a precise way we will prove
o.
5. The flfSt alternative 89 PROPOSITION 5.1. Assume(5.1)holdsforsomeXEK'R.(w). There exists positive numbers Al and l1 that can be determined a priori only in terms of the data and the number A in the definition of Q (RP, coR), such that either
(5.2)
or
As a consequence we may rephrase the first alternative in the following fonn.
5.1. Assume that (3.4) holds for some cylinder of the type of [(x,D) + Q (W, doR)] making up the partition ofQ (RP, CoR). There exists positive numbers A1 and s that can be determined a priori only in terms of the data and the number A in the definition ofQ (RP, CoR), such that either COROLLARY
or essosc
Q(pP,cop)
where
U:::;"'1 w,
"'1 == 1 -
v P E (0, RI8), T(so+S).
We regard x as the centre of a large cube
which we may assume is contained in the cube KR1- •. Indeed if not, we would have - pc 16co > R- E, i.e., w < 16r-; AREo, £ 0 =--·
2-p
We will be working within the box
[(x,D)
+ Q ( iY, 8co R)]
and will show that the conclusion of Proposition 5.1 holds within the cylinder
[(x,0)+Q(ft,2c o R)] . This contains Q (
f )P, CoR), regardless of the location of x in the cube K'R.(w).
5-(i). The p.d.e. in dimensionless form Introduce the change of variables
x
--+
x-x 2co R'
90
IV. RUder continuity of solutions of singular parabolic equations
1- _ I I
I I
I I
I
Figure 5.1
which maps l(x,O) + Q (i)P,8co R)] into Q4 == K 4 x (-41',0). Also introduce the function (5.4)
Denoting again with x and t the new variables, the function v satisfies the p.d.e. (5.5)
Vt -
div i(x, t, v, Dv) + b(x, t, v, Dv)
= 0,
in 1Y(Q4),
where i: Q4xRN+1 -+ RN and b: Q4xRN+1 -+ R, satisfy the structure conditions (5.6)
i(x,t,v, Dv)·Dv 2:
(5.8)
Ib(x,t,v,Dv)1
~; (2;y-P IDvlP -!Po'
~ ~~ (~y-P (2~J IDvlP +ch.
Here Ci, i =0,1,2, are the constants appearing in the structure conditions (Al)(A3) of Chap. II. Moreover, setting
s. The flJ'St alternative (5.9)
9
91
-= CPo - + CPl-~ + CP2, -
the function 9 satisfies (5.10)
where '"Y = '"Y(N,p,A,8 0 ,data) is a constant depending only upon the indicated quantities and bo is defined in (3.6). The numbers" and q and q, f satisfy (3.5)-(3.7) of Chap. ll. The infonnation (5.1) translates into (5.11)
v(x, t)
>i
a.e. (x, t) E Q(ho ) ==
{Ixl < ho } x {-41' < t :s; O},
where (5.12)
doR
ho = 8coR
1
(28 )¥ < 1. 0
=8 A
We regard Q( ho ) as a thin cylinder sitting at the •centre' of Q4. We will prove that the relative largeness of v in Q(ho ). spreads sidewise(l) over Q2.
4
(0,0)
-------,
2ho Figure 5.2
Proposition 5.1 will be a consequence of the following fact. LEMMA 5.1. For every II E (0,1) there exists positive numbers A* > 1 and 6* E (0,1) that can be determined a priori only in terms 0/11, N,p and the data, such that either (5.13)
or (1) For further comments on this phenomenon we refer to §l4-{i).
92
IV. H6lder continuity of solutions of singular parabolic equations
(5.14) for all time levels t E [-2 P , OJ.
Remark 5.1. The key feature of the lemma is that the set where v is small can be made arbitrarily small for every time level in [-2P , OJ.
5-0i). Proof of Proposition 5.1 assuming Lemma 5.1 In Lemma 5.1 we choose II = " o where " o is the number claimed by Lemma 4.1, and determine 6· =6· (11o ) accordingly. We let m2 be dermed by 2- m2 = 6·(IIo } and apply Lemma 4.1 with w = 1, J1. - = 0, R = 2, over the boxes (0, t)
+ K2 X {
_2 m2 (p-2)2 P ,
O} =(0, t) + Q2(O, m2}
as long as they are contained in Q2, i.e., for i satisfying (5.15)
Since (5.14) holds true for all time levels -2 P 5 t 50, each such box satisfies
meas {(x, t) E [(0, t)
+ Q2(0, m2)J
\ v(x, t) 52-m2} 5110\Q2(O, m2)\.
Therefore by Lemma 4.1 either (5.2) holds or v(x, t) ;;:: T(m 2 +1)
V(x, t) E Ql.
Returning to the original coordinates and redefining the various constants accordingly proves Proposition 5.1.
6. Proof of Lemma 5.1. Integral inequalities First we prove the lemma under the additional assumptions (6.1)
These will simplify some of the calculations and will be removed later. The weak formulation of (5.5) is (6.2)
J
Vt'p(x, t}dx +
K4
J
sex, t, v, Dv)·Dcpdx = -
K4
J
b(x, t, v, Dv}cpdx
K4
for all -4P < t < 0 and all testing functions cp E C(Q4)nC (-4P ,0; W:,P(K4)).
6. Proof of Lemma 5.1. Integral inequalities 93 Let
!
t
(6.3)
G(t)
== (6k)-(1+.p)
IIg(r)
11:14 dr,
-4P
where k and fJ are positive parameters to be chosen later, 9 is defined in (5.9) and q is the number entering in the structure conditions (5.10). We define the new unknow function (6.4)
w
== v + G(t),
and rewrite (6.2) in tenns of w. Next, by the parabolic structure(l) of (6.2), the truncation (k - w)+ is a subsolution of (6.2), i.e., for all testing functions r.p ~ 0 (6.5)
!!
!a
(k - w)+ r.p(x, t)dx +
K4
~
-!b
(x, t, v, D (k - w)+) ·Dr.pdx
K4
(x, t, v, D (k - w)+) r.pdx - G'(t)! r.p(x, t)dx.
K4
K4
In this fonnulation we take the testing function (P
r.p==
+ fJ k]P
[k - (k - w)+
I'
where (== (1 (X)(2 (t) is a piecewise smooth cutoff function in Q4, satisfying 0~( ~ 1 (6.6)
{
(=0
in Q4,
and (
== 1
in Q2;
on the parabolic boundary of Q4;
ID(11 ~ 1,
0 ~ (2,t ~ 1; (1 (x) >
the sets {x E K4
I
k}
are convex Vk E (0,1).
We use the structure conditions (5.6)-(5.8), with the symbolism _. = Ci
C. -
24p
(280A )2-P.
Set also
_!
(k-w)+
(6.7)
(6.8)
~ (w) k
-
o
[ !lik(w) = In k(l
(1) See §l-(i) of Chap. II.
ds
[k _ s + 6kjP-1' k(1
+ 6) -
+ fJ)
(k - w)+
] .
94
IV. Hl)lder continuity of solutions of singular parabolic equations
Then we obtain
!
j
~1c(w)(1'dx + Co jIDtP1c (W)jP(1'dx
K4
K4
~ C1 j
(IDtP1c (w)I()1'-l ID(ldx
K4
+ C2 22 (1'+1) (2~J
jIDtP1c (WW(1'dx K4
+p j(~1c(W»(P-I(tdx K4
By the choice (3.8) of the number 8 0 , the second term involving IDtP1c(WW is absorbed in the analogous term of the left hand side. The integral involving IDtP1c(WW-1 is treated by means of Young's inequality and the resulting term involving 1D!P1c(W)\P is absorbed in the analogous term on the left hand side. The remaining term is majorised by an absolute constant depending only upon Ci , i = 0, 1. Next, if we stipulate to take k in the interval (0, 1J, the integral involving (t is majorised by 'Y/(2 - p), where 'Y is an absolute constant depending only upOn p. Finally the sum of the last two integrals can be majorised by an absolute constant. Indeed
We conclude that there exist constants .:yo and .:y depending only upon N, p, A, 8 0 and the data, such that
7. An auxiliary proposition 95 (6.9)
!
~1c(w)(Pdx +;Yo jIDtP1c (w)I P(Pdx ~ 2 ~ p.
j K4
K4
i)
Next, since k E (0, the function tP1c(W) vanishes for alllxl :5 ho • This follows from the definition (6.4) of wand (5.11). Therefore we may apply the Poincare inequality (2.9) of Proposition 2.1 of Chap. I, to minorise the second tenn on the left hand side of (6.9). We summarise: LEMMA 6.1. There exists two constants 'Yo and 'Y that can be determined a priori only in terms of N,p, A, SO such that
!
(6.10)
~1c(w)(Pdx + 'Yo j
j K4
tP:(w) (pdx
~ 'Y,
K.
where ~1c(W) and tP1c(W) are defined in (6.7)-(6.8).
Remark 6.1. The function G(·) introduced in (6.3) is defined through the numbers k and 6 which are still to be chosen. By virtue of the structure conditions (5.10), we have
G(t)
~ 'Y(6k)-(1+r-r)
IIgll:;Q4
~ 'Y(6k)-(1+r-r) [RNICw-bo]o!r. If we choose k6 = 6* E (0, 1) depending only upon the data, we may assume without loss of generality that (6.11)
G(t) :5 'Y (6k)-(1+r-r) [RNIC w- bo ]
r-r ~ 6*2.
Indeed, otherwise for such a selection of 6*
for some positive number (J depending only upon q and p and some 'Y depending only upon the data. The number 6* will be chosen Shortly only in tenns of the data. In view of (6.11) we may regard the function w introduced in (6.4) as independent of k and
6.
7. An auxiliary proposition Introduce the quantities (7.1)
Yn
==
sup -4P
j
(P (x, t) dx,
n=O, 1,2, ....
K4n[W( ·,t)<6")
The proof of Lemma 5.1 is a consequence of the following:
96
IV. Hiilder continuity of solutions of singular parabolic equations
PROPOSITION 7.1. The number II E (0,1) being fixed, we may find numbers 6, uE (0,1) depending only upon N,p the data and II, such that/orn=O, 1,2, ... ,
either (7.2)
or (7.3) PROOF OF LEMMA 5.1: Iterating (7.2)-(7.3) gives
n=1,2, .... Since Yo::; IK41, we have only to take n=no so large that uno-I::;
112- N .
Then the lemma follows with 26· = 6no • Indeed, Yno ::; uno -1 Yo implies
I
meas {x E K2 w(x, t)
< 26·} < IIIK21.
Recalling the definition (6.4) of wand the upper bound (6.11), this in tum yields (5.14) and concludes the proof of the lemma.
7-(i). Proof of Proposition 7.1 In (6.10) we take k = 6n , n E N, where 6 E (0,1) is to be chosen. From the definition (7.1) of Yn , it follows that for every e E (0, 1) there exists to E (-4",0), such that (7.4)
/ (" (x, to) dx K4n[W(·,to)<6
n
~
Yn +1
-
e.
+1 ]
The numbers n E N and to E ( -4",0) being fixed, we consider the following two cases: either (7.5)
! / ("
(x, to) 4>6n (w (x, to» dx
~°
K.
or
(7.6)
! / ("
(x, to) 4>6n (w (x, to» dx < 0.
K.
In either case we may assume that Yn > II, otherwise the proposition becomes trivial. Also, in (7.4) we may take e arbitrarily small within the range (0,11/2).
8. Proof of Proposition 7.1 when (7.6) holds
97
7-(ii). The case (7.5) If (7.5) holds, it follows from (6.10) with k = 6n , that (7.7)
We minorise this integral by extending the integration over the smaller set
On such a set
Therefore (In 1
~ 6) I'
J
(I' (x, to) dx
J
~
K.n[w(·,t o )<6 n +1]
(P(x, to)lfIln (w (x, to)) dx.
K.
From this, (7.7) and (7.4) Yn+l ~c+C
+6)-1' .
1 ( In U
To prove the proposition in such a case, we choose 6 so small that
1+6)-"
C ( In-U
1/
~2'
Such a choice depends only upon the constants "Y, "Yo and 1/ and therefore it depends only upon the data.
8. Proof of Proposition 7.1 when (7.6) holds If (7.6) holds true, define
1(.
t. '" sup {t E (-4', t.)1 ~
(x,
t)4'•• (w (x, t»
By the definition oft., (8.1)
J
(P
(x, to) ~6n (w (x, to)) dx
K.
By the arguments of the first alternative
~
J
(I' (x, t.) ~6n (w (x, t.)) dx.
K.
98
IV. mlder continuity of solutions of singular parabolic equations
J
("(x, t.)!liln (w (x, t.» dx :5 C
K4
and 'risE [0,1].
[In +; _
J
(" (x, t.) dx :5 C
1
1
6 s ]-" '
K4n[(6 n -wl+>s6 n ]
where C is an absolute constant depending only upon N and p and the data. By the definition (7.1) of Yn • we have for all s E [0, I]
J
("(x,t.) dx:5 min { Yn ; C
(8.2)
[In 1 ~;~ s]-"}
K4n[(6 -wl+>s6 n ] n
={
if 0 :5 s < s.
Yn
C
[In 1+1-.] 1 6
-"
if s. :5 s < I,
where s. is the root of the equation [ Yn=CIn
1 6+ 6
1+
-
s.
]-" .
Solving it we find e(c/Ynl1/P - 1
(8.3)
s.
=
e(c/Ynl1/p
(1
+ 6).
Since Yn >11, we have
s. <
(8.4)
e(c/lIl l / P - 1 e(CM1/p (1
+ 6) == 0"0(1 + 6).
Next we estimate the integral on the right hand side of (8.1). By the Fubini theorem
J("(X,t.)~6n K4
(w(x,t.» dx
8. Proof of Proposition 7.1 when (7.6) holds 99
Therefore (8.1) yields / (P (x, to) ~,s" (w (x, to» dx
(8.5)
K"
~
n
p
)P_l ( [1 + 15 - s]
/1
c5 (2-
o
/ (P(X,t .. )
dx) ds.
K"n[(,s,,-w)+>s,s,,)
The last integral on the right hand side of (8.5) is estimated by means of (8.2). Taking into account the definition of s .. in (8.3), we have
/
(P(x,to)~,s" (w(x,to» dx
K"
Bo
o
c5n(2-p)
[1 + 15 _ sjP
1
y. ds n
1
c5n(2-p)
+/
1 + 15 ] -P
[
[1 + 15 - sjP-l
Gin
1 + 15 - s
ds
So
1
=
/
o
-
c5n(2-p)
[1 + 15 _ sjP-l
y. ds n
+ 15 ] -P} ds /1{y'-Gln[1l+c5-s [1+c5-sjP-l c5n(2-p)
n
Bo
where
F(Yn , c5)
=/
1
o
ds
[1 + 15 - sjP
1
_/1 (1 _£Y [In 1 +1+15 -15 s ] -P) [1 + 15ds- sjP-l . n
So
100 IV. Holder continuity of solutions of singular parabolic equations From Y n ~ v and (8.4) we have 1
F(Y. b) < / n,
-
o
[1
+ bds_ sjP-1
_ /1 (1 _Cv [In l+b-s 1+ b ]-P) [1+b-sjP-1 ds . 0"0(1+6)
These estimates in (8.5) give 1-6
(8.6)
/
(P
(x, to) cJ6n
(w (x, to» dx ~ Yn [1 - /(b)] /
K.
[1
0
bn(2-p)
+ b - sJ
p-1 ds,
where 1-6
(8.7)
/(6) / [1 o
ds
+6_
sjP-1 -
/1 (1 _Cv [In 1 +1+b -6s ]-P) [1 + bds-
_
/1
S]P-1
0"0(1+6)
ds
[1
+b-
S]P-1 •
1-6
Estimating below the left hand side of (8.6) we find
This and (8.6) yield (8.8)
Yn +1
-
e ~ Yn (1 - /(b».
We estimate /(b) below. For this let 0"1 ~ 0"0 be defined by
(see also (8.4) ) . Then integrating the first integral on the right hand side of (8.7) over the smaller interval [0"1(1 + b), 1]. we derive the estimate 1 - 0"t}2- p /(6) > -(1 2
-
(
-
2b
1+6
)2- P .
9. Removing the assumption (6.1) 101
We choose 6 so small that
1 /(6) > -(1 4
2 0'1) -P
and set 0'
= 1-
4"1 (1 - O'I) 2 -p.
Since e E (0, v /2) is arbitrary, we obtain from (8.8)
This proves the Proposition if (6.1) holds.
9. Removing the assumption (6.1) Inequality (6.10) holds in any case in the integrated form
f
(P~kdx -
K.(t)
(P~kdx + 'Yo
f
K.(t-h)
for all
t E [-4 P
We divide by h and let h t-derivative is replaced by
(d~) - f
t
t-hK4
+ h, OJ, -+
f f (Pwf(v)dx $ 'Yh,
h
> O.
0 to obtain (6.10) where the term involving the
(P(x, t)~k(W) dx
K,
==li~~~PX { f(P~k(W)dx - J(P~k(W)dx}dx. K,(t)
K,(t-h)
Define the set
S
== {t E (-4 ,0)1 (d~) - f P
(P~k(w)dx ~ o},
K.(t)
and let to be defined as in (7.4). If to E S, we have
f (Pwf(v)dx $'Y.
(9.1)
K,(to)
If to
¢ S but sup {t
< to I t E S} = to,
102 IV. ~lder continuity of solutions of singular parabolic equations
by working with a sequence of time levels tn E Sand {tn}-to, we see that (9.1) continues to hold. If to , S and T
== sup {t < to I t
E
S} < to,
we derive the two inequalities
f f
(Pl]tf( w)dx
:5 'Y,
K,,(T)
f (P~k(w)dx.
{P~k(w)dx:5
K,,(t o )
K4(T)
The remainder of the proof remains the same.
10. The second alternative We assume here that (3.5) holds true for all cylinders [(x, 0) + Q (RP, doR)] making up the partition of Q (RP, CoR). Since 8 0 ~ 1 we have
+ ,.,.
w -
280
~,.,.-
w
+2
80 '
so that we may rephrase (3.5) as (10.1)
\(X,t) E [(x,O) +Q(RP,doR)]
I u(x,t) >,.,.+ - 2~o \ < (l-lIo)IQ(RP,doR) I,
for all boxes [(x,O) + Q (RP,doR)] making up the partition ofQ (RP, CoR). Let n be a positive number to be selected and arrange that 2n ~ is an integer. Then we combine 2~N P of these cylinders to form boxes congruent to
coR Figure 10.1
10. The second alternative 103
(10.2)
Q (R", d.R)
== Kd.R x (-R", 0) d. = ( 2sow+n )~ = do (2n) !.=.I! p •
The cylinders obtained this way are contained in Q (RP , CoR), if the abscissa their 'vertices' ranges over the cube K'R.l(W)' where
'R. 1 (w) =
{A
=
{(
2(Bo+n)~} w~
2 -" -
x of
R
A)~ -1 }(w- )~ R 2 o+n
--
2so+n
B
where Ll
== (~) 2 o+n
!.=.I! p
B
-
1.
We will take A larger than 2B o+n and arrange that Ll is an integer. Then we regard Q (R", CoR) as the union, up to a set of measure zero, of Lf pairwise disjoint boxes each congruent to Q (RP, d. R). Each of the cylinders [(x, 0) + Q (RP, d. R)] is the pairwise disjoint union of boxes [(x, 0) + Q (R", doR)] satisfying (10.1). Therefore we rephrase (3.5) as
(10.3)
I(X, t) E [(x, 0) + Q (R", d.R)] I u(x, t) > p.+ - 2~o I < (1 - vo)IQ (R", d.R) I,
for all boxes [(x, 0)
+ Q (RP,d.R)] making up the partition ofQ (R",coR).
LEMMA 10.1. Let [(x, 0) + Q (R", d.R)] be any box contained in Q (RP, CoR) and satisfying (10.3). There exists a time level
t· E
such that for all S (10.4)
~ So
(-R" , - 2 R") ' Vo
+ 1,
Ix E [x + Kd.R] I u(x, t·) > p.+ - ;.1 < (II_-v:/2 ) IKd.RI·
PROOF: If (10.4) is violated for all tE (-R", -~R"). then
I(X, t) E [(x, 0) + Q (R", d.R)] I u(x, t) > p.+ - 2~o I ~
--1'RP
fix
E
[x + Kd.R]
I u(x, t) > p.+ -
;.Idt
-RP ~
(1 - vo)IQ (R", d.R)
I,
contradicting (10.3). The next lemma asserts that a property similar to (10.4) continues to hold for all time levels from t· up to O. The proof of the lemma will also detennine the numbern.
104 IV. HOlder continuity of solutions of singular parabolic equations LEMMA
10.2. There exists a positive integer n such that/or all t· < t <0.
(10.5)
Ix E [x + KdoRll 1£(x, t) > p.+ -
2B~+n I< (1 - (~ r)iKdoRI.
PROOF: Modulo a translation we may assume that x== O. Consider the logarithmic estimates (3.14) of Chap. II, written over the cylinder K do R X (t· , 0), for the function (1£ - k) + and for the levels W
k=p.+ - - .
280 As for the number c in the definition (3.12) of the function IjI we take w c=-28 0+n '
where n is a positive number to be chosen. Thus we take (10.6)
where Ht ==
esssup (1£- (p.+ K"oRX(tO,O)
~))
2
0
+
.
1be cutoff function x -+ (( x) is taken so that
{
( == 1, on the cube K(l-u)doR,
0'
E (0,1),
ID(I:5 (O'd.R)-l .
With these choices, the inequalities (3.14) of Chap. II yield for all t· < t < 0,
o (10.7)
j 1j12(x, t)dx :5 j 1j12(x, t*)dx + (ud:R)P j j K"oR
KCI-")"oR
1jI1jI~-PdxdT
tOK"oR
To estimate the various tenns in (10.7) we first observe that IjI :5 n
)P-2 ' [ 1 + InHt 'W )-1] :51'n In 2. In 2, 1jI~-P:5 '\2BW o+ n \Fo+R
We estimate the first integral on the right hand side of (10.7). For this observe that IjI vanishes on the set [1£ < p.+ - 2":0]. Therefore using Lemma 10.1, jIjl2(x,t*)dX:5 n 2 ln2 2 K"oR
(/--II:i2)
IKdoRI·
10. The second alternative 105
For the second integral we have
$
1'n uP
IKdoRI·
This estimation justifies the choice of the cylinders [(x, 0) + Q (RP, d.R)) over the boxes [(x, 0) + Q (RP, doR)). Indeed the integrand grows like 2n (2-p) due to the singularity of the equation. This is balanced by taking a parabolic geometry !!.1!::.£l
where the space dimensions are stretched by a factor 2 P • Finally the last tenn on the right hand side of (10.7) is estimated above by
where A2 = n2(so+n)bo and bo is defined in (3.6). Combining these remarks in (10.7) and taking into account (3.7), we obtain for all t· < t < 0, (10.8)
f
\li2 (x, t) dx $ n 2 102 2
C
I_-v: i2 ) IKdoRI
+ :: IKdoRI·
K(l-l7)d o R
The left hand side of (10.8) is estimated below by integrating over the smaller set
I
{ x E K(l-u)doR u(x, t) > J.I.+ On such a set, since \li is a decreasing function of
2s~+n } .
H:, we estimate
We carry this in (10.8) and divide through by (n - 1)2 102 2, to obtain for all t· < t
Ix E K(l-u)doR I 1.£(x, t) > J.I.+ $ On the other hand
2s~+n
(n:lr (11_-
I
v: i2 ) IKdoRI+ n:pIKdoRI·
106 IV. mlder continuity of solutions of singular parabolic equations
Ix EKdoR
1 u(x, t)
> J.L+ - ~+ 280 n
I
:5 Ix E K(l-CT)doR I u(x, t) > J.L+ -
2s~+n 1+ IKdoR\K(l-CT)doRI
~ Ix E K(l-CT)doR I u(x,t) > J.L+ - 28~+nl + NuIKdoRI· Therefore for all
t· < t < 0,
Ix EK(l-CT)doR 1 u(x, t) > J.L+ -
~ [(n:
lr (/--V:/2)
28~+n I +
n;p +
To prove the lemma we choose u so small that uN
Nu]IKdORI.
$1 v~ and then n so large that
Remark 10.1. Since the number 110 is independent of w and R also n is independent of these parameters.
11. The second alternative concluded The information of Lemma 10.2 will be exploited to show that in a small cylinder about (0,0), the solution u is strictly bounded above by
J.L
+
w
- 2m
'
for some m
> So + n.
In this process we also determine the number A introduced in §2 which defines the size of Q (RP, coR). To make this quantitative let us consider the box
Q ({3RP, CoR) ,
{3 -
110
- 2'
We regard Q «(3RP, CoR) as partitioned into sub-boxes [(x,O) + Q «(3RP, d.R)) where takes finitely many points within the cube KR1(w) introduced at the beginning of §10. For each of these subcylinders Lemma 10.2 holds.
x
LEMMA 11.1. For every liE (0, 1) there exist a number m dependent only upon the data and independent of w and R such that for all cylinders [(x,O) + Q «(3RP, d.R)] making up the partition ofQ «(3RP, CoR),
meas { (x, t)
E
I
[(x,O) + Q (,8RP, d.R)] u(x, t) > J.L+ -
2: }
< IIIQ({3RP,d.. R) I·
11. The se<:ond alternative concluded 107 PROOF: After a translation we may assume that (x, 0) == (0, 0). Set 81 =80 + n, and consider the energy inequality (3.8) of Chap. n written for (u - k)+, where 8
= 81,81 + 1,81 + 2, ... , m -
1,
over the cylinder Q ({J(2R)1', 2d.R). Over such a box
(u(x, t) - (JL+ - ;,)) + :$;'
a.e. (x, t) E Q ({J(2R)1', 2d.R) .
The cutoff function (x, t) - (x, t) is taken to satisfy
(== 1, { ( = 0,
on Q ({JRP, d.R), on the parabolic boundary of Q({J(2R)1', 2d. R) ,
ID(I :$ d.kp,
0:$
Ct
:$ IIJ~P.
We put these estimates in (3.8) of Chap. n and discard the first non-negative term on the left hand side. This gives (11.1)
!!ID· (u -
11'
(JL+ - ;.)) + dxdr:$
(d.~)1' (;. )"IQ ({JR1',d.R)1
Q(fJRP ,d. R)
+;
(;,y-1' G:tlQ ({JR1', d.R)! + 1'(d.R)N
P
(lt
C )
R1' (1:,,) . P
We estimate above the various terms on the right hand side of (11.1) as follows. Since 8 ~ 81 == 80 + n, ( W
28
)2-1' < (~)2-1' = ..!... 2 + cf. ao n
-
Therefore the sum of the first two terms is majorised by
(d.~)1'
(;,r
IQ({JR1',d.R)
I,
for a constant l' dependent only upon the data. The last term is majorised by making use of (3.6) of Chap. II and the definition (10.2) of d•. This gives l' (d.R)N P (1:,,)
R1' (1:,,) :$ ~pd~(~-1) RNICIQ ({JR", d.R) I P
:$
(d.~)1'A3W-boRNICIQ({JR1',d.R) I,
where A3 = 2mbo and bo is the number introduced in (3.6). Combining these remarks in (11.1) we deduce that there exists a constant l' depending only upon the data and independent of w and R, such that (11.2)
!!ID (u Q(fJRP,d.R)
(JL+ - ;.))
+1"dxdr:$ (d.~)1' G:tl Q ({JRP,d.R)I,
lOS IV. Hi>lder continuity of solutions of singular parabolic equations
for all 8 =81,81 + 1, 81 + 2, ... ,m -1. Next we apply Lemma 2.2 of Chap. lover the cube K d• R for the functions
v = u(·, t), and the levels l- + - P.
W 2.+1'
-
By virtue of Lemma 10.2
![u(.,t) < p.+ - ;']nKd•R! ~ (~YIKd.RI,
Vt E (-fJRP,O).
To simplify the symbolism we set
o
I
A.(t) == {x E Kd.R u(x, t) > p.+ - ;.} ,
A.s
=
jIA.s(T)ldT.
-fjRP
Then, with these specifications, (2.7) of Chap. I yields
(;')IA.s+1(t)1 ~~ d.R
jIDu(x,t)ldx,
VtE (-fJRP,O).
A.(t}\A.+1 (t)
First integrate both sides in dT over (-fJJlP, 0), then take the p-power and majorise the right hand side by making use of the IJl)lder inequality and (11.2). We obtain
(;. r
IA.s+1I P
~
j IDU1PdxdT)
IA.\A.+1I P - 1
-fjRPA.(T}
~ 'Y (;. From this,
J
'Y (d.R)P (
r IQ
(fJRP,d.R) IIAs \A.+1I P -
IA.+1I~ ~ 'YIQ(fJRP,d.R) l;!t IA.\A.+1I·
Adding these inequalities for 8=81, 81 +1,81 +2, ... , m - I,
To prove the lemma we have only to choose m so large that
( _'Y m- 8 1
)~ -
1
.
12. Proof of tile main proposition 109
Remark 11.1. This estimate deteriorates as p '\. l i.e., m /
00 as p
'\.1. However
the choice of m is 'stable' as p /2. To proceed we return to the box Q (PR!', CoR) and recall that it is the finite union, up to a set of measure zero, of mutually disjoint boxes [(x, 0) + Q (PR!', d.. R)]. Therefore Lemma 11.1 implies
11.1. For every vE (0, 1) there exist a number m dependent only upon the data and independent of w and R such that COROLLARY
(11.3)
meas
{(X, t) E Q (PR'P, CoR) I u(x, t) > J.I.+ - ; . } < vlQ (PR'" CoR) I.
We finally determine the size of the cylinder Q (PR!', CoR) as follows. First in Corollary 11.1 select v = Vo and determine m accordingly. Then let m2 be given by
P --
(11.4)
Vo _
2 -
2m2 ('P- 2 )
,
and assume, by taking m even larger if necessary, that m A from
~
m2. Then determine
(11.5) With these choices the box Q (PR!', CoR) coincides with the cylinder QR (ml, m2) introduced in §4. By Lemma 4.1, there exists a constant Ao dependent only upon the data and independent of w and R such that either
Ao RNK.,
wbo $
or u(x , t)
w
< ,,+ - - r2m +1
where bo is the number introduced in (3.6). We summarise: PROPOSITION 11.1. Suppose that (3.5) holdsforallcylinders [(x, 0) +Q (R!', doR)] making up the partition ofQ (R'P, CoR). There exists a constant Ao dependent only upon the data and indepetUknt of w and R such that either
(11.6)
orforaliO
essosc u $
Q(fjpP ,cop)
"'0 w,
where ~o
== 1 -
2-(m+l).
12. Proof of the main proposition The main Proposition 2.1 now follows by combining the two alternatives. Set
A == max{Ao j AI},
." == min{."o j "'l},
110 IV. ftilder continuity of solutions of singular parabolic equations
and let 0 be dermed by
.!. = (.!) l/p = (~) l/p 0-
4P
22p + 1 '
Then setting Rl == RIO both alternatives can be combined into the following statement: either w"o < ARNie
or
e880SC Q(pI',cop)
u:5 'IW.
The process can now be repeated and continued as indicated in Proposition 2.1. Indeed, by Remark 2.1, the process can be continued as long as (2.3) holds.
12-(i). Stability for p near 2 The proof of the first alternative is based on the integral inequalities (6.9) and (6.10). Because of the right hand side of (6.9), Proposition 5.1 holds with constants that deteriorate as p / 2. We briefly indicate how to prove Proposition 5.1 with constants that are 'stable' as p / 2. First, using the information (5.11 )-(5.12) we can show that there exists a positive number I. such that
meas {(x, t)
E
Q2
I v(x, t) < 2-1.} :5 lIo1Q21·
This is accomplished by the same technique as for Lemma 12.1. This technique involves constants of the type 21.(2-p). We may select p. so close to 2 that 21.(2-p) :5 2,
'tip. :5 p:5 2.
With such a choice the method can be carried out as if the p.d.e. was not singular. Next, an application of Lemma 4.1 implies that v(x, t) ~
2-(1.+1)
a.e. (x, t)
E
Ql.
The application of Lemma 4.1 over the box Q2 involves again terms of the type 21.(2-p). As remarked before, these are majorised by an absolute constant for pE
IP.,2).
13. Boundary regularity The proof of Theorems 1.2 and 1.3 regarding the regularity up to the lateral boundary of aT, is similar to the proof of the interior regularity. The few changes needed can be modelled after similar modifications presented in §11-13 of Chap. III for the degenerate case p > 2. However the proof of regularity up to t some differences.
= 0 exhibits
13. Boundary regularity III
13-(i). Regularity up to
t=O
Assume that U o is continuous with modulus of continuity say wo (·). Fix (x o , 0) E fl x {O} and R > 0 so that [xo + K 2R ] C fl. After a translation we may assume Xo = 0 and construct the cylinder
Qo (R", 2R) ==
K2R
x {O, R"}.
Set 1'+ = esssup u,
1'- =
Qo(RP,2R)
w = essosc u.
essinf u,
Qo(RP,2R)
Qo(RP,2R)
Let 8 0 be the smallest positive integer satisfying (3.1) and construct the box
Qo (dR", R) ==
KR X {O, dR"},
( W)2-"
d- -2m
'
where the number m > 1 is to be chosen. Notice that for all R> 0, these boxes are lying on the bottom of flT. Also withous loss of generality we may assume that 2~ ~ 1 so that there holds and
osc
Qo(dRP,R)
u < w. -
PROPOSITION 13.1. There exist constants '1, eo E (0, 1) and e, m > 1 that can be determined a priori depending only upon the data satisfying the following. Con-
struct the sequences Ro = R, Wo =W and 1
Rn = en R, wn+1
= max {'1Wni e R:t },
n=I,2, ... ,
and the family of boxes n=0,1,2, ....
Thenfor all n=O, 1,2, ... and·
essoscu~max{wni2essOSCUo}. Q~n) KRn
We indicate how to prove the fIrst iterative step of the Proposition and show, in the process, how to determine the number m. Set
and consider the two inequalities (13.1)
+
w
I' - 280
< 1'0+ ,
_ I'
w
_
+ 2 > 1'0 , 80
If both hold, subtracting the second from the fIrst gives
So ~
2.
112 IV. HOlder continuity of solutions of singular parabolic equations
and there is nothing to prove. Let us assume for example that the second of (13.1) is violated. Then for all 8 ~ 8 0 , the levels
k = IJ-
w
+ 28 '
satisfy the second of (4.16) of Chap. II. Therefore we may derive energy and logarithmic estimates for the truncated functions (u - k)_. These take the form (13.2)
/(U-k)~(x,t)(Pdx+ ffID(u-k)_CIPdxdT
sup
11
O
:51'
Qo(dRP,R)
/fiu-k)"-'DC'PdxdT+'Y{iiBk.R(T)'~dT}
Qo(dRP,R)
(13.3)
sup
~
r
,
0
/!li2(Dk",(U-k)_,c)C P(X)dx
O
:5 l' / / !lil!liu (D;;, (u - k)_, c) 12- PIDCI PdxdT Qo(dRP,R)
+; where
(1+ ~') ~B"R(T)11 dT f";"') , In
{
Dt and Bt,R are defined as in (4.2) and (4.4) of Chap. II.
LEMMA 13.1. For every vE (0, I), there exists a numberm>80~ 2 depending only upon the data and independent of w and R such that either
(b o defined in (3.6»
(13.4)
or
where
R
p=2
and
W d= ( 2m
)2-P •
PROOF: Consider inequalities (13.3) written for k = IJ- 2":0. As a constant c appearing in the definition of !Ii (see (3.12) of Chap. II), we tak: c = 2~. Thus we take
13. Boundary regularity 113
where
D; == lI(u -
k)-lIoo,Qo(dRP,R)·
The cutoff function ( is taken to satisfy on K p , vanishes for
Ixl =
R,
By considerations analogous to those developed in §12 we have the estimates W )1'-2
2 - 1' < ( 1tJi.1 u 2m
tJi 5 (m-s o }ln2, and
'Y
ff tJiltJi I
u 2- 1' ID(I 1' dxdr
'
:5 'YmIKpl·
Qo(dRP,R)
Moreover the last tenn on the right hand side of (13.3) is estimated above by
CW )-2 CW \2ffl)(2-1')~ . RNItIKpl:5 -ym2mbow-bo RNItIKpl,
'Ym\2ffl
r
where bo is the number introduced in (3.6). Combining these estimates in (13.3), we deduce that if (13.4) is violated, then
f
(13.6)
tJi2 (D;, (u - k)_,c) dx 5 'YmIKpl,
'
Kp(t)
We minorise the left hand side of (13.6) by integrating over the smaller set
I
{ x E Kp u(x, t) < On such a set
~- +
2: },
'
w
tJi2 ~ In2 2:
0
= (m -
So -
1)21n2 2.
2"'"fT
These remarks in (13.6) give
I{x E Kp I u(x, t) < ~- + 2:}1:5 (m _:~_I)2I K pl,
'
for a constant 'Y depending only upon the data. To prove the lemma we have only m so large that
to choose
(m -
So -
1)2
< v.
114 IV. HOlder continuity of solutions of singular parabolic equations
Remark 13.1. The process described has a double meaning. It defines a level +;' for the function u and the size of the box
p. -
d-
(-2W)2-" m
'
within which the set where u < p. - +;' is small. To conclude the proof of Proposition 13.1, choose II = 110 , where 110 is the number claimed by Lemma 4.1. Then derme m accordingly. By Lemma 4.1 and inequalities (13.2) u(x, t)
> p.- +
2:+
1'
V(x,t) E Kf x (O,dR").
Notice that no shrinking occurs in the t-direction. This is due to the fact that in (13.1) the cutoff function, can be taken independent of t.
14. Miscellaneous remarks 14-(i). Expansion of positivity A crucial fact in the proof of Theorem 1.1 is the expansion of positivity of Proposition 5.1. To focus on this phenomenon let us consider homogeneous equations with measurable coefficients of the type (14.1)
V
E
{
Vt -
L~ (0, Tj L~oc(n)) nLfoc (0, Tj w,!:(n») , 1
=0
in
nT,
where the entries (x, t) - tlij(x, t) of the matrix (aij) are only measurable and satisfy the ellipticity condition
for some A > O. In such a case, the various costants Ai and A appearing in the proof of Proposition 5.1 are all zero and the function w introduced in (6.4) c0incides with v. TIle information (5.3) has been translated into the dimensionless estimate (5.11 )-(5.12) (see Fig. 5.2). Let us think of (14.1) as defined weakly in the cylindrical domain K4 x (-4",0). The information (5.11)-(5.12) is that at the 'centre' of K4 x (-4P,O) there is a thin cylinder KhD x (-4P, 0) where v > 1. The conclusion of the arguments of §5-9 is that there exists a small positive number "Yo that can be determined a priori only in terms of N,p and A, such that v(x, t) ~ "Yo,
V(x,t) E KIX(-l,O).
14. Miscellaneous remarks 115 Thus the 'positivity' of v over K ho spreads over a full cube K 1. Actually the information (5.11)-(5.12) is only used to apply the Poincare inequality of Proposition 2.1 of Chap. I to derive the integral inequality (6.10). Precisely
j wf(v)(Pdx :5 meas {[Wk[V(X,
J] = O]n[( = I]}
K4X{t}
jIDWk(V)IP(Pdx, K4X{t}
for all t E (-4P, 0). Now to apply such an inequality it only suffices to have the information
meas{[Wk[v(X,t)] =
0]nK2}~Qo
> 0,
for some
Q
o
> 0,
In particular it is not necessary to know that the set [Wk(V) centrated in a cylinder about the origin. We summarise:
Vt E (-4P , 0).
= 0] == [v ~ 1] is con-
THEOREM 14.1. Let v be a non-negative weak solution of(14.1) in the cylindrical domain Q4(46)==K4 x (O,46)for some 6>0. Assume moreover that
meas{x E K21 v(x,t) > ko} ~
Qo,
for some positive numbers ko and Q o and all t E (26,46). Then there exists a number 'Yo ="10 (N,p, A, 6, Qo, ko ) that can be determined a priori only in terms of the indicated quantities, such that
v(x, t)
~ 'Yo,
V(x, t) E Kl x (36,46).
14-(;;). Extinction in finite time Weak solutions of (14.1) may become extinct in finite time. We refer to §2-3 of Chap. VII for a precise description of this phenomenon. The extinction profile is the set [v =O)"'\aT. Theorem 14.1 implies that the extinction profile is a portion of a hyperplane normal to the t-axis. Indeed if u(xo, to) >0 for some (xo, to) E nT • by continuity we may construct a box about (xo, to) where the assumptions of Theorem 14.1 are verified. It follows that the positivity of vat (xo, to) expands at the same time to to the whole domain of definition of v(·, to).
a
14-(iii). Continuous dependence on the operator We only remark that the comments on stability made in §14 of Chap. III. in the context of degenerate equations. carry over with no change to the case of singular equations.
116 IV. HOlder continuity of solutions of singular parabolic equations
15. Bibliographical notes Theorems 1.1-1.3 were established in [26J for the case when the principal part of the operator is independent of t. This restriction has been removed in [27J and entails the new iteration technique presented in §6-10. This technique differs substantially from the classical iteration of Moser [81.82.83J or DeGiorgi [33J. It extracts the 'almost elliptic' nature of the singular p.d.e. as follows from the remarks in §14-(i). We will further discuss this point in Chap. VII in the context of Harnack estimates. The method is rather flexible and adapts to a variety of singular parabolic equations. For example it implies the HOlder continuity of solutions of singular equations of porous medium type. To be specific. consider the p.d.e. Ut -
div a(x, t, u, Du)
+ b(x, t, u, Du) = 0,
in {h,
with the structure conditions (At>
a(x, t, u, Du)·Du ~ Co lul m -
(A 2 )
la(x, t, u, Du)1 ~ C 1 lul m - 1 IDul + 'PI (x, t), Ib(x, t, u, Du)1 ~ C2 1D lul m 12 + 'P2(X, t).
(A3)
1
lDul 2
-
'Po(x, t),
mE (0,1),
We require that u E L::
(0, Tj L~oc( ll»)
and
lul m E L~oc ( 0, Tj W,!:( ll») .
The non-negative functions 'Pi, i = 0,1,2. satisfy (A4) - (As) of §I of Chap. I with P = 2. Further generalisations can be obtained by replacing sm-l, S > 0 with a function cp( s) that blows up like a power when s - 0 and is regular otherwise.
Results concerning doubly non-linear equations bearing singularity and/or degeneracy are due to Ivanov [52.53.54J and Vespri [l02J. A complete theory of doubly singular equations. however. is still lacking.
v Boundedness of weak solutions
1. Introduction Let u be a weak solution of equations of the type of (1.1) of Chap. II in aT. We will establish local and global bounds for u in flT. Global bounds depend on the data prescribed on the parabolic boundary of flT. Local bounds are given in tenns of local integral nonns of u. Consider the cubes Kp C K 2p . After a translation we may assume they are contained in fl provided p is sufficiently small. For 0 ~ tl < to < t ~ T consider the cylindrical domains
The local estimates are of the type.
(1.1)
~
lIulloo,Q. ,; ( 1 +
If lUI'.) .. ,
where the numbers q" i = 0, 1, are detennined a priori in tenns of p and N and the constant 'Y is detennined a priori in terms of the structure conditions of the p.d.e. and Ql. Unlike the elliptic theory, the estimate (1.1) discriminates between the degenerate case p > 2 and the singular case 1 < p < 2. To illustrate this point, consider local weak solutions of the elliptic equation {
u E w:1'''(fl) loe ,
p>l
div IDul,,-2 Du = 0,
in fl.
118 V. Boundedness of weak solutions
These solutions satisfy the following estimate for any p exists a constant 'Y -y (N,p, e), such that
=
> 1. For every e >
°
there
Consider now the corresponding parabolic equation (1.2)
U {
E
Ut -
Gloe (0, Tj L~oc(D)) nLfoe (0, Tj WI!;: (D») , p > 1, div IDul,,-2 Du
= 0,
in DT,
and the cylindrical domain Q~p == K 2p X (0, tl. Assume fmt that p > 2. Then for all eE (0, 21 there exists a constant -Y='Y (N,p, e) such that for all tt~s~t
For the singular case 1 < p
< 2 a local sup-estimate can be derived only if u
is
SJljJiciently integrable. Introduce the numbers
A,. == N(P- 2) +rp,
(1.4)
r ~ 1,
and assume that u E Lfoe(DT) for some r ~ 1 such that A,. > 0. Then there exist a constant 'Y='Y(N,p, r) such that for all tt<s=t there holds
When 1 < p < 2 such an order of local integrability is not implicit in the notion of weak solution and it must be imposed. The counterexample of § 12 of Chap. XII shows that it is sharp.
2. Quasilinear parabolic equations Consider quasilinear evolution equations of the type (2.1)
Ut -
div a(x, t, u, Du) = b(x, t, u, Du)
in 'D'(DT ).
The functions a: nT x R N +1 -+ R N and b: nT x RN +1 -+ R, are measurable and satisfy
2. Quasilinear parabolic equations 119 (B l)
a(x, t, u, Du) . Du ~ GoIDul" - colul 6 - rpo(x, t),
(B 2)
la(x, t, u, Du)1 :$ GlIDul,,-l + cllul6~ + rpl(X, t), Ib(x, t, u, Du)1 :$ G2IDul"¥ + c2lul 6- l + rp2(X, t)
(B3)
for p > 1 and a.e. (x, t) E ilT . Here Gi , C;, i = 0,1,2. are positive constants and 6 is in the range
N+2
p:$6
The non-negative functions rpi, i=O,I, 2. are defined in ilT and satisfy
(Bs) where
-41 = (1-11:
p
0
)-N +p'
11:0 E (0,1].
For fELl (ilT) and h E (0, T) we let !h denote the Steklov average of function u is a local weak sub(super)-solution of (2.1) in ilT if u E
(2.2)
Gloc (0, Tj L~oc{il»)nLfoe
f.
A
(0, Tj WI!:(il») ,
and for every compact subset IC of il. (2.3)
f {!
uhrp+[a(x, T, u, DU)]h ·Drp- Ibex, T, u, DU)]h rp} dx:$ (?)O
K:x{t}
for all 0 < t :$ T - h and all testing functions (2.4)
rp E Gloe (O,TjL2(1C»)nLfoc (O,TjWJ'''(IC»,
rp ~ O.
The statement that a constant 'Y = 'Y (data) depends only upon the data means that it can be determined a priori only in terms of the numbers N, p, 4, 6,11:0, the constants Gi , C;, i=O, 1,2. and the norms ~
8-1
IIrpo, rpr ,rp;r 114,UT'
2-(i). The Dirichlet problem Consider the boundary value problem Ut -
(2.S)
{
div a(x, t, u, Du) = b(x, t, u, Du),
u(·, t)18U = g(., t), u(·,O) = u o ,
in ilT, a.e. t.E (0, T),
120 V. Boundedness of weak solutions
We retain the structure conditions (Bd-(B6). and on the Dirichlet data 9 and 1£0 we assume
9 E L oo (BT), 1£0 E L2(n).
(2.6) (2.7)
The notion of weak solution is in (2.5) of Chap. II.
Remark 2.1. Unlike the assumption (Uo ) in Chap. II. we do not assume here that 1£0 E Loo(n).
Accordingly. our estimates of the norms
111£(', t)lIoo.n
deteriorate as
t'\,O.
2-(ii). Homogeneous structures Local and global sup-bounds. take an elegant form for solutions of equations of the type (2.8)
Ut - diva(x, t,u,Du) {
(2.9)
= 0,
in
nT,
P> 1
a{x,t,u,Du)· Du ~ CoIDuI P , la{x, t, u, Du)1 ~ C1IDul p - 1,
for two given constants 0 < Co ~ Cl. The lower order terms are zero and the principal part has the same structure as (2.10)
Ut - div IDul p - 2Du = 0 or Ut - (IU:r:iIP-2u:r:;),c;
= O.
Because of the structural analogy with (2.10) we will refer to (2.8)-(2.9) as equations with homogeneous structure.
3. Sup-hounds We let u be a non-negative weak subsolution of (2.1) and will state several upper bounds for it. The assumption that u is non-negative is not essential and is used here only to deduce that u is locally or globally bounded. If u is a subsolution. not necessarily bounded below. our results supply a priori bounds above for u. Analogous statements hold for non-positive local supersolutions and in particular for solutions. The estimates of this section hold for P in the range
(3.1)
p> max {I j
The case \ < p
::2}'
~ max { 1 j ;~2}
i.e.•
~2 == N(P -
2) + 2p>0
will be discussed in §5. Let 6 and
numbers appearing in the structure conditions (Bl)-(B6) and set
"'0 be the
3. Sup-bounds 121
N+2
(3.2)
q=p--,
N
The range of 6 in (B4) is p:$ 6 < q. We will assume that (3.3)
max{pj 2} :$ 6
< q.
This is no loss of generality by possibly modifying the constants Ci and the functions 'Pi, i = 0, 1, 2. We also observe that owing to (3.1), the range (3.3) of 6 is non-empty. In the theorems below we will establish local or global bounds for s0lutions of (2.1). However precise quantitative estimates will be given only for the case (3.4)
i
= 0,1,2.
In this case we may take Ko = 1 in (B6) and K=p/N.
3-(i). Local estimates THEOREM
3.1. Let (3.1) hold. Every non-negative. local weak subsolution U 0/
DT is locally bounded in DT . Moreover. if 'Pi E LOO(DT)' i = 0,1,2. there exists a constant 'Y = 'Y (data) such that V [(x o, to) + Q (PP, p)] CDT and VUE(O,I).
(2.1) in
sup
(3.5)
U
[(zo.to)+Q(upP ,up»)
:$'Y((I-u)-(N+P)+IQ(PP,p)I)~ (
ffU6dXdT)~
1\1.
[(zo,to)+Q(pp ,p»)
3-(ii). Global estimates: Dirichlet data 3.2. Let u be a non-negative weak subsolution o/the Dirichlet problem (2.5) and let (2.6) hold. Then u is bounded in Dx(e, T), Ve E (0, T). Moreover, if'Pi E LOO(DT)' i=O, 1,2. there exists a constant 'Y = 'Y (data). such that/or all
THEOREM
O
~pu(
.• t) $0:,"9 + 1
(ti + D,!. (/[U'dxdT) ~ AI
/fin addition the initial datum U o is bounded above. then
(3.6)
~u(.• t)" max {S:,"9; ~p ...} +1 (l!u'dxdr) ~ A1.
122 V. Boundedness of weak solutions THEOREM 3.3 (THE WEAK MAXIMUM PRINCIPLE). Let u be a non-negative weak subsolution of the Dirichlet problem (2.5) for equations with homogeneous structure as (2.8)-(2.9). Then
(3.7)
sup u aT
:5 max {ess sup 9 i ess sup u o } ST
.
a
Remark 3.1. The weak maximum principle holds for equations with homogeneous structure for all p> 1. As a particular case, Theorems 3.1-3.3 give a priori sup-estimates for nonnegative weak solutions of (3.8)
Ut -
div IDuI P - 2 Du =
al U 6 - l
+ a2!p,
at
E R,
i
= 1,2,
where
N+2
1
1
"4=
p
N+p(I-lt o ),
ltoE(O,I].
These conditions on the lower order terms are optimal for a sup-bound to hold as it can be seen from the 'linear' case p = 2. Set p = 2 and a2 = 0 in (3.8). For a local weak solution uE V,~c(flT) to be locally bounded, 6 must not exceed 2Nh2. Likewise if al = 0, the forcing term II' must satisfy A
where
N+2
q > -2-.
These are classical and optimal results for the linear case p=2 (see [67]).
4. Homogeneous structures. The degenerate case p > 2 Here we consider non-negative local or global subsolutions of equations with the homogeneous structure (2.8)-(2.9). These structures reveal the basic difference between the degenerate case p> 2 and the singular case 1 < p < 2.
4-(i). Local estimates THEOREM
4.1. Every non-negative. local weak subsolution u of (2.8)-(2.9) in
flT is locally bounded in flT . Moreover for all E E (0,2] there exists a constant
'Y depending only upon the data and E, such that V [(x o, to) + Q (8, p)]cflT and VuE (0, 1),
4. Homogeneous structures. The degenerate case p> 2 123
(4.1)
= p",
Remark 4.1. If 9 (4.1) is dimensionless but it is not homogeneous in 1.£. In the linear case p = 2, (4.1) holds for any positive number e. In our case, e is restricted in the range (0,2]. It is of interest to have sup-estimates that involve 'low' integral norms of the solution. The next theorem is a result in this direction. Even though it is of local nature, it will be crucial in characterising the class of non-negative solutions in the strip RN x (0, T). (1) THEOREM 4.2. Let u be a non-negative, local subsolution u 0/(2.8)-(2.9) in nT . There exists a constant 'Y = 'Y (data), such that V [(xo, to) + Q (9, p)] c nT and Vue (0, 1), P/2
(4.2)
sup [(z .. ,t.. )+Q(a9,ap»)
1.£
:5
'Y";9/PP N(p+l)+p (1 _ u) 2
(
sup t .. -9
1\
f
1.£(x, 'T)dx
)
[z .. +Kp)
( p")~ 9 .
4-(ii). Global estimates for solutions of the Dirichlet problem Consider a non-negative weak subsolution of the Dirichlet problem (2.5) for equations with homogeneous structure and let (2.6) hold. If the initial datum 1.£0 is also bounded above, then the weak maximum principle estimate (3.7) holds true. If however 1.£t is not bounded, it is of interest to investigate how the supremum of 1.£ behaves when t -+ 0. THEOREM 4.3. Let 1.£ be a non-negative weak subsolution o/the Dirichlet problem (2.5) and let (2.6) hold. There exists a constant 'Y = 'Y (data), such that Vte(O,T),
(4.3)
(1) See §7 flf Chap VI and §2 of Chap. XI.
~
= N(P -
2) + p.
124 V. Boundedness of weak solutions Results of this kind could be used to construct solutions of the Dirichlet problem with initial data in L 1 (fl) or even finite measures. Indeed the regularity results of Chap. III supply the necessary compactness to pass to the limit in a sequence of approximating problems.
4-(iii). Estimates in L'T==RN x (0, T) Consider a non-negative weak subsolution 1£ of (2.8) in the whole strip L'T. By this we mean that 1£ is a local weak subsolution of (2.8) in flT for every bounded domain fl C RN. To derive global sup-estimates, we must impose some control on the behaviour of 1£ as Ixl-.oo. We assume that the quantity
- . Jp>-Ie
1I1£II{r,t} = O
(4.4)
1£(x,r)
dx,
-2)
P
~=
N(P - 2) + p,
Kp
is finite for some r > 0 and for all t e (0, T). The subsolution 1£ at hand is not necessarily bounded. However it is locally bounded and and as lxi- 00 it grows no faster than Ixl;;!,. This is the content of the next theorem.
4.4. Let 1£ be a non-negative subsolution 0/(2.8) in L'T, and assume holds. There exist a constant 'Y = 'Y (data), such that/or all te (0, T),
THEOREM
(4.4)
sup
(4.5)
PROOF:
p~r
111£(·, t)lIoo,K ~ "'("Ii 1I1£II{~~} p
A
pl'1(p-2)
t-~.
Apply Theorem 4.2 with the choices
(x o, to) == (0, t),
(J
= t,
(1
= 1/2,
P ~ r,
and p replaced by 2p. It gives
111£(·, t)lIoo,Kp
Ie
pi' P
-2)
~ "'("Ii ( O
JP>-I(
P/2
1£(x, r) ) -2)
P
At
_~ p-
•
Kp
Remark 4.2. The assumption (4.4) is not restrictive. We will show in Chap. XI that it is necessary and sufficient for a non-negative solution of (2.8) to exist in L'T· The right hand side of (4.5) blows up as t '\, 0 at the rate of at least t - ~ . Such a rate is not optimal. However the advantage of Theorem 4.4 is that it does hold for all t E (0, T). The purpose of the next theorem is two-fold. It gives an optimal estimate of how the local sup-bound for 1£ may deteriorate as either Ixl - 00 or t'\,O.
5. Homogeneous structures. TIle singular case 1 < p < 2 125 THEOREM 4.5. Let 1£ be a non-negative subsolution 0/(2.8) in ET and assume (4.4) holds. There exists constants 'Y. and 'Y depending only upon N,p and the constants Ci , i=O, 1 in the structure condition (2.9), such that
(4.6)
for all
°< t < 'Y.llull~;':}
111£(·, t)lIoo,K p ~ 'Y
pp/(,,-2) tNI>'
and/or all p ~ r pI>'
Ilull{r,t}' A = N(p -
2)
+ p.
Information of this kind are of interest in investigating the behaviour of the solutions for t near zero and in studying the structure of the non-negative solutions in ET. (1) The functional dependence in (4.6) is sharp as it can be verified from the explicit Barenblatt solution (4.7)
B(x,') =
{1-~. (.l~~ t} ~,
d -L.
'Y" = (~) ,,-1
p;
2,
t>o
p>2.
The function 8 solves the Cauchy problem
(4.8)
{
div IDul,,-2 Du = 0, in RN x (0, 00), 8(·,0) = M60 ,
Ut -
where 60 is the Dirac mass concentrated at the origin, and
M == 118(·, t)1I1,RN,
'TIt> 0.
The i~itial datum is taken in the sense of the measures, i.e., for every cp E Co(RN)
f
8(x, t)cpdx
--+
M cp(O),
as t '\. 0.
RN
°
°
For t > and for every p> we have
5. Homogeneous structures. The singular case 1 < p < 2 The estimates of §3 are valid for solutions 1£ E L1oc( DT ) as long as p (1) See Chap. XI.
> max { 1; : : 2 } .
126 V. Boundedness of weak solutions In this section we will show that weak solutions uEL'oc(nT), r~ I, are bounded provided
p > max { 1;
;:r}.
Such integrability condition to insure boundedness is sharp. In §12 of Chap. XII we produce a solution of the homogeneous p.d.e. (1.2)
JZ2}'
the that is unbounded.(I) Thus in the singular range 1 < p < max {I; boundedness of a weak solutions is not a purely local fact and, if at all true, it must be deduced from some global information. One of them is the weak maximum principle of Theorem 3.3 and Remark 3.1. Another is a sufficiently high order of integrability.
5-(i). Local estimates A sharp sufficient condition can be given in terms of the numbers ~r =
(5.1)
N(P- 2) +rp,
We assume that u satisfies u E L'oc(nT) , for some r ~ 1 such that ~r > o.
(5.2)
1be global information needed here is (5.3)
{
u can be constructed as the weak limit in L,oc (nT ) of a sequence of non-negative bounded subsolutions of (2.8).
The notion of weak subsolution requires u to be in the class
By the embedding of Proposition 3.1 of Chap. I, we have
N+2 N
q=p--.
Therefore ifpis so close to one that ~q ~o, the orderofintegrability in (5.1)-(5.2) is not implicit in the notion of subsolution and must be imposed. (1) The notion of solutions that are not in the function class (2.2) is discussed in Chap. XII
5. Homogeneous structures. The singular case 1
5.1. Let u be a non-negative local weak subsolution 0/(2.8)-(2.9) in and assume that (5.2) and (5.3) hold. There exists a constant 'Y = 'Y (data, r). such that V [(xo, to) + Q (9, p)] CflT andVuE (0,1). THEOREM
flT
(5.4)
Remark 5.1. If 9 =pP, (5.4) is dimensionless but it is not homogeneous in u.
5-(;;). Estimates near t=O Fix tE (0, T) and let us rewrite (5.4) for the pair of boxes KTP x
(ut, t),
Kpx (0, t).
5.1. Let u be a non-negative local weak subsolution of (2.8)-(2.9) in flT and let (5.2)-(5.3) hold. There exists a constant 'Y = 'Y (data, r). such that for all O
(5.5)
supu(·, t) ~ K"p
'Yt-N/>'r
(1-0')
¥.E r
Remark 5.2. Assume that (5.2) holds with r
(5.6)
= I, i.e.,
2N p> N+1·
°
Then the behaviour of the supremum of u as t '\. is formally the same as that of solutions of the Dirichlet problem for degenerate equations as in Theorem 4.3.
5-(;;;). Global estimates: Dirichlet data A peculiar phenomenon of these equations is that, unlike their degenerate counterparts, local and global estimates take essentially the same form. This appears, for example, by comparing (5.5) with the next global estimate.
128 V. Boundedness of weak solutions
THEOREM 5.2. Let 1£ be a non-negative weak subsolution of the Dirichlet problem (2.5) and let (2.6) and (5.2)-(5.3) hold. Theruxists a constant "1 = "1 (data, r), such that for all 0 < t $ T,
s:r u(·.') ,;
(5.7)
s~f
9+
,Ni>.
(It dxd-.-)"' u'
>.
6. Energy estimates The proof of the sup-bounds stated in the previous sections is based on local and global energy estimates similar to those of §3 of Chap. II.
6-(i). Local energy estimates If [( X o , to) + Q (9, p) 1C !1T we let ( denote a non-negative piecewise smooth cutoff function vanishing on the parabolic boundary of [( x o , to) + Q (9, p) I.
PROPOSITION 6.1. Let 1.£ be a non-negative local weak solution of (2.1) in !1T and let (B.)-(B6) hold. There exist a constant "1 = "1 (data), such that V [(xo, to) + Q (9, p}lc!1T andfor every level k > 0 (6.1)
jrJriD (1£ - k)+ (I"dxdr
/ (1.£ - k)! ("(x, t)dx + "1- 1
sup to-8
[(zo,to )+Q(8,p»)
[zo+Kp)
$"1 //(1.£ -
k)~ ID(I"dxdr +"1
[(zo,t o )+Q(8,p»)
//(1£ - k)! (,,-1(t dxdr
[(zo,to )+Q(8,p»)
}7It.;(1+1C)
to
+"1 / / 1£6 X. [1.£ >
01 dxdr + "1
[(zo,t o )+Q(9,p»)
where It =
Ito
{
/IAt,p(r)ldr to-8
Ii and IAk,p(r)1
== mess {x E [Xo + Kp]
I u(x, r) > k} .
In (6.1) the integral involving 1.£6 can be eliminated if Ci
= 0,
i
= 0,1,2,
and
b(x, t, 1.£, Du)
== 0.
Moreover the last term can be eliminated if!pi == 0, i =0, 1, 2. PROOF: The proof is very similar to that of Proposition 3.1 of Chap. II. First we may assume that (xo, to) == (0, O) modulo a translation. Then in (2.3) we take the testing functions
6. Energy estimates 129
cP = (Uh - k)+ (II, where Uh is the Steklov average of u. All the terms are estimated as in §3 of Chap. II, with minor modifications, except the integrals involving the lower order terms b(x, t, u, Du). For these, we let h - 0 and use the structure condition (B3) and Young's inequality to estimate
I jlb(x, r, u, Du) (u - k)+ (1Ildxdr Q(9,p)
(6.2)
:5 C2 I liD (u - k)+ IP¥ (u - k)+ (/dxdr Q(9,p) +C2 Ilu 6 - 1 (u-k)+(Pdxdr+ Ilcp2(U-k)+(PdXdr Q(9,p) Q(9,p)
:5
~o I liD (u Q(9,p)
k)+ IP(Pdxdr + 'Y I Iu6 ')( [(u - k)+ > 0] dxdr Q(9,p)
+ 'Y I I
cpr')( [(u - k)+ > Q(9,p)
0] dxdr .
Thus we arrive at
sup I (u - k)! (P(x, t)dx + 'Y- 1 r riD (u - k)+ (IPdxdr -9
:5'Y II(u -
k)~ ID(IPdxdr + 'Y II(U -
Q(9,p)
k)! (P-l(t dxdr
Q(9,p)
+'Y I I u6 ')([(u - k)+ >0] dxdr + 'Y I I~')([(u - k)+ >0] dxdr, Q(9,p)
Q(9,p)
where ..r;.
~
6 = CPo +;;!:r+ CP1 CP2·
By the HOlder inequality and (B5)-(B6),
II~dxdr :511~lIq,nT Q(9,p)
0 {
}
~(1+IC)
IIA/c,p(r)ldr -9
Remark 6.1. Inequality (6.1) for the function (u - k)_ holds true for local supersolutions of (2.1) and k:5 o. Remark 6.2. Unlike inequalities (3.8) of Chap. II, the levels k here are not restricted.
130 V. Boundedness of weak solutions
6-(ii}. Global energy estimates: Dirichlet data PROPOSITION 6.2. Let U be a non-negative weak sub-solution of the Dirichlet problem (2.5), let (2.6) hold and let k satisfy
k
(6.2)
~
supgAO. ST
There exists a constant 'Y = 'Y (data), such that for every non-negative function t--+(t) eCl[O, T] andfor every O
sup j(u - k)! (T) dx +
O<.,.
n
I liD (u -
11
k)+ IP(T) dxdT
n.
~'Y j(Uo-k)!(O)dx+'Y jj(u-k)!(t(T)dxdT n
If
+'Y
n,
(t
)~(1+IC)
u6x [u > k] (T)dxdT+'Y ~IA1c'P(T)I(T)dT
where 1£ =1£0 Ii, and
e U I U(X,T) > k}.
IA1c,p(T)1 == meas {x
In (63) the integral involving u 6 can be eliminated if £1=0,
i=0,1,2,
and b(x, t,u, Du) ==0.
Moreover the last term can be eliminated ifIPi==O, i=O, 1, 2. A similar statement holds for the truncatedfunctions (u - k)_ provided k < infgVO.
(6.2)'
PROOF:
-
ST
If k satisfies (6.2), by Lemma 2.1 of Chap. I (uhht) - k)+ E W!,P(U),
VO
Therefore the testing functions
are admissible in the weak fonnulation of the Dirichlet problem.
7. Local iterative inequalities 131
7. Local iterative inequalities The common element in the proof of the sup-bounds stated in §3 and 4 is a set of
iterative inequalities. We will derive them, starting from the energy inequalities of §6. Modulo a translation, we may assume that (xo, to) coincides with the origin. Fix q E (0, 1) and consider the sequences Pn = qp +
(l-q) 2n p,
(In = q(J +
(l-q) 2n
(J,
n = 0,1,2, ... ,
and the corresponding cylinders QniEQ ((In, Pn). It follows from the definitions that Qo = Q «(J, p), and Qoo = Q (q(J, qp) . Consider also the family of boxes
where for n=O,I,2, ... _ . Pn + Pn+l 3(1 - q) Pn = 2 =qp + 2n+2 p,
For these boxes we have the inclusion
n = 0,1,2, ....
Qn+lCQnCQn,
Introduce the sequence of increasing levels k
n
=k-~ 2n '
where k is a positive number to be chosen. We will work with the inequalities (6.1) written for the functions (u - kn+l)+, over the boxes Qn. The cutoff function ( is taken to satisfy ( :ani~hes_on the parabolic boundary of Qn, { (= 1 10 Qn, 2"+2 ID'"I':. < - {1-a)p'
0
< < - (t -
2"+2
(l-a)B"
With these choices, (6.1) yields (7.1)
sup
j(u - kn+l)! (P(x, t)dx + {{ID (u - kn+l)+
JJ Q..
-8,,
-y2np
(I
P
dxdT
{{
~ (1- q)ppp JJ (u - kn+tl~ dxdT Q..
+ (1 ~:)(J j j(u -
kn+l)! dxdT
Q..
+-y jju6x [(u - kn+tl+ > 0] dxdT+-yIA n+llllh(1+ IC ), Q..
132 V. Boundedness of weak solutions
where whe have set
IAn+11 == meas{(x,t) e Qn I u(x,t) > kn +1}' The last two tenns can be eliminated for equations with homogeneous structure. First we observe that for all 8 > 0
II(u -kn)~
(7.2)
dxdT
-
~ II~u kn)~ X [u > kn+1]dxdT
Qn
Qn
~ (kn+1 - kn)' IAn+11
k'
= 2(n+1),IAn+ll. Then we estimate
To estimate the integral involving u 6 , first write 2n+1 - 2 kn = kn+1 2n+1 _ 1 .
Then estimate below (7.5)
II(u -kn)~ Qn
dxdT
II(U -kn)~ [u > ~ IIu ~:: =~)6 ~ 'Y: II >
~
X
kn+1] dxdT
Qn
6
(1-
X[u > kn+l]dxdT
Qn
u6 X [u
n6
Qn
Finally
kn+1] dxdT.
7. Local iterative inequalities 133
We combine these estimates into (7.1) to derive the following basic iterative inequalities
Moreover the last two tenns can be eliminated for equations with homogeneous structure. To proceed, construct a non-negative piecewise smooth cutoff function (n in Qn, which equals one on Qn+l, vanishes on the lateral boundary of Qn and such that ID(nl $ 2n+2 /(1 - u}p. Then the function (u - kn+l)+ (n vanishes on the lateral boundary ofQn and by the multiplicative inequality of Proposition 3.1 of Chap. I,
(7.7)
/ /(u -
~
x
kn+l)~ (!dxdT
$ 'Y (_sup
/
(u _ kn+l)! dx) i
-Bn
~
(i/ID(U-kn+d+IPdXdT+ Qn
i/(U-kn+l)~ID(nIPdxdT). Qn
Remark 7.1. The estimates in (7.2)-(7.5) and the inequalities (7.6), (7.7) are valid for any number 6 ~ max{pj 2}. The structural restriction 6 = q does not play any role in the derivation of (7.6) and (7.7).
134 V. Boundedness of weak solutions
8. Local iterative inequalities (p > max {1;&~2} ) Introduce the dimensionless quantities (8.1)
Yn =
H
(u -
kn)~ dxdr,
n = 0,1,2, ....
Qn
We will derive an iterative inequality for Yn by estimating the right hand side of (7.7) by (7.6). We assume first
(8.2)
p
N+2 > max { 1 j N2N} + 2 ' max{Pi 2} ~ 6 < q == P -r
and estimate
We estimate the last integral by (7.7) and in turn estimate the right hand side of (7.7) by the inequalities (7.6) and (7.3). We arrive at the recursive inequalities (8.3)
Yn+l
< -
-ybn ~ k~(q-6) (1 - O')P~
b ( PN(J)~* + -y k~(q-6) n
1...2.
.A!* Y,!'+~*
y'l+,N +"Vbn n,
I. (pN(J)'"
(1
-y.
)l+~"
k6 n
,
where and
(8.4)
.A.. .. ( (;) kqz(P-'l + (~) f
kqz<,-<»).
The last two tenns in (8.3) can be eliminated for solutions of equations with homogeneous structure.
9. Global iterative inequalities 135
9. Global iterative inequalities We let u be any non-negative weak subsohition of the Dirichlet problem (2.5) and assume that (6.2) holds so that u satisfies the energy estimates (6.3). Fix 0 < t ~ T and introduce the sequence of increasing time levels
tn = ut (1 - 2: ),
u E (0,1),
n = 0,1,2, ... ,
and the cutoff functions iftn+l~T~t
if tn < T < tn+l
Introduce also the sequence of increasing levels k kn = sup 9 + k - -2' ST
n
k > 0 to be chosen
n = 0, 1,2, ... ,
and write (6.3) for the functions (u - kn+d + and the cutoff functions (n to obtain t
(9.1)
sup
t,,+1
/(u - kn+l)! dx + II
/
"n
J"
(u - kn+l)+ IPdxdT
t,,+lll
~ 'Y:: /
t
/(U - kn+d!
t"ll
t
dxdr +
'Y / /u6x [u > kn+lJ dxdT t"ll
where we have set
The last two terms of (9.1) can be eliminated for equations with homogeneous structure. Moreover the last term can be eliminated if in the structure conditions (B 1 )-(B3), !Pi ::0, i=O,I,2. If !Pi E LOO(nT), i=O,l, 2, then It= N' Proceeding as in (7.2)-(7.5), we estimate
136 V. Boundedness of weak solutions t
(i)
t
f f(U - kn+l)! dxdT ~ 'Y2:;~:2) f f(U - kn)~ dxdr, t"a
(ii)
(iii)
t"a
t
t
~a
~a
f fU6 x. [u > kn+lJ dxdr ~ 'Y2n6 f f(U - kn)~ dxdT,
j
IAn+l(r)ldT
~ 'Y:;6 jiU -kn)~ dxdT. t"a
t"
Combining these remarks in (9.1) we arrive at the recursive inequalities
Remark 9.1. The structure restriction 6 < q does not play any role in the derivation of (9.2). This inequality holds for all 6 ~ max{p; 2}.
9-(i). Global iterative inequalities. The case p>max
{I; J~2}
Next we assume that the numbers p and 6 are in the range (8.2). We apply the multiplicative embedding inequality of Proposition 3.1 of Chap. I, and proceed as in the case of the local inequalities. This process is indeed simpler. since (u - kn)+(·,t)EW!'P(O) fora.e. tE(O,T). Setting t
(9.3)
Yn ==
f fa
t"
we obtain (9.4)
(u -
kn)~ dxdr,
10. Homogeneous structures 137
t .. ).
where b = 26(1+ In these. the last two tenos can be eliminated for solutions of equations with homogeneous structure as in (2.8)-(2.9). Moreover the last teno can be eliminated if. in the structure conditions (B 1 ) - (B 3 ). 'Pi == 0, i = 0, 1,2. If 'Pi ELOO(nT), i=O, 1,2. then It=p/N. Suppose now that the initial datum 1.1.0 in (2.7) is bounded above and let us take in (6.3)
kn = max { S~! 9 j
s~p 1.1.0 } + k -
2:'
n
= 0,1,2, ... ,
where k > 0 is to be chosen. Then the first integral on the right hand side of (6.3) is zero and we may take ( == 1. In such a case. we arrive at an inequality analogous to (9.1). where the first integral on the right hand side is eliminated and where the integrals are all extended over the whole nt • Proceeding as above we find that the quantities
Yn ==
H
(1.1. -
kn)~ dxdr
nc
satisfy the recursive inequalities
Yn +1:5
(9.5)
-ybnlntl*~ 1+*~ k~(9-6) Yn
n
+ -yb
Inti K! ( k16 Yn )1+K~
b = 26(1+~~). For equations with homogeneous structure. all the tenos on the right hand side of (9.5) are zero.
10. Homogeneous structures and 1 < p < max { 1; J~2 } Let 1.1. be a non-negative local weak subsolution of (2.8)-(2.9) in that 1.1. satisfies
(10.1)
1.1.
E L[oc(.f1T ),
nT . We assume
for some r ~ 1 such that Ar>O.
The numbers Ar have been introduced in (5.1). We also assume that 1.1. can be constructed as the weak limit in L[oc (nT) of a sequence of bounded subsolutions of (2.8). By possibly working with such approximations we may assume that 1.1. is qualitatively locally bounded. Below. we will derive iterative inequalities similar to (8.3) but involving the L[oc -nonos as well as local sup-bounds of u. If 1
J~2}' we have q < 2. If (10.1) holds for some r E [1, 2).
A2
then > 0 and p > max { 1 j J~2}. Therefore it suffices to assume that (10.1) holds for some r > 2. In such a case we have (10.2)
r>q,
138 V. Boundedncss of weak solutions
In (7.6) we discard the last two terms in view of the homogeneous structure of (2.8) and, owing to Remark 7.1, set also 6 =r. We obtain (10.3)
J
sup
-9.. <e
(u-k n +1)!(x,t)dx+ ffID(u-kn+d+IPdxdr
11
K~..
Define
Yn =
i
Q..
H
(u -
kn)~ dxdr,
n= 0,1,2, ... ,
Q..
and estimate
Yn +1
~ lIull:'~(9,p)
H
u 9 dxdr.
Q.. We majorise the right hand side by means of (7.7) and in turn estimate the right hand side of (7.7) by (10.3). We arrive at the recursive inequalities (10.4)
y.
r- 9 < (1 _ u)f(N+p) "(bn II u Il oo,Q(9,p) 8 AIf y'1+f n ,
n+1 -
where
(10.5)
8, '" { ( ; ) k-(.-p)"i'
+ (~) ~ k+')"i' }.
The recursive estimates (10.4)-(10.5) have a global version. Let u be a nonnegative weak subsolution of the Dirichlet problem (2.5) and assume that u E L oo (0, Tj Lr(o» , for some r>2 satisfying
~r >0.
Then the quantities Yn defined in (9.3) with 6 = r satisfy (10.6)
,,(bnlOel f r-9 1 1+f . Yn +1 ~ k(r-2)~ lIulloo,O>cCe.. ,t) t~ Yn
II. Proof of Theorems 3.1 and 3.2 The starting point in the proof of Theorem 3.1 is the inequality (8.3). We take (J = p" and stipulate to choose k ~ 1. Recalling that max{pj 2} ~ 6 < q, the quantity .Ak in (8.4) is majorised by 2. To simplify the presentation consider first the case
II. Proof of Theorems 3.1 and 3.2 139
i
= 0,1,2,
so that we may take K. = N' With these choices, (8.3) yields
It follows from Lemma 4.2 of Chap. I that Yn - 0 as n -
00,
provided k is chosen
to satisfy k
where Yo
=
H
ulidxdT
= max{ko i I},
= ckiq-li)/f (1 - u)-(N+p) + IQ (PP,p) I) -1,
Q(PP,p)
for a constant C depending only upon the data. This in tum implies
The general case of K. E (0, N) is proved by a minor modification of these arguments. It suffices to rewrite (8.3) as Yn +1 :5
k~~::li) (1- u)-(N+p) + IQ (PP,p) I)~" {y~+~N + y~+~,,}
and follow the iteration process of Lemma 4.1 of Chap. I. To prove Theorem 3.2 we refer to the global recursive inequalities (9.4). As before, we take k ~ 1 and consider frrst the case of 'Pi E VlO(flT ), so that K. = pIN,. Choosing u = 1 we arrive at Yn +l:5
'Ybnlfltl~N ( k~(q-li)
1+t
_l!.U) ~N Y l+~N p
It follows from Lemma 4.1 of Chap. I that Yn - 0 as n to satisfy
n
00,
.
provided k is chosen
for a constant C depending only upon the data. This in tum implies that for all
O
140 V. Boundedness of weak solutions
The general case of I( E (0, It) is proved by a minor modification of these arguments. The second part of Theorem 3.2 is proved exactly the same way, by starting from the recursive inequalities (9.5).
12. Proof of Theorem 4.1 We refer back to the iterative inequalities (8.3) and discard the last two tenns because of the homogeneous structure of the p.de. in (2.8)-(2.9). Since the resulting inequalities hold for all p ~ 6 < p N12 , we take 6 = p and rewrite them as (12.1)
Yn+l
=
'Y bn ~
(1- 0')"
k
m Ynl+m ,
-Jb AI:
where Yn are defined in (8.1) and AI: are defined in (8.4) with 6 =p, i.e.,
We stipulate to take k so large that of the two tenns making up AI: the first dominates the second, i.e., (12.2)
k> -
p!, (-P") (J
so that
AA;
~
(J
204,
04= pp'
It follows from Lemma 4.1 of Chap. I that Yn -
Yo ==
H
uPdxdr
°
as n -
= CA- 1 (1 -
00
if we choose k from
u)(N+P) k 2 ,
Q(B,p)
where C is a constant depending only upon 'Y, b, N and p. For such a choice and (12.2),
(12.3)
esssup u Q("B,,,p)
~
~(
'YVA
(1 - 0')
HUPdxdr)
I"
(~);6 .
Q(B,p)
This estimate proves the theorem for E = 2. Fix E E (0, 2) and consider the increasing sequences Po up,
=
and for n= 1, 2, ...
12. Proof of Theorem 4.1 141 n
n
(12.4)
L2-
Pn = up+ (1- u)p
i,
(In
= u(J + (1 -
u}(J
L 2-
i,
i=l
i=l
and the corresponding cylinders Q(n) ==Q ((In, Pn). By construction (12.5)
Q(o)
== Q (u(J, up)
and
Q(oo)
== Q ((J, p) .
Set (12.6)
Mn = esssupu Q(")
and write (12.3) for the pair of boxes Q(n) and Q(n+1). This gives
"(2n~:: ffUPdxdr)! AA,.!-p
Mn $
(
(1 - u)
<
-
Q(,,+l)
M¥"(2n~ VA ( ffUP-2+Edxdr)! AA~ ~1
~
(1 - u)
.
Q(fJ,p)
If1]E (0, I), the right hand side of this inequality is majorised by ~
d= 2 • , where
Combining these estimates we arrive at the recursive inequalities
n = 0,1,2, .... From these, by iteration n
Mo $1]nMn+1 +BdL(1]d)i,
VnEN.
i=O
fa
We choose 1] = so that the sum on the right hand side can be majorised by a convergent series and let n - 00 to obtain
sup Q(ufJ,up)
u$
,,(Af
~
(1 - u) •
142 V. Boundedness of weak solutions
13. Proof of Theorem 4.2 The proof of the theorem is a consequence of the following: 13.1. Let u be a non-negative local sub-solution of (2.8)-(2.9) in . nT, and let p > 2. There exists a constant "Y = "Y (data), such that V [(x o, to) + Q (0, p)]CnT and Vu E (~, 1), PROPOSITION
H
uPdxdT $ (1-"Y
(13.1)
)Np
[(zo,to)+Q(O",O'p»)
fu(x, T) dx) P
(sup
to-'
U
1\
[zo+Kp)
( PP)~ 0 .
PROOF: We may assume that (x o, to) =(0,0), and having fixed uE (!, 1), consider the increasing sequences {p,,} and {O}" introduced in (12.4) and the corresponding cylinders Q("). Let (x, t) - ,,, be a non-negative piecewise smooth cutoff function in Q(,,+1) that equals one on Q(") , vanishes on the parabolic boundary of Q(,,+1) and such that
The function (u'"K,t) vanishes on aKp"+l' Therefore by the embedding inequality (3.1) of Chap. I applied with m= 1, (13.2) / /(u,,,)p( ¥ )dxdT Q"+l
The constant "Y depends only upon the data and it is independent of p, 0 and n. The energy estimates for solutions of (2.8) give
H
2
IDu("IPdxdT $ (1 2"P u)PPP
Q .. +l
H
2 uPdxdT + (1 "Y_ "u)O
Q"+l
$
~(1 ~;)'9
[(;
H
u2dxdT
Q"+l
)lt
uPdzdTIl
(~t],
where we have estimated the second integral by HOlder inequality. Without loss of generality we may assume that
14. Proof of Theorem 4.3 143
(;) HulldxdT > (~);;!"
for all n = 0,1,2, ... ,
Q"+1
otherwise the Proposition becomes trivial. Combining these remarks with (13.2) and setting
we obtain the recursive inequalities
By the interpolation Lemma 4.3 of Chap. I, we conclude that there exists a constant 'Y, depending only upon the data such that
H UlldxdT~
Q(tr9.trp)
'Y N (1 - u) II
sup
(
fU(X,T)dx)1I
-9
14. Proof of Theorem 4.3 We may assume that the boundary datum is non-positive, by possibly replacing U with w =u-supg. ST
We start from the global iterative inequalities (9.4) and discard the last two terms on the right hand side since the p.d.e. has the homogeneous structure (2.8)-(2.9). Taking 6 = p we obtain y.
n+l ~
.!!±.I!......IL
(ut)N"+2"
where Yn are defined in (9.3) and (14.1)
l tl m y'l+m n ,
'Ybn f1
kWH
144 V. Boundedness of weak solutions
It follows from Lemma 4.1 of Chap. I that Yn - 0 as n -
Yo
00
if k is chosen from
= f/U PdxdT:5 C(CTt)~ Intl-1kP/p , at
for a constant C depending only upon 'Y, b, p and N. Thus for all CTt < T :5 t
(14.2)
IIU(·.T)II~.n'; u(N+P~.tN" (lluPdzdT) pl.
Consider the decreasing sequence of time levels
tn. t 2-' tn = - - 2 4 ;=0 '
L
and apply (14.2) over the expanding domains n x {tn+ 1, t}, with CT taken from
CT
i.e.,
2-(;+1) = 1 -"~ LJ,-o
1 + ,,~+1 2-(i+l) LJ,=o
~ 2-(n+1).
Setting also
Mn =
sup lIu(·, T)lIoo,a, t .. <-r
we obtain from (14.2)
where d=2(N+p)/",. By the interpolation Lemma 4.3 of Chap. I we conclude that
)..=N(p - 2) + p.
15. Proof of Theorem 4.5 Even though the theorem is of global nature, our starting point is the recursive inequality (12.1). We begin by observing that in the proof of Theorem 4.1 the
15. Proof of Theorem 4.5 145
choice of k ~ 1 was made to guarantee that Ale could be majorised by a quantity independent of k. Here we stipulate to choose k satisfying
1 k ~ "2supu("O), K"p
and in (12.1)(1) replace Ale with the larger quantity
aA. = ( ; )
+
(~) f (,"PK'~ U(.,O») (P-2)"t'
The numbers p and (1 being fixed, we let () be so small that, of the two terms making up A .. , the second dominates the first, i.e.,
(}<~(
(15.1)
-
)P-2
2 sUPK"p
u(·, 0)
The knowledge of such a () at this stage is only qualitative. It is part of the proof to give an upper estimate for all the positive numbers () for which (15.1) is verified. With these choices, the recursive inequalities (12.1) imply
_ 'Ybn
y.
n+l -
k-~ Amy'l+m .!'!±I!"
(1 - (1)PN+2"
n
We proceed now as before and arrive at an analog of (12.3); namely, there exists a constant'Y dependent only upon the data such that for all (1 E (0,1)
sup
(15.2)
u
Q(ulJ,up)
If () and choices,
SUPK"p
~ 'Yff~ HUPdxdrf'A m~. (
(1 - (1)
Q(IJ,p)
u(·,O) satisfy (15.1), it follows from (15.2) and the indicated
(J N/2p
Supu(.,O)
~ 'Y(PP/ )~ (Supu(.,O~ (1 - (1)
K"p
)
Therefore for p.=N(p - 2)
\~"p
(2-p)~ N (
'l
)1/2
l1 u p dxdr J(lJ,p) J
+ p2, P/IJ.
(15.3)
supu(.,O)~ K"p
'YL(N) (pp(})N/1J. +p
(1 - (1),.
(
l1 uPdxdr ) J J Q(IJ,p)
(1) The inequalities (12.1) are written over the cylinders Q(6n , Pn) introduced at the beginning of §7.
146 V. Boundedness of weak solutions This inequality holds for all fJ,p,u for which (15.1) is verified. It also holds for any pair of boxes
[(x o, to) + Q (fJ, p»)
[(x o, to) + Q (ufJ, up)] ,
and
with arbitrary 'vertices' provided they are contained in ET • Fix any te (0, T) and introduce the boxes
Kp/2 x Ut, t}.
and
We rewrite (15.3) and (15.1) in terms of these cylinders, for which u= t. LEMMA 15.1.
Forallte(O,T)andp>Ojorwhich -(,,-2)
t
(15.4)
~
21'-1",
(
sup u(x, t)
)
,
K,,/3
tMreholds (15.5)
~~~
u(x, t)
~ (~) 'Y
N/IA
(jf
UpdxdT) p/IA
t/'lK"
For r > 0 introduce the quantity (15.6)
~=N(P-2)+p.
f(t)= sup {TN/Asupllu(.,T)lIoo,K,,}, p~.,.
O<.,.
p'tbs
By possibly working within the time. interval (e, T) and then letting e '\, 0, we may assume that f(t) is finite. This follows from Theorem 4.4. Let t* e (0, T) be the largest time level for which
t p/ A ~ 2" [f(t)]-(p-2) ,
(15.7)
VO < t ~ t*.
The knowledge of t* is only qualiwive. Shortly we will find a quantitative upper bound for t*. Here we remark that owing to the definition of f (t) the condition (15.4) holds for all p > r and all t e (0, t*). Consequendy (15.5) holds for all t e (0, t*]. We estimate the integral on the right hand side of (15.5) as follows t
ff
UPdxdT
~ p~
t/'lK,.
f t
(lI u(.,T)lI oo p'tbs
,K,,)
f
1'-1
t/2
K,.
2)~ PP£:, { sup < _ (t
O<.,.
T
X {
~
= (~).
U(X,T) dxdT pN+'tbs
N/A
sup p~r
lIu(.,Tlloo,K,. .....IL pP-'2
f ~:'1
sup sup o<.,.
"
p~
fp-l(t)
Ilull{r,t}
P-
},,-1
dxdT}
16. Proof of Theorems 5.1 and 5.2 147
where the nonn
11·II{r,t} is defined in (4.4). Putting this estimate in (15.5) gives t)< "Y p-t!:rJ !(t)P(P-l)/" IlluII P{r,t} /" . .sup u(x, t N />.. K ,,/2
We divide by (p/2)p/(p-2) and multiply by t N />... Then take the supremum for p> r and use the fact that t E (0, t·) is arbitrary to deduce
'VO < t:S t*, i.e.,
'VO < t :S t·.
(lS.S)
Thus it follows from (15.7) that (15.S) continues to hold for all 0 < t :S t., where
t.
= "Y. II u 11 -(p-2) {r,t"} .
16. Proof of Theorems 5.1 and 5.2 We first prove Theorem 5.1 for the case when the assumptions (5.1 )-(5.2) hold for some 1:S r:S 2. In such a case we have p> max { 1; J~2}' and we may use the iterative estimates (S.3). In these we discard the last two tenns and take 6 = 2. We also stipulate to take 1 k> - sup u - 2 Q(ulJ,up) and arrive at
y.
<
n+l -
"Y bn A~Ny'l+~N !( 6) -"'tT n , (1- u)P,-W-k, q-
6 = 2,
! JII±,
where
.Au = { (~) p'P
(p-6)~ [
sup Q(ulJ,up)
u]
p
+ (~)
Ii.} p
,
6=2,
and Yn are defined in (S.l). By Lemma 4.1 of Chap. I, Yn -+ 0 as n -+ 00, provided we choose k from
Yo
==
if
u 2d.xdr = C (1- u)N+p A;lk~(q-6),
Q(IJ,p)
for a constant C depending only upon the data. This implies
148 V. Boundedness of weak solutions
(16.1)
sup
11.:5
Q(fTlJ,fTp)
(Hu
'Y~~
~
(1 - u)"N"('9-Tf
2dxdT)
~
6=2.
,
Q(',p)
We conclude the proof for the case r E [1, 2) by means of an interpolation process similar to that of Lemma 4.3 of Chap. I; namely, consider the sequences Pn, (In and the corresponding cylinders Q(n) == Q ((In, Pn), introduced in (12.4)-(12.5). Define also the numbers Mn as in (12.6), and write (16.1) for the pair of boxes Q(n) and Q(n+1). This gives
where
~ = { (;) M~-6)~ + (~) ~} .
Consider the two terms making up An. If for some n dominates the second, we have
(16.3)
Mn<
-
(
= 0, 1,2, ... the first term
(J ) 1/(2-p)
-
pP
and there is nothing to prove. Otherwise, (16.3) fails for all n=O, 1,2, ... and
An:5 2
(~) ~ ,
n=O,l,2, ....
We deduce from (16.2)
Mn :5
'Y2np~ ~ (PP)~ ~ Mn+1 7i
(1 - u)P ,-
(H 11.
r
dxdT
)
~
Q(n+l)
The proof is now concluded as in Lemma 4.3 of Chap. I. The proof of Theorem 5.1 for the case r > 2 is based on the fCCursive inequalities (10.4). As before, we stipulate to take
1 k> - 2
sup
u.
Q(fT',fTp)
and majorise BIc by
B Ic where
< k(2-r)~BfT' _
17. Natural growth conditions 149
B.
~ { ( ; ) [Q(:~pA'-2)~ + (~) ~}
With these choices, we obtain from (10.4) the recursive inequalities
<
y;
n+l -
II II r - q
'"'(bnB!
(1 _ u)i(N+p)k(r-2)!!p u
y;l+i
oo,Q(8,p) n
.
By Lemma 4.1 of Chap. I, Yn -+ 0 as n -+ 00, provided
y;o
= jfurdXdT =
-
C(I-
u)N+PB-lk(r-2)~llull(q-r)~ u oo,Q(8,p) '
Q(8,p)
for a constant C depending only upon the data. Thus (16.4)
sup
u
Q(u8,up) (r 2jfR+pj
<
'"'(
- (1 _ u);:!,
B(r 2jfN+pj
u
lIull~~
oo,Q(8,p)
(
jfurdXdT
)
Q(8,p)
Let Q(n) =Q(On, Pn) and Mn be defined as in (12.4)-(12.6). Then from (16.4)
where
B.~ {(;)M~-2)~ + (~)~}
The proof is now concluded as in the case r E [1, 2). The proof of Theorem 5.2 is essentially the same. IfrE [1,2), it follows from the recursive inequalities (9.4). If r > 2, we start from the global inequalities (10.6).
17. Natural growth conditions Consider the Dirichlet problem (17.1)
{ Ut -
divlDulp - 2 Du = IDuI P , in nT, p> I,
ul r = I E LOO(r),
where r denotes the parabolic boundary of nT. The lower order term has the 'natural' or Hadamard growth condition with respect to IDul (see [48]). The notion
150 V. Boundedness of weak solutions
of weak solution is that of §2 of Chap. II. Here we stress that if we merely require that IDul E V(flT), the testing functions must be bounded to account for the growth of the right hand side. The problem we address here is that of finding a sup-bound for a solution u. It is known that weak solutions of (17.1) in general are not bounded, not even in the elliptic case (see [15]). This is due to the fast growth of the right hand side with respect to IDul. On the other hand the existence theory is based on constructing solutions as limits, in some appropriate topology, of bounded solutions of some sequence of approximating problems. The limiting process is possible if one can find a uniform upper bound on the approximating solutions. Therefore the main problem regarding sup-estimates for solutions of (17.1) can be formulated as follows. Assuming that a weak solution u of (17.1) is qualitatively bounded, find a quantitative VlO (flT) estimate, depending only upon the data. In such a form, the problem was fmt formulated by Stampacchia [93] in the context of elliptic equations. THEOREM
17.1. Let u be a bounded weak solution of (17.1). Then
07.2)
lIulloo,aT :S F
=IIflloo,r·
PROOF: By working with u+ and u_ separately, we may assume that u is nonnegative. Set M = esssupu.
aT
If M
> F, in the weak formulation of (17.1) we take the testing functions (u - k)+,
where
=M -
k
E ~
F,
for some E>O,
modulo a Steklov averaging process. These are admissible since they vanish on the parabolic boundary of flT and are bounded. We obtain esssup f O
(u - k)! dx +
S1x{r}
f f ID (u JJ aT
k)+ I"dxdr
:s ffID(u-k)+I"(U-k)+dxdr aT :S E f f ID (u - k)+ I"dxdr.
aT Thus if EE (0,1), we have (u - k)+ =0 in flT and esssupu:S M -
aT
E.
This contradicts the definition of M and proves the theorem. COROLLARY 17.1. Assume f == o. Then (17.1) does not have any non-trivial bounded weak solution.
17. Natural growth conditions 151
17-(i). General structures More generally we may consider the Dirichlet problem
(17.3)
u E C (0,T;L2(n))nLP (0, T; Wl,p(n)) , { Ut - div a(x, t, u, Du) b(x, t, u, Du) in nT,
=
ul r = f E LOO(r), where the p.d.e. satisfies the sbUcture conditions
a(x, t, u, Du) . Du ~ ColDul P -
(Bi) (B;)
The lower order tenDs have the Hadamard 'natural' growth condition. Here Ci , i = 0,2, are given positive constants and the non-negative functions
(Bs) where
P
1
(B 6 )
-: = (I-lI:o)-N .. q +p
11:0
E (0,1).
THEOREM 17.2. Let u be a qualitatively bounded weak solution of(17.3) in nT. There exists a constant C that can be determined quantitatively a priori only in terms of the data. such that
PROOF: As before we may assume that u is non-negative. If M is the essential supremum of u in nT, we may assume that M > 2I1flloo,r; otherwise there is nothing to prove. In the weak formulation of (17.3), we take the testing function (u - k)+, where IIflloo,r ~ k < M.
This is admissible, modulo a Steklov averaging process, since it is bounded and it vanishes in the sense of the traces on the parabolic boundary of nT • Calculations in all analogous to those in §4-(ii) of Chap. II give (17.4)
sup O
I l1x{t}
(u - k)! dx + Co Jr [ ID (u - k)+ IPdxdT 2 J
II + II ~ C2
aT
ID(u - k)+ IP(u - k)+dxdT
aT
{ k) + IP2 (u - k)+} dxdT.
'Y
aT
152
v. Boundedness of weak solutions
Here and in what follows we denote with 'Y a generic positive constant that can be detennined a priori only in tenns of the data. Next choose k = M - 2e where eE (0,1) is so small that M -2e ~ IIflloo,r, and
II
C2
ID (u - k)+ IP (u - k)+ dxdT
aT
II ID ~ ~o II ~ 2C2e
(u - k)+ IPdxdT
aT
ID (u - k)+ IPdxdT.
aT Thus we may take 2e
= min {lIflloo,ri ~
g: }.
Combining these calculations in (17.4), we arrive at
sup O
I
(u - k)! dx + fr [ ID (u - k)+ IPdxdT
1
aT
12x{t}
~ 'Y
II
"oX [u > k] dxdT.
aT By RUder inequality and (B5HB6 ) the last tenn is majorised by
where 'Y is a constant depending only upon the data and
I u(x,t) > k}.
Ale == {(x,t) E nT Consider the sequence of increasing levels kn
=M
e
- e - 2n
'
n
= 0,1,2, ... ,
and the corresponding family of sets An == {(x,t) E
nT
I u(x,t) > len}.
TIlese remarks imply that for all n E N
sup O
I 12x{t}
(u - kn
)! dx + 11[[ ID (u -
kn )+ IPdxdT
aT
< _ 'Y IAn 14
(1+11:) ,
for a constant 'Y depending only upon the data. From this and the multiplicative inequality of Proposition 3.1 of Chap. I,
17. Natural growth conditions 153
i.e., for all n=O, 1, 2, ... , -p~ n IA 11+1< IAn+l I _< 'Ybn E , It follows from Lemma 4.1 of Chap. I that IAn 1- 0 as n -
00
if
In this case we would have u ~M - e
a.e. aT
which contradicts the definition of M. Now
i.e.,
IAol ~ (!) P j1ulPdxdr. n If the right hand side is less than 'Y. we have a contradiction. Thus
To prove that lIullp,nT is bounded above only in terms of the data, we may assume, modulo a shift that involves the supremum of the boundary data, that u is a bounded non-negative weak solution of (17.3) vanishing on r in the sense of the traces. In the weak formulation of (17.3), take the testing function
where Q is a positicve parameter to be chosen. We may also assume without loss of generality that Ut E L2 (aT). We obtain
154 V. Boundedness of weak solutions
j! aI(j
o
(eOS
-1)ds) dxdr +
Q
0
:S O2
at
IltDulPeOUdxdr + 'Y 11(1 + CPo) eoudxdr, at
for a constant 'Y
III DulPeoudxdr at
= 'Y (data) and for all t E (0, T). We choose =202 and set Q
to obtain
IIwllt-p(aT) :S 'Yo + 'Yl
II (1 + CPo) wPdxdr, aT
for two constants 'Yi = 'Yi (data) , i=O, 1. Next by (Bs) - (B6 ),
11(1 +CPo) wP-dxdr aT
Moreover since w(·, t) vanishes om on for a.e. t E (0, T), by the embedding of Proposition 3.1 Chap. I,
IIwll:~.aT :S 'Yllwllt-p(aT)'
'Y = 'Y (data) .
Combining these remarks in (17.5) we conclude that there exist two constants Co ,C1 , depending only upon the data such that
IIwll:~.aT :S Co +Clln,.I~lIwll:~.aT· If T is so small that, say then
IIwll:~.aT :S 2Co • For arbitrary T> 0, the argument can be repeated up to covering the whole a finite number of steps.
flT
in
18. Bibliographical notes 155
18. Bibliographical notes The sup-bounds of §3 are essentially due to Porzio [87]. They follow a parabolic version of DeGiorgi iteration technique (see [67]) and remain valid even in the 'linear' case p = 2. An effort has been made to trace the dependence of the various constants upon the size of the domains where the estimates are derived. We have also computed how the various estimates deteriorate when t -+ O. In the case of homogeneous sbUctures for degenerate equations (see §4), the interpolation estimate (4.1) is of particular interest. It reveals a behaviour dramatically different from the linear case p= 2. An estimate of this kind (i.e., for small e) had been proved by Moser [83] for solutions of linear parabolic equations with measurable coefficients. Generalizations to quasilinear equations with 'linear' growth p = 2 are in [7,97]. The global estimates in ET of §4-(IV) are taken from [41]. We have given a different and simpler proof. For the porous medium equation with power-type non-linearities, estimates of the same nature have been proved by BtSnilan-Crandall-Pierre [10]. Analogous estimates for general non-linearities appear in [4]. Still in the context of the porous medium equation, rather precise local sup-estimates have been recently obtained by Andreucci [3]. For equation with singular structure (1 < p < 2), the theory of local and global boundedness has started only recently in [42] and [43]. Improvements to equations with general structures are in [87]. The results of Theorems 5.1-5.3 are sharp. They will play a central role in the Harnack estimates of Chap. VII. The integrability condition (5.2) is sharp as shown by the counterexample in §13 of Chap. XII. The arguments of §17 appear in [101].
VI Harnack estimates: the case p > 2
1. Introduction We will establish a Harnack-type estimate for non-negative weak solutions of degenerate parabolic equations of the type (1.1)
U {
E
Gloc (0, Tj L~oc(ll}) n Lfoc (0, Tj WI!;:(ll)) , p> 2,
Ut -
div IDulp-2 Du = 0, in llT.
Since the equation is invariant by the scaling x - hx, t - hPt, h > 0. it may seem plausible that the Harnack estimate of Hadamard [50] and Pini [86],(1) would hold in the geometry of the cylinders (1.2)
This is not the case, as one can verify for the explicit solution (x, t) - 8(x, t) introduced in (4.7) of Chap. v. Let (xo, to) be a point of the free boundary {t= Ixl>'}, and let p> 1. Then if to is sufficiently large, the ball Bp(x o ) taken at the time level to - pP intersects the support of x - 8 (x, to - PP) in a open set. Therefore sup 8(x,to - PP) > Bp(zo)
°
and
This reveals a gap between the elliptic theory and the corresponding parabolic theory. Indeed non-negative weak solutions of (1) See (2.2) in the Preface.
2. The intrinsic Harnack inequality 157
div IDul p - 2 Du
= 0,
uE
w,!:(n),
p
> 1,
satisfy the Harnack inequality, (2) whereas solutions of the corresponding parabolic equation (1.1) in general do not. Let u be a non-negative local solution of the heat equation in nT. Then for all e> 0 there exists a constant 'Y depending only upon N and e, such that for every cylinder Qp(xo, to) C nT and for every uE (0, 1),
sup
(1.3)
Q .. ,,(Xo,to)
u
< -
'Y
N
2
(1 - u).;¥
(
ff u dxdr ) ~ , E
Q,,(xo,to)
where Qp(xo, to) is defined by (1.2) with p=2. This local sup-bound of the solution in tenns of the integral average of a small power of u, is a key fact in Moser's proof of the Harnack estimate. An estimate of this kind does not hold for solutions of (1.1) and it is replaced by the more structured inequality (4.1) of Chap. V. A study of [83) however reveals that (1.3) continues to hold for sufficiently smooth solutions of (1.4)
With this in mind one may heuristically regard (1.1) as it were (1.4) written in a time scale intrinsic to the solution itself and, loosely speaking, of the order of t [u(x, t)]2- p. Next we observe that (2.2) in the Preface is equivalent to (1.5)
The Harnack estimate of Krylov and Safonov [64) for non-divergence parabolic equations is given precisely in this fonn. This suggests that the number [u(xo, t o)]2-Pis the intrinsic scaling factor and leads to conjecture that non-negative solutions of (1.1) will satisfy the Harnack inequality with respect to such an intrinsic time scale.
2. The intrinsic Harnack inequality The following theorem makes rigorous the heuristic remarks of the previous section. THEOREM
2.1. Let u be a non-negative weak solution off1.1). Fix any (xo, to) E
fl.r and assume that u(xo, to) > o. There exist constants 'Y> 1 and C > 1, depending only upon N and p, such that (2.1) (2) See [82,92,96J.
158 VI. Harnack estimates: the case p> 2
where (2.2)
provided the cylinder (2.3)
Q4p(9) == {Ix - xol < 4p} x {to - 49, to + 49}
is contained in nT •
t o+ 0
I.
I p
4p Figure 2.1 Remark 2.1. The values u(xo, to) are well defined since u is locally Wider continuous in nT. Remark 2.2. The constants "'( and C tend to infinity as p -
00.
However they are
'stable' asp'\.2, i.e., lim ",(N,p), C(N,p)
",,"2
= "'(N, 2), C(N, 2) < 00.
Therefore by letting p _ 2 in (2.1) we recover. at least fonnally, the classical Harnack inequality for non-negative solutions of the heat equation. Such a limiting process can be made rigorous by the C,~ (nT ) estimates of Chap. IX. In Theorem 2.1 the level 9 is connected to u(%0' to) via (2.2). It is convenient to have an estimate where the geometry can be prescribed a priori independent of the solution. This is the thrust of the next result which holds for all 9 > O. THEOREM
such that
2.2. There exists a constant B > 1 depending only upon N and p.
3. Local comparison functions 159 (2.4)
V (xo, to) E nT. Vp, (J
u(xo, to)
> 0 such that Q4p«(J) c nT,
~ B { (~) ~ + (;) NIp [B!?!o) u(·,to + (J)f/J],
where A = N(P - 2) + p.
(2.5)
Remark 2.3. Inequality (2.4) holds for all p E (2,00), but the constant B is not 'stable' as p '\,2, i.e., lim B(N,p) = 00. p'\.2
In (2.4) the positivity of u(xo, to) is not required and (J > 0 is arbitrary so that Theorems 2.1 and 2.2 may seem markedly different. In fact they are equivalent, i.e., PROPOSITION 2.1.
Theorem 2.1 <=> Theorem 2.2.
In view'of Remark 2.3, the equivalence is meant in the sense that (2.1) implies (2.4) in any case and (2.4) implies (2.1) with a constant 'Y = 'Y( N, p) which may not be 'stable' as p'\,2. A consequence of Theorem 2.2 is COROLLARY
2.1. There exists a constant B > 1 depending only upon N and p,
such that (2.6) V(xo, to) E nT, Vp, (J
ju(z,t.)dx S
> 0 such that Q4p«(J) c nT,
B{ (~t + ( ; t P [U(Z.,t.+9)J VP }
B,.(zo)
2-(i). Generalisations All the stated results remain valid if the right hand side of (1.1) contains a forcing term f, provided (2.7)
q> (N +p)/p
and / is non-negative. We will indicate later how to modify the proofs to include such a case.
3. Local comparison functions Let p> 0 and k > 0 be fixed and consider the following 'fundamental solution' of (1.1) with pole at (x, t):
160 VI. Harnack estimates: the case p> 2
(3.1)
_
Blc,p (x, tj x, l) ==
~ kpN { ( Ix - xl ) ;f-r } PSN/>'(t) 1 - S1/>'(t) +'
where A is defined in (2.5) and (3.2)
S(t) = b(N,p)kP- 2pN(p-2)(t -l) + p>',
b(N,p)
= A ( p~ 2 )
t
~
f,
P-l
By calculation, one verifies that Blc,p (x, tj x, l) is a weak solution of (1.1) in RN x {t > l}. Moreover for t = f it vanishes outside the ball B p (x) and for t > f the function x- Blc,p (x, tj x, l) vanishes, in a C1 fashion, across the boundary of the ball {Ix - xl < S1/>'(t)}. One also verifies that
Blc,p (x, t; x, l) ~ k, and that forf~t~t*, the support of Blc,p (x,t;x, l)
D* == {IX -
xl ~ Sl/>'(t)} x [f, t*],
is contained in the cylindrical domain Q* == BS1/~(t.) (x) x [f, t*]. If u is a non-negative weak solution of (1.1) in Q* satisfying
u(x,l) ~ k
for
Ix - xl < p,
then
u(x, t) ~ Blc,p (x, t; x, l) ,
'v'(X,t)EQ*.
This is a consequence of the following comparison principle.
nT satisfying u,v E C (O,T; L2(n») n £P (O,T; W1,p(n» n C (liT) { u ~ v on the parabolic boundary of nT. Then u~v in nT . LEMMA
3.1. Let u and v be two solutions of (1.1) in
PROOF: We write the weak form of (1.1) for u and v in terms of the Steklovaverages, as in (1.5) of Chap. II, against the testing function
[(v - U)h]+ (x, t)
=[
*!
t+h
1
(v - u)(x, T)dT +'
h E (0, T), t E [0, T - h).
Differencing the two equations and integrating over (0, t) giv\ 'S
3. Local comparison functions 161
j[(v n
=
-2//n.
U)hJ! (x, t)dx -
j[{V n
U)hJ! (x,O)dx
[lDvl p - 2 Dv -IDuI JI - 2 Du] h·D [{v - U)h]+ dxdr.
As h -+ 0 the second tenn on the left hand side tends to zero since (v - u) + E C (liT). Applying also Lemmas 3.2 and 4.4 of Chap. I we arrive at
/{v n
u)~{x, t)dx
= -2
(IDvIJl-2 Dv -IDuI JI - 2DU) ·D{v - u)dxdr :5 O.
//
n.n(v>u)
3-(i). Local comparison junctions: the case p near 2 The next comparison function is a subsolution of (1.1) for p > 2 and for p < 2 provided p is close enough to 2. For definiteness let us assume p E [2,5/2J and consider the function (3.3)
t ~
(3.4)
t,
where the positive numbers II and ~(II) are linked by
~(II)
(3.5)
= 1-
v(p - 2) . P
Introduce the number
(3.6)
p(v)
= 4(1 + 2v)/{1 + 4v),
and observe that (3.7)
1
1
4 :5 ~(v) :5 2'
for
p E [2, p(v)J .
LEMMA 3.2. The number v> 1 can be determined a priori only in terms of N aTul independent ofpE [2, 5/2], such that Qk,p is a classical subsolution of
!Qk,P - div (lDQk,pIJl-2 DQk,P) :5 0 PROOF:
For (x, t) ERN x {t>t}, set
in RN x {t >
t}.
162 VI. Harnack estimates: the case p> 2
IIzll ==
t(~) ~~) , :F == (1 - IIzll;!r ) + '
a ==
(p ~ 1)
2.
Then, by calculation,
(3.8)
£* (g/c,P)
= -v:F;!r + NaP-1:F - -LaP-1Ilzll;!r + ~(v)a:F;!r IIzlI;!r. p-l
Introducing the set
£1 ==
[
--Z..1( N(P-l»)] IIzlIFT ~ 2 1+ N(P-l)+p
,
we have
:F < p - 2[N(P - 1) + p]
in £1,
and therefore by (3.5)
r: (g/c,P) :5 -v:F;!r + NaP- 1 + ~(v)a:F;!r - aP -
1
(N + -L) IIzll;!r p-l
:; ...-1 [-\(v) (N(P -"1) + p) o!t - 2(P~ 1)1 [H N(P + ~2(P"":'~~I)1 -"1)
:;; .p--l
:5 aP-
1
p
t-
(~ 2(P~ 1») < O. -
Within the set
£2
1( N(P - 1) )] == [ IIzll ;!y < 2 1 + N(P _ 1) + p p-
,
we have
:F> P - 2[N(P-l)+p] It follows from (3.5) and (3.7) that
p );!r £* (g/c,p) :5 -v ( 2[N(P _ 1) + p] p
:5 -v ( 2[N(P _ 1) + p]
);!r
+ aP - 1 N + ~(v)a a
-2
+ p [NpaP + 1] .
4. Proof of Theorem 2.1 163
Choosing (3.9)
V==
~ [NpaP-2+1] (2[N(P-l)+P])~,
max
pe[2,5/2) P
P
we have in either case
One verifies that for t = t (hc,p (XI tj X, f) $ k,
and that for t $ t $ t*, the support of (hc,p, 'R,*
== {Ix - xl < L'''(II) (t) } x[t I t*],
is contained in the cylindrical domain C* ==
{Ix - xl < L'''(II)(t*)} x[t
I
t*].
Therefore if u is a solution of (1.1) in C* such that U(X,
f) ~ k
then U(X,
t) ~ (h:,p (x, tj ft, f)
in C*.
Remark 3.1. The same proof shows that gkr is a sub-solution of 0.1) also for P< 2, providedp is close to 2. Precisely ifpE (4 - p(v),2) ..
4. Proof of Theorem 2.1 Let (xo, to) E nT and p > 0 be fixed, assume that U(XOI to) > 0 and consider the box
where C is a constant to be detennined later. The change of variables X-Xo
x---P
I
maps Q4p into the box Q == Q+ U Q- , where
Q+ == B4 X [0, 4C),
Q- == B4 X (-4C, 0].
164 VI. Harnack estimates: the case p> 2
(0,0) I
, I
•
--------------------~ 1
c/o
-4
4
Figure 4.1 We denote again with x and t the new variables. and observe that the rescaled function
v(x,t)
tpp) = U (x o,1 t) U ( xo+px, t o+ [U (xo,to )IP-2 0
is a bounded non-negative weak solution of {
div (lDvlp-2 Dv) v(O, 0) = 1.
Vt -
=
°
in Q
To prove the Theorem it suffices to find constants 'Yo E (0, 11 and C > 1 depending only upon N and p such that
ill! v(x, C) 2: 'Yo· Construct the family of nested and expanding boxes T
E (0, I],
and the numbers
M.,. == sup v, Q...
N.,. == (1- T)-fl,
T
E [0,1),
where {J > 1 will be chosen later. Let To be the largest root of the equation M.,. = N.,.. Such a root is well defined since Mo = No. and as T /1. the numbers M.,. remain bounded and N.,. /00. By construction
Since v is continuous in Q there exists within Q.,." at least one point. say (i, t). such that v (i, t) = N.,." = (1 - To)-fl. The next arguments are intended to establish that within a small ball about i and at the same time-level f the function v is of the same order of (1- To)-fl. For this we make use of the R))der continuity of v and more specifically of Lemma 3.1 of Chap. III.
4. Proof of Theorem 2.1 165 Set
R = 1- 'To 2 ' and consider the cylinder with 'vertex' at (x, t)
By construction [(x, t)
+ Q (RP, R)] c Q!:tfa
and therefore
v ~ N~ = 213 (1- 'To)-p == w.
sup [(z,i)+Q(RP ,R»)
If A is the number determined by Proposition 3.1 of Chap. Ill, we may choose {J> 1 so large that (213 fA) > 1. Therefore the cylinder
[(x, t) + Q (aoRP, R)] , is contained in [(x, t)
:0 == (~r-2
= [213 (1
~'To)-pr-2 > 1
+ Q (RP, R»), and osc
[(z,i)+Q(ooRP,R»)
v < w. -
It follows that [(x, t) + Q (aoRP, R)] can be taken as the starting box in Lemma 3.1 of Chap. III. We conclude that there exist constants "( > 1 and 0:, Co E (0,1) such that for all rE (0, R].
We let r (4.1)
=u R and then choose u so small that for all {Ix - x I < u R}, v(x, t) ~ v (x, t) - 211+1"((1 - 'To)-pu Ot = (1 - 2p+1,,(uOt ) (1 - 'To)-p 1 = 2(1 - 'To)-p.
The various constants appearing in Proposition 3.1 and Lemma 3.1 of Chap. III, in our context, depend only upon N and p and are indePendent of v, (1) therefore the number u can be determined a priori only in terms of N, p and (J. We summarise: (1) See §3-(I) of Chap. III.
166 VI. Harnack estimates: the case p> 2 LEMMA
4.1. There exist a number u E (0,1) depending only upon N,p and P
such that
(4.2) Remark 4.1. The location of (x, t) and the number 'To (and hence R) are determined only qualitatively. However in view of (4.1) the number u is quantitatively determined as soon as P> 1 is quantitatively chosen.
4-(i). Expanding the positivity set We will choose the constants P> 1 and C> 1 so that the qualitative largeness of v(·, t) in the small ball BcrR(X) turns into a quantitative bound below over the full sphere Bl at some further time level C. This is achieved by means of the comparison functions of§3. Assume first thatpE [2, p(II)], where II is the number determined in Lemma 3.2, and consider the function (ik,p introduced in (3.3). with the choices 1
k
(4.3)
= -(I-or) 2 0
_
~
p=uR. '
At the time level t=C the support of x-(ik,p (x, Cj x, t) is the ball
where 'Y
= 'Y(u, II ) = 21(U)"¢l '2 .
Choose II
P = A(II) Since
Ixl <
1 and
t
and
E (-1,0]. these choices imply that the support of x -
(ik,p (x, Cj x, t) contains B2. and by the comparison principle
inf vex, C) ~ inf (ik,p (x, Cj x, t)
ZeBI
ZeBI
~
2-(1+2&1)
(i) "¢l {1 -
(~ ·)6}~ p-
== 'Yo· The various constants depend only upon N and p and are 'stable' as p'\. 2.
5. Proof of Theorem 2.2 167
Turning to the case p ~ p( 1/ ), we consider the comparison function Bk,p (x, t; x, l) introduced in (3.1)-(3.2), with the choice of the parameters k and p as in (4.3). At t =C the support of Bk,p (., C; x, l) is the ball
I_ - ;;1'< {b[~(I-
TO)--r-'(aR)N(P-') (C - i) + (aR)'}
= {b-yP-2 (1 where
-y(N, (3)
="21 (0') 2 N
T o )(N-.BHp-2)
(C -l)
+ (O'R)>-},
( )P-l
b=~ -p-
and
o
p-2
Choosing (4.4)
{3=N
and
3>-
C
= b-yp-2'
we see that the support of Bk,p (-, C; x, l) contains B2, and by the comparison principle, (4.5)
inf v(x,C) ~' inf Bk,p(X,C;x,l)
ZeBI
ZeBI
~ (2)-(1+'i') Gt {1==
Gtt'
-Yo·
Remark 4.2. These estimates involving the comparison function Bk,p hold for all p > 2. However as p'\. 2, the constant -Yo in (4.5) tends to zero. The purpose of introducing an auxiliary comparison function gk,p for p near 2 is to have the constants under control as p approaches the non-degenerate case p = 2. We also remark that gk,p is a subsolution of (1.1) only for p close enough to 2.
5. Proof of Theorem 2.2 Let (x o, to) E nT, p> 0 and (J> 0 be fixed so that the box Q4p((J) is contained in nT. We may assume that (x o, to) coincides with the origin and set u. == u(O,O). If C and -y are the constants detennined in Theorem 2.1, we may assume that (5.1)
Indeed otherwise
B
== (2C);f-J
,
168 VI. Harnack estimates: the case p> 2
and there is nothing to prove. By Theorem 2.1 and (5.1) u* ~ 'Yu(x, t*),
Consider the 'fundamental solution' Bk,p with pole at (0, t*) and with k='Y-1U*. By the comparison principle, at the level t=(J, we have (5.2)
u«,6);'
;t.; {1- (Sl~~I(t)t
r
~
"Ixl < p,
where
Here .x and b are defined in (2.5) and (3.2) respectively. It follows from (5.1) that
('Y~~2 + 1) p>' ~ S(t) ~ ('Y:- 2 + 2~ )u~-2 ( ; )
p>'.
Therefore (5.2) gives
pp)N!>.
( u(x,(J)~u~!>'"9
'Y1,
'Y1 == 'Y1(N,p),
and the theorem follows with
B
= max { 'Y-;>'!p; (20) ~ } .
We have shown that Theorem 2.1 implies the estimate of Theorem 2.2. To prove the equivalence of Proposition 2.1, assume that (2.4) holds true for all (J> 0 such
that Q4p«(J)
c nT . Choose
Then if Q4p«(J) c nT, (2.4) gives u(xo, to) ~
2B N (p-2)! >. in(f ) u(·, to + (J). Bp
Zo
5-(i). About the generalisations The only tools we have used in the proof are the HOlder continuity of the solutions of (1.1) and the comparison principle. The integrability indicated in (2.7)
6. Global versus local estimates 169
guarantees the local HOlder continuity.(l) Moreover the comparison principle remains applicable since J ~ o.
6. Global versus local estimates The assumption that· the cylinder Q 4p( 8) be contained in the domain of definition of the solution is essential for the Harnack estimates of Theorems 2.1 and 2.2 to hold. Indeed the function (x, t) - 8(x, t) introduced in (4.11) of Chap. V does not satisfy (2.4) for Xo = 0 and to arbitrarily close to zero. This is not due to the pointwise nature of (2.1) and (2.4). A Harnack inequality, with to arbitrarily close to zero, fails to hold even in the averaged form (2.6). To see this let 1£ be the unique weak solution of the boundary value problem in Q:= (O,I)x(O,oo), for all t ~ 0,
1£t - (I1£z IP-21£z)z = 0 { 1£(0, t) = 1£(1, t) = 0 1£(,,0) = 1£0 E C:'(O, 1) 1£o(x) E [0,1], "Ix E (0,1)
('P)
and 1£o(x) = 1 for x E
U, i)·
We claim that -1 1£ 1£t > - - - p-2 t
(6.1)
in V'(Q).
Let us assume (6.1) for the moment. Since 0 ~ 1£ ~ 1, by the comparison principle (6.1) implies that _ - (11£.IP 21£_) < •
• z -
1 (p - 2)t'
t> O.
At any fixed level t, the function x-1£(x, t) is majorised by
v(x,t) =
-yx6
~,
ti=I
OE ( P-l -p-,1 ) ,
(-y0)P-1 (1 - 0)(P - 1)
~ ~2' p-
Indeed 1
- (Ivz IP- 2vz)z ~ (p _ 2)t Therefore for every 0 E (
and
v(O, t)
= 0,
v(l, t)
> O.
7' 1) there exists a constant C =C (0), such that C(O) 1£ (!, t) ~ t 1/(p-l)'
(1) See the structure conditions in §1 of Chap~ II, Theorem 1.1 of Chap. III and Theorem 3.1 of Chap. V.
170 VI. Harnack estimates: the case p> 2
Now assume that (2.6) holds for to=O, x o 1 ~ canst
=!, 9=t and p= 1. Then for t>I
(cp!-, + c*) -- 0
as t - - 00.
The proof of (6.1) is a particular case of the following
6-(i). Regularising effects PROPOSITION 6.1. Let u E V (0, Ti WJ,P(I1» be the unique non-negative weak solution of
{
(6.2)
Ut - div IDul p - 2 Du = 0, u(·, 0) = U o E L2(11),
in I1T' p> I, U O ~O.
Then ifp>2. -1 u
(6.3)
Ut> - - -
- p-2 t
in 1)'(11) a.e. t
>
0,
andifI
Ut
1 u < -- - 2-p t
in 1)'(11) a.e. t > O.
PROOF: We only prove (6.4). By the homogeneity of the p.d.e., the unique solution v of (6.2) with initial datum
v(·,O)
= kp!-,uo,
k > 0,
is given by (x, t) - - v(x, t) = kp!-, u(x, kt).
If k ~ 1, v(·, 0) ~ U o and v(·, t) ~ u(·, t) in 11, Vt E (0, T). Fix t E (0, T) and let k = (1 + ') for a small positive number h. Then u(x, t
+ h) -
u(x, t)
= u(x, kt) - u(x, t) = k~kp!-,u(x,kt) - u(x,t) = k~v(x,t) - u(x,t) ~ (k~ - I)u(x, t).
By the mean value theorem applied to ( k ~ (6.5)
1),
h =.! u(x, t) u(x,t + h) - u(x,t) ~ 2 _ p (1 +e)2-lit-
for some ( e (0, ~).If h <0, and Ihl « I, we have k < 1, v(·,O) ~uo and (6.S) holds with the inequality sign reversed. Divide by h and in (6.5) take the limit in 'D'(I1) as h-+O.
7. Global Harnack estimates 171 Remark 6.1. In the proof of the proposition, the homogeneity of the operator and the positivity of the initial datum, are essential.
7. Global Harnack estimates The averaged Harnack estimate (2.6) holds with to arbitrarily close to zero for non-negative local solutions of (1.1) in the strip ET == R N X (0, Tj, i.e.,
{u
(7.1)
E
Cl~ (0, T; L~oc(RN)) n L:oc (0, T; Wj!;:(RN)) ,
Ut - dlV
(/Du/ p - 2 Du)
=
°
10
ET •
THEOREM 7.1. Let u be a non-negative solution 01(7.1) in E T . There exist a constant B> 1 depending only upon N and p. such that
(7.2) 'V (xo, to) E ET , 'V p, (J >
f
u(x,
°such that to + (J < T,
to)dx~B{ (~)~ + ~)N/P[B!?L) u(·, to + (J)] AlP}.
Bp(XD)
Inequality (7.2) is more general than (2.6) in that the value u(Xo, to + (J) is replaced by the infimum of u over the ball Bp(x o) at the time level to + (J. In (7. 1) no conditions are imposed on x-u(x, t) as /x/- 00 and no reference is made to possible initial data. The only global information is that the p.d.e. is solved in the whole strip ET. Nevertheless (7.2) gives some control on the solution u as /x/- 00, namely, COROLLARY
(7.3)
7.1. Every non-negative solution 01 (7.1) in ET satisfies
'VxoERN, sup
'Vr>O,
sup
O<.,.S;T-E p?r
'VeE (O,T)
! ~dx~
Bp(XD}
UXT P
P
B --.L
e~
[
1+
T r
(-) P
--.L p::-2
1Alp
u(xo,T-e)
to=TE (0, T-e), divide by p~ and take the supremum of both sides for p ~ rand T E (0, T - e).
PROOF: Apply (7.2) with
172 VI. Harnack estimates: the case p> 2
8. Compactly supported initial data The proof of (7.2) will be a consequence of the following: PROPOSITION
8.1. Let v be a non-negative solution 0/ the Cauchy problem
V
(8.1)
{
E C (R+j L2(RN))
n V (R+'i W1,P(R N )) ,
-divIDvl p - 2 Dv=0 inEoo ==RNxR+, EC(Br) /orsomer>O, ( ) v',O = Vo ~ oand { _ . RN,-B
Vt
=0
In
r'
There exists a constant B=B(N,p) > 1. such that for all 8>0,
Inequality (8.2) can be regarded as a special case of (7.2) when additional infonnation are available on the initial datum. Basic facts on the unique solvability of (8.1) are collected in §12. We assume the Proposition for the moment and proceed to gather a few facts about v. LEMMA 8.1. For each t E R+, the function x--+v(x, t) is compactly supported in RN, i.e.,
(8.3)
V T E R +, 3 R
= R(T) > 0
supp{v(·,t)} C
BR(T),
Vt
such that E (O,T).
Moreover the 'mass' is conserved, i.e., (8.4)
!
v(x,t)dx
= !vo(x)dx,
aN
vt ~O.
Br
PROOF: Consider the function 8k,p introduced in (3.1)-(3.2), with p = 2r and (x, f) == (0,0). For t=O and Ixl
8 ..... (.,0; 0, 0)
~ ~
+-m
p } l=l
supVo, Br
provided k is chosen sufficiently large. By the comparison principle, v 5 8k,p' The second statement follows from the first by integrating the p.d.e. over RN x (0, t). Remark 8.1 (Existence of solutions). In view of (8.3), a solution of the Cauchy problem (8.1) can be detennined by fixing any T > 0 and solving the p.d.e. in
8. Compactly supported initial data 173 the bounded domain nT == B R(T) X (0, T) with homogeneous boundary data on Ixl = R(T) and the same initial conditions as in (8.1). It follows from the comparison principle of Lemma 3.1 that the solution is unique. This construction and Proposition 6.1 also give the regularising inequality (8.5)
Vt
-1 v
> ---p-~t'
in V' (Eoo) ,
and the estimate supv(·, t) $ SupVo'
(8.6)
RN
COROLLARY
(8.7)
-Br
8.1. The quantities
-
IIvll r = sup sup
J
v(x,t)
p>'-!(
tER+ p?,r
P
-2)
dx,
A = N(p - 2) + p,
Bp
(8.8)
f(t) = sup
{7'N!~sup IIv(.,r)lIoo,B
O
p?,r
p }
pp!(p-2)
are finite. Moreover there exists constants 'Y. and 'Y depending only upon N,p. such that (8.9)
for all 0 < t < 'Y. Ilv"I~-P, pp!(p-2)
IIv(.,t)lIoo,Bp $'Y PROOF:
tN!~
andfor all p ~ r,
Wv"I~!~,
A=N(p-2)+p.
The estimate (8.9) is the content of Theorem 4.5 of Chap. V.
8-(i). Proof of Theorem 7.1 assuming (8.2) It suffices to prove (7.2) for (xo, to) == (0,0) and (J E (0, T). Fix p = r > 0 and consider the Cauchy problem (8.1) with initial datum if x E Br, if x E R
Nr•
\B
By the results of Chaps. III and V, the solution u is locally bounded and locally HOlder continuous in E T • Therefore up to the translation that maps (xo, to) into the origin, u(·, 0) is continuous in B r . By (8.3) the comparison principle of Lemma 3.1 can be applied over the bounded domain B R(T) X (0, T) to yield v $ u. Then (7.2) follows from (8.2).
174 VI. Harnack estimates: the case p > 2
9. Proof of Proposition 8.1 9.1. There exists a constant 'Y ='Y(N, p) such that
LEMMA
for all 0 < t < 'Y.llvll~-P and for all p ~
(9.1)
T,
t
I I IDvlp-1dxdT
~ 'Yttp1+;!J Ilvlll!+~.
o Bp PROOF: The calculations below are fonnal in that they require v to be strictly positive. They are made rigorous by replacing v with v + e and lelling e - O. Let x - (x) be a non-negative piecewise smooth cutoff function in B2p that equals one on Bp and such that ID(I ~ 1/p. By the HOlder inequality t
I jlDvIP-1(P-1dxdT
o Bap
~I
t
I(T7IDvIP-IV-27(P-l) (T-7 v2 7)dxdT
o Bap
~
(i
(i T-~v2~
c!
I Tl/PIDVIPV-2/P(PdxdT)
P
o Bap
I
1
dxdT)
P
0 Bap
== [J1 (t)] ~ [J2(t)]; . To estimate J1(t), in the weak fonnulation of (8.1) we take the testing function I{J == tl/pvl-2/p(p to obtain
P
2(P-l)
It;v2~(PdX + P -
t
p
2 IIT;IDvIPv-2/P(PdXdT 0 Bap
B2p
t
=p I IT;IDvIP-2vl-2/P(P-IDV.D(dxdT
o Bap t
+ 2(P 1-1)
If.1
7"P - 1V2c! p (PdxdT
o B 2p
.
In the estimates below 'Y denotes a generic positive constant that can be detennined a priori only in tenns of N and p and that might be different in different contexts. By Young's inequality and the structure of the cutoff function (
9. Proof of Proposition 8.1 175
I t
P
11'; IDvlp-2Vl-2/p(P-1 Dv· D(dxd1'
o Bap
We conclude that there exists a constant "Y="Y(P) such that
Estimating L i , i = 1, 2, separately we have L1
:::;"Ypl+~/t 1''4!-1 (1'N/>. IIv(" 1')IIoo'Bap)(P-a~P+l) (2p)~
\
o :::;
"Ypl+~
I t
LI h
1'£t!-1 [/(1')] (p-a~Ptl)
v(x, 1') dx) d1'
p~
Ilvll r d1'
o :::; "Y pl+~
t£t! [/(t)] (p-a~Ptl) Ilvll r .
By Corollary 8.1
I(t) :::; "Y Ilvll~/'\ Therefore for all such t, L1 :::; "Y pl+~ tf :::;
(tlllvll~-2) f IIvll!+~ 1+ E.::.! "Y p1+-;=f tx Ilvll r ---x-. ).
1.
Next L2 :::;
"Ypl+~ It1'*-1 (1'N/>' IIv("(2p)~ 1')lIoo,Ba ~ p )
o
:::; "Y pl+~
t* Ilvlll!+~ .
On the other hand J2(t)
== L2 and the Lemma follows.
LI h
v(x,1') dx) d1' p~
176 VI. Harnack estimates: the case p> 2
Remark 9.1. The estimates above show that 'Y ='Y( N, p) /00, as p \. 2. Remark 9.2. The proof is independent of the fact that the initial datum is of compact support and that v is a solution in the whole 1:00 • The lemma continues to hold for every non-negative solution in 1:T for some T > 0, provided the quantities
sup
sup
Or
lIu(·, T)lIoo,B~ 2 ppl(p-)
are finite for all E E (0, T). The conclusion will hold for all times
o < t :5 'Y.llullr,T-E.
(9.1)'
Remark 9.3. Lemma 9.1 is independent of the homogeneous structure of the p.d.e. Indeed it continues to hold, in the same form, for equations with homogeneous structure as in (2.8)-(2.9) of Chap. V, provided the analog of Corollary 8.1 is in force. A version of this Corollary can be proved, by essentially the same technique, for solutions of equations with general structure such as (2.1) of Chap. V.
Remark 9.4. The functional dependence upon t on the right hand side of (9.1) is optimal, as shown by the following example. The family
(9.2)
.>. -NI.>.
Br(x,t)={t+r)
'Yp =
p-2 ( ~1)~ -p-'
{l-'Y
p (
p
IX I (t+r'>']
...L..}~ p-l
11'>')
,
+
> 2; t, r > 0,
solves (8.1) with initial data Brh 0) supported in the ball B r. By calculation we have, for all p ~ r
! t
!IDBrIP-ldxdT = 'Y(N,p) [t + r'>'] 11'>' - r,
o B2(1 where 'Y(N, p) is an explicit constant independent of rand t. The assertion follows by letting r -+ O. Let Eo denote the integral average of the initial datum, i.e.
(9.3)
By the conservation of mass (8.4)
10. Proof of Proposition 8.1 continued 177
Ilvll r
=sup sup p-~fv(x,t)dx teR+ p?r
B"
$ sup sup teR+ p>r
p-~ f
-
v(x, t) dx
RN
--r-~Eo·
Therefore Lemma 9.1 can be rephrased as LEMMA
(9.4)
9.2. There exists a constant "Y="Y(N,p) such that for all 0
~
(9.5)
rP
< t < "Y. EC- 2 '
andfor all p ~ r
t
1
fflDvlP-ldxdT $"Y
(:p)X (;)~ E~+~.
OB"
1O. Proof of Proposition 8.1 continued Let ( be the standard cutoff function in B2r that equals one on B r . In the weak formulation of (8.1) take x -+ (P (x) as a testing function and integrate over B 2r X (0, to), where (10.1)
and e is a small positive constant to be chosen. Making use of (9.5) we obtain
f B2..
v(x, to) dx
~
2- N
f
B..
Vo dx -
~
to
f
f
IDvl p - 1dx
OB2 ..
~ 2- N Eo - "Y (e"Y.)l/~ Eo -- 2-(N+l) E 0, for the choice
We summarise:
178 VI. Harnack estimates: the case p> 2 LEMMA 10.1. There exist a constant c. that can be determined a priori only in terms of N and p. such that
f
v(x, to) dx
~ 2-(N+l) Eo,
B2r
Since v is continuous, there exists some x E B2r such that (10.2) Next we apply the Harnack estimate of Theorem 2.1. For this, construct the cylinder
Q_ B
=
(-)
46r X
{ X
4C (6r)"
4C (6r)"
to - [v(x, t o)],,-2 ' to + [v(x, to)]"
} 2
'
where C is the constant determined in Theorem 2.1 and 6 > 0 is to be chosen. Such a box is contained in Loo if
t > o -
4C(6r)"
v x,to )],,-2·
[(_
Using (10.1) and (10.2) we see that this is the case if
r"
,. --> E~-2 -
4C(6r)"2(N+l)(p-2)
g-2
E'II
Therefore Q C Loo for the choice
It follows from Theorem 2.1 that
"{Ix - xl < 6r} (10.3)
_
at the time level t v(x, t) ~
-y-1
= to +
4C(6r)" [v(x, to)]
,,-2
v(x, to)
~ 2-(N+1)-y-1
Eo
== coEo· Therefore we have located a ball or radius 6r about x and at the time level f where v is bounded below by eoEo. In view of(10.1) and (10.2) the time level tis bounded above by (l0.4)
11. Proof of Proposition 8.1 concluded 179
11. Proof of Proposition 8.1 concluded Let (J > 0 be fixed. We may assume that (11.1) Indeed otherwise
Eo:5 B
( 8);;!-' rP
,
and (8.2) becomes trivial. We will expand the bound below on v given by (10.3), up to the time level 8 over the ball B r • Consider the 'fundamental solution' 8/c,p introduced in (3.1)-(3.2), with pole at (x, t) and p=6r,
By the cQmparison principle, (11.2)
vex, t) ~ 8 co E o .6r (x, tj x, t) ,
V x E R N , Vt ~ f.
Let us estimate below the right hand side of (11.2) at the time level t=8. First by (10.4) and (11.1) - 1 8-t>-8 - 2 ' therefore the support of X-+8co E o .6r (x, 8j x, t) will cover the ball B 4r about the origin if
8(8)
= b(eoEo)p-2 (6r)N(p-2) (8 -
t) + (6r)'\
~ ~ (c oE o)P-2 (6r)N(p-2) 8 ~ (8r)".
This will occur if (11.3) We may assume that (11.3) is in force and estimate above 8(8) :5 b [eo6Nt-2 E:- 2 r N(p-2) 8 + 6'\r N(p-2) r P
:5 {b [eo 6N
t- 2 + ~:} ~-2 r N(p-2) 8
== 61 E:- 2 r N(p-2) 8. We return to (11.2). These estimates imply that for all x E Br for t =8
180 VI. Harnack estimates: the case p> 2
Therefore
f
Eo ==
Vo dx :$ 6;>'lp
(
0 ) NIII [
rP
W! v(·, 6}] >'111 ,
Br
and the Proposition follows with B=max{(2Bl}~;
BF; 6;>'11I}.
12. The Cauchy problem with compactly supported initial data 1be proof of (7.2) is based on comparing u with the unique solution of the Cauchy problem
(12.1)
V E C (R+; L2(RN}) n V (R+; Wl'II(RN» , { Vt - div IDvllI-2Dv = 0 in !Joe == RN xR+, EC(Br} forsomer>O, v (,,0) = Vo ~ 0 and { _ . RN\B = 0 10 r.
Such a problem plays a role also in the theory of Harnack estimates for nonnegative weak solutions of (1.1) in the singular case 1 < p < 2. The Cauchy (RN) and all p > 1 will be studied in problem for general initial data in Chaps. XI and XII. To render the theory of Harnack inequalities self-contained we briefly discuss the unique solvability of (12.1) for all p > 1. First, the notion of solution is:
Lloc
(a) For every compact subset JC c RN and for every T > 0, u is a local solutions of the p.d.e. in JC x (0, T), in the sense of (1.2)-(1.4) of Chap. II. (b) v(·, t}
Vo in L2(RN}.
-+
PROPOSITION
12.1. There exists a unique solution to (12.1) for all p > 1.
PROOF: For n = 1, 2, ... let Bn be the ball of radius n about the origin and consider the boundary value problems
E C (R+; L2(Bn» n V (R+; W:'II(Bn» Vn,t - div IDvn lp-2 DVn = 0 in Bn x (0, n), Vn
(12.2)
{
vn(-,O} = Vo E L2(Bn}.
12. The Cauchy problem with compactly supported initial data 181
The functions Vn vanish in the sense of the traces on Ixl = n. We regard them as defined in the whole Eoo by extending them to zero for Ixl > n. The problems (12.2) can be uniquely solved by a Galerkin(l) procedure and give solutions Vn satisfying (12.3)
The sequence {v n }n6N is equibounded(2) in E oo , and uniformly Holder continuOUS(3) in Ex (e, 00) for all e > O. In the weak formulation of (12.2), we take the testing function (v n + e )p-2Vn , modulo a Steklov average. Letting e - 0 gives
VneN.
(12.4)
Therefore
Vn e LP (R+; W1,P(R N )) uniformly in n. A subsequence can be selected and relabelled with n such that Vn - v uniformly on compact subsets of Eoo and weakly in LP (R+; Wl'P(RN)). The limit v is in the function space specified by (12.1), it is HOlder continuous in Ex (e, 00) for all e > 0, and it satisfies the p.d.e. weakly in Eoo. To prove this we select a compact subset IC c RN and some T > O. Then if n is so large that IC C B n , we write (12.2) weakly against testing functions supported in ICx (0, T). The limiting process can be carried on the basis of the previous compactness and the non-linear term is identified by means of Minty's Lemma. (4) It remains to show that v takes the initial data Vo in the sense of L2(RN). Let 1/ e (0, 1) be arbitrary and let VO,'l be a mollification of Vo such that
!lvo - VO,'l!l2,RN
--+
0
1/ '\. O.
as
In the weak formulation of (12.1), take the testing function Vn - vo,'l modulo a Steklov average. If n is so large that supp[VO,'l] C B n , we obtain
jlvn - VO,'l12(t)dx :::; !lvo -
VO''lIl~,RN + 'Y j
RN
t
jIDVo''lIPdxdT,
"It> 0,
ORN
for a constant 'Y depending only upon p. Letting n - 00,
IIv(·, t) -
voll~,K: :::; 211vo - VO''lIl~,RN + 'Y j
t
jIDVo''lIPdXdT,
"It> 0,
ORN (1) See J.L.Lions [73] or Ladyzhenskaja-Solonnikov-Ural'tzeva [67]. (2) By the weak maximum principle of Theorem 3.3 of Chap. V. (3) By the HOlder estimates of Theorem 1.2 of Chap. III and Theorem 1.2 of Chap. IV. (4) See G. Minty [78].
182 VI. Harnack estimates: the case p> 2
for all compact subsets IC eRN. From this, ]~ IIv(" t) - Vo1l2,K: = 211vo - Vo,,,1I2,RN,
for all '1 E (0,1).
To prove uniqueness we first write the p.d.e. satisfied by the difference w =VI - V2 of two possibly distinct solutions originating from the same initial datum vo , i.e.,
C (R+;L2(RN») nLP (R+;Wl,P(RN »), { Wt - div (IDvIIP-2 DVI - IDV2l p - 2DV2) = 0, in 1:00 , w(',O) = 0, in L~oc(RN). wE
(12.5)
In the weak formulation of (12.5), take the testing function w(, modulo a Steklov average, where x --+ (x) is a non-negative piecewise smooth cutoff function in the ball B2R that equals one on BR and such that ID(I ~ 1/R. This gives, for all t>O,
! t
~!IWI2(t)dX + Bil
!
OB2Il
! t
=-
!
OB21l
1be second integral on the left hand side is non-negative(l) and it is discarded. 1berefore
!
IwI2(t)dx
~ IIwll p,l;ao (IIDVlllp,l;; + IIDV2l1p,l;ao)P-1
Bil
Uniqueness follows letting R --+ 00. A similar argument proves the following weak comparison principle. LEMMA 12.1. Let Vi, i = 1,2, be two weak solutions to (12.1) originating from bounded and compactly supported initial data Vo,i, i= 1, 2, satisfying Vo,l ~Vo,2' Then VI ~ V2 in 1:00 , PROOF: In the weak formulation of the difference w = VI - V2, take the testing function w+( modulo a Steklov average.
(1) See Lemma 4.4 of Chap. I.
13. Bibliographical notes 183
13. Bibliographical notes In the classical work of Moser [81,82,83], ~e HOlder continuity is implied by the Harnack estimate. Conversely we use the HOlder estimates of Chaps. III and N to establish a Harnack inequality. This point of view, even though not explicitly stated, is already present in the work of Krylov and Safonov [64]. The results of §2 have been established in [40]. A version of these holds for non-negative weak solutions of the porous medium equations u E C' oc
(13.1)
{
!
(0, Tj L~oc(fi») , urn
u - Llurn
=0
in fiT, m
E
L~oc (0, Tj W,!;;(fi») ,
> 1.
In particular the intrinsic Harnack estimate takes the form
13.1. Let u be a non-negative weak solution 0/ (13.1). Fix any (xo, to) E fiT and assume that u(xo, to) > O. There exist constants 'Y > 1 and C> 1, depending only upon Nand m, such that THEOREM
(13.2)
provided the cylinder
Q4p(9) == {Ix - xol < 4p} x {to - 49, to + 49} is contained in fiT. A version of Corollary 2.1 for (13.1) appears in [6]. For the remaining results we refer to [40]. The 'fundamental solutions' Blc,p are due to Barenblatt [8]. The comparsion function (ilc,p for p close to 2, is introduced in [40]. The technical device of the family of expanding cylinders Q.,. in §4 appears in Krylov -Safonov [64]. The regularising effects of Proposition 6.1 are due to Benilan and Crandall [9].
VII Harnack estimates and extinction profile for singular equations
1. The Harnack inequality We will investigate the local behaviour of non-negative solutions of the singular p.d.e.,
(l.1)
{
u E Ut -
C,oc
(0, T; L~oc(n») n Lfoc (0, T; W,!:(n)),
= 0,
div IDuIJI-2 Du
1
in UT.
Weak solutions of (1.1) exhibit an intriguing behaviour. Even though in general they are not locally bounded,(I) they might become extinct after a finite time. It turns out however that the Harnack inequality of Theorem 2.1 of Chap. VI continues to hold provided p satisfies the further restriction 2N
(1.2)
N
+ 1
We will show that such a range of p is optimal for a Harnack estimate to hold. The extinction in finite time, the Harnack inequality and the LOO-estimates are linked by the range (1.2) of the parameter p. THEOREM 1.1.
Let u be a non-negative weak solution of (1.1 ) and let (1.2) hold.
Fix any (x o, to) E nT and assume that u(xo, to) > O. There exist constants 'Y> 1 and cE (0, 1). depending only upon Nand p. such that (1.3) (I) See §5-(IV) of Chap. V.
1. The Harnack inequality 185 where (1.4)
provided the cylinder (1.5)
Q4p(0) == {Ix - xol < 4p} x {to - 40, to + 40}
is contained in
nT.
Remark 1.1. The statement of Theorem 1.1 is the same as that of Theorem 2.1 of Chap. VI except that now the constant c is 'relatively small'; that is, the positivity of u(xo, to) spreads over the ball Bp(xo) but is preserved only for the 'relatively small' time c [u(xo, t o)]2-Ppp. Remark 1.2. As p '\. J~l' the constant "Y tends to infinity and c tends to zero. However these constants are 'stable' as p /' 2. i.e.,
lim "Y(N,p), c- 1 (N,p) = "Y(N, 2), c- 1 (N,2) <
p/2
00.
Therefore the classical Harnack inequality for non-negative solutions of the heat equation can be recovered by letting p /' 2 in (1.3). The limiting process can be made rigorous by the (nT ) estimates of Chap. IX.
C,7/:
t o+ a
I
1 p
4p Figure 1.1 Fix (xo, to) E nT, assume that u(xo, to) 'paraboloid' of two sheets "{s.,,} (x o, to)
>0
and construct the truncated
== { s ~ It - tol > c [u(xo, to)]2-PIx - xol PIJP } ,
where c is the number claimed by Theorem 1.1 and IJ and s are positive parameters. A consequence of (1.3) is the following:
186 VB. Harnack estimates and extinction profile for singular equations COROLLARY 1.1. Let u be a non-negative local weak solution of (1.1) and let p be in the range (1.2). There exist constants c E (0,1) and 'Y > 1 that can be determined a priori only in terms of N and P. such that
°
(1.3')
V(Xo, to) E nT, v 8 > such that P{s,4} (XO, to) u(xo, to) ~ 'Yu(x, t), V(x, t) E P{s,l} (xo, to).
C
nT,
In particular for solutions of the Cauchy problem, we have COROLLARY 1.2. Let u be a non-negative local weak solution of (1.1) in RN x R + and let p satisfy (1.2). There exist constants c E (0, 1) and 'Y > 1 that can be determined a priori only in terms of N and p, such that (XOI to) ERN X R + .
(1.3")
Figure 1.2
l-(i). Harnack estimates of 'elliptic' type The p.d.e in (1.1) is singular in the sense that the modulus of ellipticity of its principal part becomes infinite at points where IDul = O. At these points the 'elliptic' nature of the diffusion dominates the 'time-evolution' of the process; that is, the positivity of u at some point (xo, to) 'spreads' at the same time level over the
1. The Harnack inequality 187
full domain of defmition of x -+ u(x, to). This is the content of Theorem 14.1 of Chap. IV and holds for non-negative solutions of (1.1) for the whole range 1 < p < 2. When p is in the range (1.2), such a property can be made quantitative and takes the form of an elliptic Harnack inequality. THEOREM 1.2. Let u be a non-negative weak solution off1.1) and let (1.2) hold. Fix any (x o, to) e nT and construct the cylinder
(1.6)
{
Q4p(O) == {Ix - xol < 4p} x {to - 40, to + 40}, 0= c [u(x o , t o )]2- p pP, c> 0,
where c is the constant of (1.4). There exist a constants.'Y > 1, depending only upon N and p, such that (1.7)
provided Q4p(O)
c nT . The constant'Y /00 as eitherp'-.. J~l
or p/2.
Remark 1.3. The strict positivity of u(xo, to) is not required and the Harnack estimate (1.7) holds at the same time level. Remark 1.4. While Theorem 1.1 is •stable' as p /2, this is not the case of Theorem 1.2. Indeed (1.7) fails for solutions of the heat equation. To verify this consider the heat kernel in I-space dimension
r(x,t)
= ~e-~, v41rt
xeR,
and apply (1.7) for the sequence of points (x o, to) == (n, 1), n 1.2 were to hold for p= 2, we would have for some p> 0
r(n, 1)
~ 'Y r(n
e N. If Theorem
+ p, 1).
Letting n -+ 00 we get a contradiction.
1-(U). Generalisations The theorems generalise to the case when the right hand side of ( 1.1) contains a forcing term I(x, t, u) provided (1.8)
o ~ I(x, t, u, Du) ~ lo(x, t) + F u,
for a constant F and a function fo satisfying (1.9)
N+p
q>--. P
188 VII. Harnack estimates and extinction profile for singular equations
2. Extinction in finite time (bounded domains) PROPOSITION 2.1. Let n be a bounded domain in R N and let u be the unique non-negative weak solutior. of
u E C (R+; L2(n)) n LP (R+; wJ,p(n)) , { Ut - div IDul p - 2 Du = 0,
(2.1)
u(·,O) =
Uo
1
nT,
Uo ~
E LOO(n),
O.
There exists a finite time T· depending only upon N, p and u o , such that
u(·, t) == 0,
(2.2)
forall t
~
T·.
if
max
Moreover
(2.3)
O
-
'1 .11 Uo 11 22 -nP Iurll N!p-2l+2p 2l1i
' 2- p "', •• 11 U o 11 s,n,
8
= N(2-p) P
{I ;Z2}
if I
where '1. and '1•• are two constants depending only upon Nand p.
Remark 2.1. There is an overlap in the range of p in the two estimates of (2.3). For 1 < p < ;~l' N ~ 2, the upper estimate of T· does not depend upon the measure of
n.
n
PROOF OF LEMMA 2.1: The solution of (2.1) is bounded in x [0,00) and U>lder continuous in x [e, 00) for all e > O. Assume first that p ~ ;~l. In
n
keeping with the notation of Chap. V we let
Ar == N(P - 2) + rp, Vr> 1 and A == N(p - 2) + P for r
= 1.
In the weak formulation of (2.1) take u as a testing function, modulo a Steklov average. This gives (2.4)
By the U>lder inequality and the embedding of Corollary 2.1 of Chap. I, we have
These remarks in (2.4) yield the differential inequality
!lIu(.,t)lb,n +'Yll1u(.,t)II~:r: where By integration
$ 0
in V'(R+),
2. Extinction in finite time (bounded domains) 189
IIu(·, t)II~1f , :::; IIuoll~1f , -
(2.5)
(2 - p)'Ylt,
as long as the right hand side is non-negative. From this,
(2.6)
and
0<
r* :::; 1'''lnl ii lIuoll~:6.
Remark 2.2. The estimate (2.6) is 'stable' as P '\. J~l' i.e., as ~ '\. O. As p / 2, the boundary value problem (2.1) tends to the corresponding boundary value problem for the heat equation(l) for which there is no extinction in finite time. Accordingly, letting p / 2 in (2.6) gives
lIu(·, t)II2,n
:::;
lIuo ll2,n e-th2InI2/N,
where l' is the constant of the embedding of Corollary 2.1 of Chap. I. Next we take
p in the range
2N I
N~2.
In the weak formulation of (2.1) we select, modulo a Steklov average, the testing function u s - 1 , where
2-p
s=N-- > 1. P
(2.9)
This gives
where
1'2 == (s-l)
(s+
~_ 2»)"
By the embedding of Corollary 2.1 of Chap. I and the specific choice of s, we have (1) If u(p) are the solutions of (2.1) and u is the solution of (2.1) with p = 2, the
convergence takes place in the sense (p) --+ U, U"'i U (p) ,U"'i
. In
Co<
[-n X (E,OO )] ,vE > 0 , \oJ
i
= 1,2, ... ,N,
and uniformly in [0, TI, "IT> Estimates of u~) in Co<
o.
(ax (E, 00») uniform in p>2N/(N +2) will be given in Chap. X.
190 Vll. Harnack estimates and extinction profile for singular equations
We conclude that
d dt lIu(·, t)II.,n
+ '13I1u(·, t)II::t:
~ 0 in Z>'(R+), '13 == '1-P '12.
From this, by integration
{ II (., )1I.,n - II olls,n u
t
< u
1
}
~ ( ) 2-P'13 t lIuolI!1 +
This in tum implies (2.3).
JZl
and are 'stable' as p '\.1. However we cannot infer the convergence of (2.1) to a boundary value problem, in some reasonable topology, since the Hinder estimates of Chap. IV deteriorate as p'\.I and (2.1) only gives IDul EL1(nT) uniformly in p.
Remark 2.3. These estimates deteriorate as p /'
Remark 2.4. Proposition 2.1 holds for solutions of variable sign. The only modification in the proof occurs in the case 1
JZ1'
2-(i). The Harnack inequality and the rate of extinction 1be extinction profile is defined as the set 8[u>O] n noo. By Theorem 16.1 of Chap. IV the extinction profile of the solution of (2.1) is the portion of hyperplane n x {t =T*}. The Harnack estimate cannot hold in a 'parabolic geometry' independent of u, say, for example, within a cylinder of the type
Qp(xo, to) == Qt(xo, to)
u Q;(xo, to),
Q;(xo, to) == Bp(xo) x {to ± PP}·
Indeed if (x o, to) belongs to the extinction profile and p is so small that Qp(x o, to) C oo , the solution u of (2.1) is positive in Q;(xo, to) and it vanishes identically in Qt(xo, to). 1be intrinsic geometry of the Harnack inequality (1.3) implies an estimate of the rate of extinction of u( ., t) as t /' T*. We let
n
M = lIull oo ,nco •
2.1. Let u be the unique non-negative weak solution 0/ (2.J) and let p be in the range (1.2). There exists a constant '1 depending only upon Nand p, such that/or all (x, t) E n x (~. , TOo)
LEMMA
3. Extinction in finite time (in a,N) 191
T.}~ (T* -t)~ . ~
u(x,t):S 'Ymax { M 2 -p; [dist{x,8n}]p
(2.7)
PROOF: Fix x En and ~.
:S t :S T*, assume that u(x, t) > 0 and set
T* 4p == min { dist{x,8n}; ( 2M2-p
(2.8)
)l/P} .
We apply (1.3) over the ball Bp(x) and the cylinder
Q4p(X, t)
== B4p(X) X {t - [u(x, t)]2- P(4p)P, t + [u(x, t)]2-P(4p)P} .
By virtue of the choice (2.8) such a cylinder is contained in noo and (1.3) holds for it. We must have
T* - t ~ c[u(x,t)]2- p pp,
otherwise, by the Harnack estimate, u(x, t) tum implies the lemma.
= 0 against the assumption. This in
3. Extinction in finite time (in RN) PROPOSITION
3.1. Let u be the unique non-negative weak solution ofthe Cauchy
problem
(3.1)
Then, if 2N l
(3.2)
N~2,
there exists a positive number T* depending only upon N, p and U o such that u(·, t)
== 0,
"It ~T*.
Moreover (3.3)
o < T* < 'V·*llu 11 2s,B.,.' -p -
I
0
2-p
8=N--, P
for a constant 'Y** depending only upon N and p. PROOF: The solution of (3.1) can be constructed as the uniform limit in Eoo of the sequence {un}nEN of the solutions of the problems in bounded domains(l) (1) See §12 of Chap. VI.
192 VII. Harnack estimates and extinction profile for singular equations
Un E C (R+; £2 (Bn)) n lJ' (R+; WJ,P(Bn ») {
!
,
Un - div IDun l p - 2DUn = 0 in Bn x (0, n)
un(-,O) = Uo E £2(Bn ).
T:
If p is within the range (3.2). by Proposition 2.1. the extinction time of Un is estimated independent of meas Bn. Moreover since Un ~ Un+l' we also have T: ~ T:+l' This proves (3.3). Remark 3.1. The proposition holds for data of variable sign. Also U o need not be of compact support and it would suffice to assume 8=N~.
The proof is the same except for making precise in what sense the solutions of (3.1)n converge to the solution of (3.1). (2)
3-(i). The range (1.2) is optimal for a Harnack estimate to hold Fix (xo, to) E RN X (0, T·), where to is so close to T· to satisfy T.
(3.4)
C
- to < 4P to,
where c is the constant appearing in (1.4). Now choose p>O so large that (3.5)
By the choice (3.4), the box
Q4p(Xo, to) == B4p(Xo)
x {t o- [u(xo, t o)]2- p (4p)P, to+[u(xo, t o)]2- p (4p)p} is contained in Loo. If (1.3) were to hold for 1 < p constants c and 'Y independent of p, it would give
o < u(xo, to)
~
'Y
inf
zEB.. (zo)
<
J~l' N ~ 2, for some
u(x, T·) = O.
Remark 3.2. The choice (3.5) is possible in the whole Loo.
The same arguments imply that if p is in the range (1.2), no extinction in finite time can occur, for solutions of (3.1) . Within such a range, the Harnack estimate holds. Therefore if a finite extinction time T· were to exist, the choices (3.4)-(3.5) would give u(x, T·) >0. (2) We have
Un
E U(L'oo) uniformly in n. However such an order of integrability is not
sufficient to guarantee that the solutions Un are bounded. The number ).8 = N(P- 2) + sp is zero and the condition (5.1)-(5.3) of Chap. V are violated. Questions of convergence for general initial data in Lloc
(RN)
will be discussed in Chap. XII.
4. An integral Harnack inequality for all I < p < 2 193
4. An integral Harnack inequality for all 1 < p < 2 A weak integral form of (1.3) holds for any non-negative weak solution of (1.1) for p in the whole range 1 < p < 2, and it is crucial in the proof of the pointwise estimate (1.3). In the estimates to follow we denote with 'Y = 'Y(N,p) a generic positive constant, which can be determined a priori only in terms of N and p and which can be different in different contexts. PROPOSITION 4.1. Let u be a non-negative weak solution of (1.1) and let 1 < P < 2. There exists a constant 'Y = 'Y( N, p) such that
'V(xo, to)
E
fl oo , 'Vp> 0 such that B4p (x o) c fl, 'Vt > to
(4.1)
Since 1 < p < 2, the number A = N(P - 2) + p might be of either sign. The proposition can be regarded as a weak form of a Harnack estimate, in that the Ll-norm of u(·,t) over a ball controls the Ll-norm of U(·,T) over a smaller ball, for any previous or later time. It could be stated over any pair of balls Bp(xo) and Bqp(xo) for q E (0, 1). The constant 'Y ='Y(N,p, q) would depend also on q and 'Y(N, p, q) / 00 as q / 1. Remark 4.1. The proof shows that the constant 'Y(N,p) deteriorates as p /2. The proof depends on some local integral estimates of the gradient IDul which we derive next.
4-(;). Estimating the gradient of u PROPOSITION 4.2. Let u be a non-negative weak solution of (1.1) and let 1 < p< 2. There exists a constant 'Y='Y(N,p) such that
'V(xo, to)
E
floo, 'Vp > 0 such that B4p(Xo)
c fl, 'Vt > to, 'Vv > 0, 'Vq E (0,1),
there.. holds t
(4.2)
j j(T - to); (u + v)-~ IDulPdxdT toB.... (zo)
<
'YP
- (1 - q)p
[1
+
(t - to) vP-2] (~);. p>'
pP
!tf.::!l
x { sup to::5T::5t
ju(x,T)dX +VPN } B .. (zo)
"
,
194 VII. Harnack estimates and extinction profile for singular equations t
(4.3)
~j fiDUIP-ldXdT toB.. p(zo)
t
(4.4)
~j jlDulP-1dxdT toB.. p(zo)
Remark 4.2. The estimates (4.1)-(4.4) have been stated 'locally'. However they continue to hold for to 0, i.e. for cylinders Bp(xo) x (0, t) carrying the 'initial
=
data'.
Remark 4.3. The constant 'Y( N, p) in (4.2)-(4.4) tends to infmity as p /2. PROOF OF PROPOSITION 4.2: We translate the coordinates so that (xo, to) coincides with the origin and will work with non-negative weak solutions of
(4.5)
Ut -
div IDul p - 2 Du = 0, in B 4P x (0, (0),
p E (1,2).
<
Fix 0' E (0, 1) and let x -+ (x) be a non-negative piecewise smooth cutoff function in Bp that equals one on Blip and such that ID(I ~ 1/(1 - O')p. In the weak formulation of (4.5) take the testing function
v> 0, modulo a Steklov averaging process. We obtain for all t >
°
4. An integral Harnack inequality for all I < p < 2 195 t
(4.6)
2; P
j j T~ (u + v)-~ IDuIP(PdxdT OBp
.If
(u + v) ~ " (x, t)(Pdx
2(P P_ 1) t"
Bp t
+ 2(P ~ 1) j
T~-l (u + v) 2(,,;1) (PdxdT
j
OBp t
+p j j T~ IDul p - 2 Du·D((P-l (u + V)l-~ dxdT. OBp
We estimate the various tenns on the right hand side in tenns of the quantity
8 == sup
.
ju(x,T)dx.
O<.,.
Bp
Since pE (1, 2) we have 2(P;1) < 1. Therefore by the HOlder inequality (I·)
p
t.lj(u+v)~ I"Pdx " ..
2(P-l)"
Bp
J.
~
'5:,y t" p "
(
-
Bp
( p>'t)~ (8 + vpN) ~ " .
t
:5 'Y
~
sup ju(x, T)dx + v pN O<.,.
='Y P
)
sup j (u + v) ~ " (x, T)dx j T" -ldT o<.,.
o
Next, by Young's inequality
. - -
Bp
196 VII. Harnack estimates and extinction profile for singular equations t
(iii)
III .,.; IDulp- Du·D(,p-l 2
p
(U
+ V)l-a dxd.,.1
OBp
t
~ ~¥ II .,.; (u + V)-a IDuIP(Pdxd.,. OBp t
+
(1
2~;ppI' II.,.; (u + v)p-2 (u + v)
2('1'-1)
dxd.,..
OBp
This last integral is estimated above by p "Y(N,p)
(1 - u)p
(.!.-)
Vp - 2
(.!.-) ; (8 +
VpN)
2(,;1) •
p>'
pi'
Combining these estimates in (4.6) proves (4.2). To prove (4.3), write (4.2) with (x o, to) == (0,0) and select v 2 - p = (t/ PP). Then by th~ HOlder inequality t
(4.7)
IIIDUIP-1dxd.,. OB"p
t
{f
= JJ
H=1 'T'P
I'
(u+v)-"12=1 IDulp-1.,.-p.12=1 (u+v)"12=1 " dxd.,. I'
I'
OB"p
2=1
,; (ijT*(U+ v)-iIDUIPdzdT) " (il.,.;_l(U + v)~ dxd.,.) ~OB"p
.1 I'
OB"p
The last integral is estimated above by
(a)
( t)f;{
"YP p>'
(t)~pN}~
8+ pi'
~ "YP
( t.) ~ (p>.t );!1c.ll 8 I'
+"YP p>'
.
The first integral on the right hand side of (4.7) is estimated by (4.2) with the indicated choice of v and it is majorised by the same quantity on the right hand side of (a), apart for a factor (l-U)I- P. Combining these remarks in (4.7) proves (4.3). Finally (4.4) follows from (4.3) by a further application of Young's inequality.
4. An integral Harnack inequality for all 1 < p < 2 197
4-0i). Proof of Proposition 4.1 Assume that (Xo, to) coincides with the origin and consider the family of expanding concentric balls n
Bn == {Ixl < Pn} ,
Pn = P
L 2-
i,
n = 0,1,2, ....
i=O
We have Bo
== Bp and Boo == B2p' Introduce also the 'intermediate'
spheres
and let x -+ 'n (x) be a piecewise smooth non-negative cutoff function in Bn that equals one on Bn and such that ID'nl ~ 2n+2/ p. In the weak formulation of (4.5) take 'n as a testing function to obtain
for any two time levels 'Tl and'T2 in that
[0, tJ. We take as 'T2, a time level in [0, tJ such
We also set
Sn == sup
ju(x, 'T)dx.
O
Since'Tl E [0, tJ is arbitrary, (4.8) implies
Next we apply (4.3) over the pair of balls Bn and Bn+l for which (1 - u) ~ 2-(n+2). This and Young's inequality give t
2n;2
j jIDuIP-1dxd'T ~ ",bn ( ; )
.1 P
(Sn+l) 2(,;1) +",bn
(;l) 6
-P
0Bft
~eSn+l+",(N,p,e)bn
( t)J!; pl
'
valid for every eE (0, I), for some constant ",(N,p, e) depending only upon N, p
198 VII. Harnack estimates and extinction profile for singular equations
and E. Combining these estimates, we conclude that for every E E (0,1) there exists a constant 'Y( N, p, E) such that
s.. !> <Sft+! +1(N,p,<)
{z +.(~) r-.}
b".
11le Proposition now follows from the interpolation Lemma 4.3 of Chap. I.
5. Sup-estimates for &~2 < p < 2 We now combine the Ll:: -estimates of §5 of Chap. V with the integral inequality (4.1). If p is in the range (1.2), we may take r = 1 in (5.1)-(5.4) of Chap. V and rewrite the latter as
(5.1)
lIu(·, t)lIoo,Bp(zo)
~
'Y (t -
to)-~ (
J T)dx~ u(x,
sup to~.,.~t
pI)..
B Sp / 2 (zo)
+'YC~to)~ . This and Proposition 4.1 imply LEMMA
5.1. Let u be a non-negative local weak solution of (1.1) in noo and let
(1.2) hold. There exists a constant 'Y(N,p) such that
V(xo, to) E noo , Vp> 0 such that B4p(Xo) c
(5.2)
sup u(x, t) zeBp(zo)
~
'Y (t -
+'Y
to)-~
(inf
to~"'9
n,
"It> to
J T)dx~
u(x, B2p(Zo)
pI"
C~to)~.
The constant 'Y( N, p) tends to infinity as either p '\, J~1 or as p / 2.
Remark 5.1. The lemma continues to hold also for to = 0, i.e., for cylinders Bp(xo ) x R+ carrying the 'initial' data. The peculiar feature of this estimate is that the supremum of the solution over a ball at some time level is bounded above by the Ll-nonn of u over a larger ball at either the same time level or some 1uture' time. This is in contrast with the behaviour of non-negative solutions of the heat equation. Accordingly, the constant 'Y(N, p) deteriorates as p / 2.
6. Local subsolutions 199
5-(;). A special/orm 0/(5.2) We will use this fact in the following form. Let u be a non-negative weak solution of the p.d.e. in (1.1) in some space-time domain and let p be in the range (1.2). Let R>O and assume that the cylinder
Q4R
== B4RX{-4, O}
is all contained in the domain of definition of u. Then
6. Local subsolutions As in the degenerate case, the proof of Theorem 1.1 is based on expanding the positivity set of the solution u by means of suitable comparison functions. Let b, k, IJ. be positive parameters satisfying (6.1)
Consider the cylindrical domain with annular cross section
Q(8) ==
(6.2)
{:-1 k
P- 2
< IxlP< I} x{O, 8},
and the function
(6.3)
LEMMA
6.1. Assume that p is in the range (1.2), i.e., A == NCp - 2) + p > O.
Then the constant b=b(N,p) can be chosen a priori only dependent upon Nand p, so that
(6.4)
'rIk > 0, 'rI IJ. > 0 > satisfying (6.1), Wt - div(lDwl p - 2 Dw) ~ 0 a.e. in Q(8),
8=min{ (2~)P-l
IJ.j
k2 -
P}.
Remark 6.1. The proof below shows that the constant b> 1 is •stable' as p /,2.
200 YD. Harnack estimates and extinction profile for singular equations
e
o
Figure 6.1
PROOF OF LEMMA 6.1: The function x-!li(x,t} is radial and decreasing with respect to Ixl. so that writing (6.4) in polar coordinates we have
C(!Ii} == !lit - div(ID!liI,,-2 D!Ii} = !lit
+ ( N ; 1) (_!li,},,-l
_ (p _ 1)( _!li,),,-2!1i",
where
p=lxl,
!Ii'
= ~!Ii ; dp
We write (6.5)
( _!li,)2-" C(!Ii)
= (_!li,)2-"!lit + 'R.(!Ii) ,
'R.(!Ii) == N - 1 (-!Ii') - (p - 1}!Ii" , p
and calculate 'R.(!Ii) as follows. First we set
~ (IXI")~ IIzll =kFIb -t ; k FF-;
W=--l
Then by direct calculation
F 11
= 1 + IIzll
= (1 -lxI 2 );!r.
6. Local subsolutions 201
(6.6)
=
We calculate the expressions 1/1' w'v + wv' and 1/1" = w" v + 2w'v' + wv" from (6.6) and combine them into 'R.(I/I) to obtain (6.7)
'R.(I/I) :5 -P-(I - p2);;!-r 2-p -IIzll 2-p:F
- (p - 1) [ - P
(w) M {N p2 :F
1
+ 1] + p-1 } :F
+ 2pw(I- p2)~ {(N - 1) _ ~ M + I}. p-I
2-p:F
Rewrite the first factor in braces on the right-hand side of (6.7) as
M).
P{ ... } = ( N 2- p :F We will impose on /I z II to be so large that (6.8)
N-_p_M
This is possible since N(p - 2) + p == A> O. The second term in braces on the right hand side of (6.7) is negative if we choose IIzll to satisfy (6.8). If N = 1,2, this is a direct consequence of (6.8). If N ~ 3,
(P_I){N-I _ ~M p-I 2-p:F
+I} =
M)
(N- _P_ 2-p :F
+ -p- (3 2-p
2p)M :F
+ (p- 2).
The first term is negative in view of (6.8) and the second is negative since p > ;:~l > ~ if N ~ 3. We drop the last negative term on the right hand side of (6.7) and estimate
202 VII. Harnack estimates and extinction profile for singular equations
R.(!li) :::;
(6.9)
_Pv (w) 1l:.ll [N __P_1l:.ll] . 2-p p :F 2-p :F 2
We return to (6.5) and estimate above the term ( -!li,)2-"!lit. First using (6.6)
Also
!lit = _1_ vw ~. 2-p :F t
1l:.ll
Therefore
(-!li')2-"!lit :::;
~ (~)2-" vwM~. 2-p
:F t
p
We combine this with (6.9) into (6.5) and set
= (-!li')2-"[!lit _ div(ID!liI,,-2 D!li)] (2 -
C* (!li)
p):Fp2 vwllzll
to obtain
C*(!li) :::; "YW2-"pP
t
+ P[N - -p-
M] .
2-p :F
From the definition of w and II z II
w 2-"pP = ( t
and .c*(!li)
IIzll
1 + IIzll
),,-1 1-" <- 11'-1 6
< ...::L + -p- 11'-1 2- p
_I_
[-A + E-] . :F
We will choose II z II so large that ~ :5 ~,and then select 6 from "y p A -----=0 11'-1 2- P 2 .
We will have ~
==
I41T :5 ~ if for example IIzll > ¥, i.e. if k 2 -" V-I
Ixl" (~) ,,-1' > t > O.
From the construction of the cylinder Q(fJ) in (6.2), we have
k2 -" V-I
Ixl" ~ ",.
Therefore to prove the lemma it suffices to take
A
o < t :::; fJ = ( 2p
),,-1
",.
7. Tune expansion of positivity 203
7. Time expansion of positivity The next subsolution of (1.1) will be employed to expand the set of positivity of u in the direction of increasing t. Set (7.1)
k~
g;(x;t) := Re(t)
R(t):= kP - 2 t
+ pP,
{
1-
( IX I,,);2:r}2 R(t) +'
F:= l-lIzlI;2:r,
IIzll:=
t
l:).
e
Here k and p are positive parameters and > 1 is a number to be chosen independent of k and p. For t = 0 the function g;(., 0) is supported in the ball B p and for t > 0 the support of x -+ g;( x, t) is the 'expanding' ball
We will consider g; only within the domain
The function g; is continuous in 1){k,e}. vanishes on the 'lateral' boundary of and it is of class Coo in the interior ofV{k,e}.
V{k,e}
LEMMA 7.1. The number of N and p so that
Remark 7.1. The constant
e= e( N, p) can be determined a priori only in terms
eis 'stable' as p /
2.
PROOF OF LEMMA 7.1: By direct calculation within 1){k,e} we have
ek,,-lppe
2
2
k"-I~
-L
g;t = - Re+l(t) F + p-l ~+l(t)Fllzllp-l, 2p k~ (IXI);2:r x Dg; = - p _ 1 ~(t)F R(t) lxi' IDg;I,,-2 Dg; = _ ( ~ ) p- 1
,,-1 [ kPPe F ],,-1 -=--Re(t)
_ div 1vg;1,,-2 Dg; = (~)"-1 _1 p-l R(t) Setting
R(t) ,
[k~ F]"-1 ~(t)
(N _pIlZII;2:r) . F
204 VU. Harnack estimates and extinction profile for singular equations the previous calculations give (7.3)
C· (4))
= {- f.F + _2_lIzll~ p-l ~)P-l [~ ]p-2( _ IIZIl~)} + (p _ 1 R£.(t) F N P F .
Introduce the two sets
£1 == {(X,t) E V{k.£.}
I:F < 6},
£2 == {(X,t) E V{k.£.}
IF
~ 6}.
Here 6 is a small positive number to be determined so that. within £1. the last term on the right hand side of (7.3) is negative. i.e.•
With such a choice we have in £1 (7.4)
c- (4)) ~
_2_ p-l
+ (~)p-l p-l
(N _~)6 ,
where in estimating the term containing R(t) on the right hand side of (7.3) we have used the fact that 1 < p < 2. We determine 6 so that the right hand side of (7.4) is non-positive and observe that such a choice can be made independent of f.. Next. having determined 6. within £2 we have (7.5)
c- (4)) ~ -f.6 + _2_ + N (~)p-l p-l
p-l
[nt.(t) ~] 2-1' pPt. 6
Within the range (7.2) of t we estimate
nt.(t) 1]2-1' ( 1)£.(2-1') -2 (e)2-p [--< 1+ 61' < . pPt.6 -6
e
We substitute this estimate in (7.5) and choo~e is non-positive.
eso large that the right hand side
8. Space-time configurations Locally bounded weak solutions of (1.1) are locally R;lder continuous in the interior of their domain of definition. "ripE (0,1). This is the content of Theorem 1.1 of Chap. IV. The proof consists of controlling the essential oscillation of a local solution over a family of nested and shrinking cylinders. Such a control is established in Proposition 2.1 of Chap. IV. by working with cylinders whose 'space dimensions' are rescaled in terms of the solution itself. As observed in Remark 2.2
8. Space-time configurations 205
of Chap. IV, such a geometry is not the only possible. A version of Proposition 2.1 holds for an intrinsic parabolic geometry where the scaling occurs in the 'time dimension'. We restate the proposition for such a geometry in the context of (1.1) and in a form convenient for the proof of the Harnack inequality. Let 1.£ be a local weak solution of (1.1). Fix (x, t) E nT and suppose that we can fmd a cylinder of the type (8.1)
[(x, t) + Q (aoR", R)] == {Ix - xl < R} x {l- aoR"} , ao == (~) 2-",
where A is an absolute constant, R is so small that [(x, t) and w is any positive number satisfying
(8.2)
+ Q (aoR", R)] c
nT,
11.£1 :5 w.
sup [(z,l)+Q(ooRr> ,R)]
PROPOSITION 8.1. There exist constants eo, 1/ E (0,1) and C, A > 1 that can be determined a priori depending only upon N and p, satisfying the following. Construct the sequences Ro = R, WO = W
Rn =C-nR,
Wn+l
= TJWn,
n = 1,2, ... ,
and the boxes
Q(n) == {Ix - xl
_
< Rn} x {l- anR~, l}, an -
( Wn)2-" A .
Thenfor all n=O, 1,2, ... and
essosc 1.£ :5 W n · Q(n)
A consequence is the HOlder continuity of 1.£ at (x, t). A particular case is LEMMA 8.1. There exist constants "'( > 1 and a E (0, 1) that can be determined a priori only in terms of the data, such that for all 0< p :5 R
essosc u(x, t) :5 "'(Wo
Ix-zl
(RP)Q .
Remark 8.1. This is a version of Lemma 2.1 of Chap. IV, stated for a 'fixed time'
l. Remark 8.2. The constants A and C depend only upon N and p and are independent of u. Moreover they are 'stable' as p /2. This follows from the remarks of §3-(I) of Chap. IV.
206 VB. Harnack estimates and extinction profile for singular equations
9. Proof of the Harnack inequality We let u be a non-negative local weak solution of (1.1) in nT and let p be in the range (1.2). Let (xo, to) E nT, assume that u(Xo, to) > 0 and consttuct the cylinder
Q4p(Xo, to) == {Ix - xol < 4p}
x {to - [u(x o•t o)]2-" (4p)", to + [u(xo, t o)]2-" (4P)"}. where we assume that p is so small that Q4p(X o, to) c
nT . The change of variables
x-xo
x~--,
p
Q+ == B4 x [0,4"), Denoting again with x and t the new variables, the rescaled function
is a bounded non-negative weak solution of {
div IDvl,,-2 Dv v(O,O) = 1.
Vt -
=0
in Q,
To prove the theorem it suffices to determine constants c and "Yo in (0, 1), depending only upon N and p such that (9.1)
inf v(x, c)
zEBl
~ "Yo.
9-(i). Locating the sup of u in Q For TE (0,1) consttuct the family of nested expanding cylinders
and the numbers
M.,. == supv, Q..
Here 6 E (0, 1) is a small number to be chosen later and has the effect of rendering '/lat' the boxes Q.,..
9. Proof of the Harnack inequality 207
Remark 9.1. This construction is similar to that in the proof of Theorem 2.1 of Chap. VI. The cylinders QT however are 'thin' in the t-dimension. Also the exponent of (1 - T) in the definition of NT is fixed and depends on the singularity of the p.d.e. For T = 0, we have Mo = No. Moreover as T / ' 1 and since v E L~(Q). Therefore the equation MT = NT has a largest root, say To, which satisfies
Since v is HOlder continuous in Q, it achieves the value MTo at some point (x, f) E QTo and sup v(x,f) $ 2~(1- To)-~. Iz-zl
(9.2)
LEMMA 9.1. There exist a positive number e that can be determined a priori only in terms of N and p, such that
v(x , f' )-~ , OJ -> ~(I-'T. 2 0
'v'lx - xl < e(1 - To).
Remark 9.2. The proof employs the estimates of Lemma 5.1 in the form (5.3). Therefore e "\. 0 as p /' 2. PROOF OF LEMMA 9.1: Construct the box (x, f)
-
+ Q4R == {Ix - xl < 4R} x {t -
I-To 4, fl, where 4R = -2-'
Apply to such a box the estimate (5.3) with the appropriate change of variables to obtain sup v(x, t) $ "Y( !v(x, f)dx)"/>' + Iz-zl
"YR-~,
'Vf - 1 $ t $
t.
B2R
In view of (9.2) and the definition of R sup v(x,t) $ "Y1(1- To)-~,
'v't -1 $ t $ t,
Bi!.=;al where "Y1 ="Y1 (N, p) is a constant that can be determined a priori only in terms of N and p. Next consider the cylinder
208 VB. Harnack estimates and extinction profile for singular equations By virtue of such a construction we have sup v ~ "Yl(1- TO)-~' QRo
The 'vertical size' of QRo is larger than
Therefore QRo satisfies the space-time configuration of (8.1 )-(8.2). We conclude that
'v'0 < p < Ro,
'v'lx - xl < p,
vex, f) ~ v (x, f) -
at the level f
"Y"Yl(1 - To)-r-;
(;J
°
Since v (x, f) = (1 - To)-~, by taking p='TIRo. 'TIE (0,1) we find
vex, f) ~ (1 -
To)-r-; (1- "Y"Yl'1°), ~
'v'lx - xl ~ '1Ro == '1(8"Yl p )-1(1_ To) and the lemma follows by taking '1 so small that enough (1 - "Y"Y1 '1°) then choosing
9-(i;).
= ! and
Time-expansion o/positivity
The previous arguments are independent of the number 6. We will now detennine 6. LEMMA 9.2. There exist small positive numbers Co, 6 that can be determined a priori only in terms of N and p, such that
(9.3)
vex, t) ~ co(1 -
To)-r-;,
'v'lx - xl < e(1 - To), 'v'6 ~ t ~ 26.
PROOF: Consider the comparison function ~ (x - x; t - f) in the domain V{k,(} (x, defined in (7.1)-(7.2) with the choices,
p = e(l- To). The function ~ is a subsolution of (1.1) for a time interval
For t = f by virtue of Lemma 9.1, v ~ ~ (x - x; 0). Therefore by the comparison principle
9. Proof of the Harnack inequality 209
(x-i)"-R(t.l) ~
__+-______________-,2
·&to 1
Figure 9.1
v
~~
in {Ix -
xl P < R(t -l)} x {O < (t -l) < 36}.
In particular for 6 < t - t < 36 and Ixl :5 e(l - To),
vex, t) ~
~(l-TO)-~ [lie + 1](
{ 1-
(ae );;!T}2+ ae + 1
== co(l - To)-~. The location of t in the box Q".o is only known qualitatively. However, as (t -l) ranges over [6, 36], the intervals [t + 6 < t < t + 36] have the common intersection [6 :5 t :5 26] and the lemma is proved.
Remark 9.3. The number 6"" 0 as p / 2. This follows from Remark 9.2 and the choice of 6 above.
9-(iii). Sidewise expansion of positivity We will expand the positivity set of v over the ball {Ixl < I} at the time level t=26. For this we will prove that there exist a constant 'Yo ='Yo(N,p) such that
vex, 26) ~ 'Yo,
'v'lx - xl < 2.
Consider the comparison function (9.4)
I/t
( X-x.~) 3'3P'
introduced in (6.3), in the annular cylindrical domain
210 VII. Harnack estimates and extinction profile for singular equations
{e(l- To) < Ix - xl < 3}x {6,26}. The number k is given by
k
= co(l- To)-r-;,
where Co is determined in Lemma 9.2. The parameter JJ here can be chosen by imposing
11'-1 e1'
i.e.•
u<--r2-1' 31'. Co
Wechoose JJ = min {
1 11'-1 e1' }
4;
~-1' 31'
'
and pick (J according to the second of (6.4). By further restricting either JJ or the number 6 of Lemma 9.2 we may assume that (J =6. The function t[I in (9.4) vanishes for Ix - xl=3 and fort=6. Moreover for Ix - xl =e(l - To) and 6
X -
x
t -
6) 5
-3-' 3P
Co(l -
-I!.-
To)-r-p
5 v(x, t),
by Lemma 9.2. Therefore by the comparison principle. we have for t
= 26 and
Vlx-xl<2
9-(iv). Proof of Theorem 1.1 for p near 2 The proof is very similar to that of Theorem 2.1 of Chap. VI for p close to 2. We only indicate the main differences. As before. construct the family of expanding cylinders Q.,. == {Ixl < T} x {-T, O} and the numbers
where f3 is a positive number to be chosen. The definition of the numbers N.,. differs from that in §9 since f3 is arbitrary. Let To E [0, 1) be the largest root of the equation M.,. = N.,.. so that
10. Proof of Theorem 1.2 211
If (x, t) is a point in QTo where v achieves the value M To ' we have P I I - To - I - To ( ) ~2 p( I-To )-; vX,t x-xl<-2-; t - -2-
(9.5)
Let and consider the box
c
From the definitions of QT and Ro we have Q 0 (x, t) IIvlloo,Qo(z,l)
Q!:t;a, so that by (9.5),
~ 2P (1 - To)-P
Therefore Qo (x, t) satisfies the space-time configuration (8.1)-(8.2). It follows that
\fIx - xl < Ro, By taking
Iv(x, t) - v (x, t) I < -y2P(1 - To)-P
(~)
Q
p=eRo and then e sufficiently small we have
LEMMA 9.1'. There exists a small positive number e E (0, I) that can be determined a priori only in terms of N, p such that I _ 1 v(x ,OJ f\ > P \fIx - xl < e(1 - To) P~ P + . - -(I 2 - ~) 0,
Remark 9.5. The constant e depends upon f3 but it is 'stable' as p / 2 since no use has been made of (5.3). The proof can now be completed by expanding the positivity set of v with the aid of the comparison function g,.,p introduced in §3-(i) of Chap. VI.
10. Proof of Theorem 1.2 Fix a point (x., t.) E aT assume that 1.£ (x., t.) cylinder Q8R
(x., t.) ==
>
°
and for R> 0, construct the
{Ix - x.1 < 8R}
x {t. - c [1.£ (x .. , t .. )]2-" 8R", t.
+ c [1.£ (x .. , t.)]2-" 8R"} ,
where c=c(N,p) is the number appearing in Theorem 1.1, and R is any positive number such that Q8R (x., t.) is contained in the domain of definition of u. We first establish an auxiliary proposition, then we will prove that it implies the theorem.
212 VII. Harnack estimates and extinction profile for singular equations PROPOSITION 10.1. There exists constants C=C(N,p) and,,=,,(N,p) that can be determined a priori only in terms of N and p, such that
(10.1)
sup u(·,to) ~ u(xo, to) ~ C
C- 1
inf B"Re Z.)
B"ReZ.)
u(·,to),
where xo=x. and
(10.2) PROOF:
The change of variables
t - t.
x-x. x- - -
t -
R '
[u (x., t.)] 2 PRP
=
maps QSR (x., t.) into Q Bs x (-8,8). Denoting again with x and t the new variables. the rescaled function
v(x, t)
=u (1x., t. ) u (x. + Rx, t. [u (x., t.)]2- PtRp)
is a bounded non-negative solution of Vt -
{
div IDvl p - 2 Dv
v(O,O)
=0
in Q.
= 1,
We first prove that there exist a quantitative constant C (10.3)
1
C
~
v(x, c)
~
C,
= C (N, p). such that
v Ixl < 1.
By the Harnack inequality (1.3) (10.4)
v(x, c) ~ 'Yo,
Vlx/ < 2,
for a quantitative constant 'Yo = 'Yo (N, p). This proves the estimate below in (10.3). For the estimate above we require the following lemma. LEMMA 10.1. There exists a quantitative constant" E (0,1) depending only upon N and p, such that
(10.5) PROOF:
v(x, -c)
~
2/'Yo,
Vlx/ < 2".
For x ranging over the ball B4 consider the closed truncated 'paraboloid'
t + c ~ c[v(x. _c)]2- p Ix -
xiI',
-c
~
t
~
O.
By the Harnack estimate (10.6)
v(x, -c)
~ 2- v(x, 0), Ix - xiI' < [v(x, _C)]p-2 , 'Yo
10. Proof of Theorem 1.2 213
and in particular v(O, -c} =
lho. Since v is HOlder continuous, the set {x
I u(x,-c} <2ho}
is non-empty and contains a ball about the origin. We claim that in particular it contains the ball B 2 f/' where (277}P
= (~) 2- p •
If not, there would exist some x E B2f/ such that v(x, -c} = the ball Ix - xlP < [v(x, _c}]p-2 = (277t covers the origin, and (10.6) for
2ho. It follows that
x=O gives
-2 = v(x,-c) $ ~o
1
-. ~o
The contradiction proves the lemma. To prove the estimate above in (10.3), we combine the quantitative bound (10.5) with Proposition 4.1. This gives
We return to the original coordinates and write the estimate above in (10.3) as
u(x,t o} $ Cu(x.,t.},
'v'lx - xol
< 77R,
XO
== x •.
Since, by Corollary 1.1, u (x., t.) $~u(xo, to} the left estimate in (10.1) is proved. The estimate below in (10.3) reads
On the other hand the estimate above in (10.3) for x = x. gives u(xo, to} $ Cu (x., t.). Combining these last two estimates proves the bound above in (10.1) and the proposition follows.
1D-(i). Proof of Theorem 1.2 Fix (x o, to) E fh and p > O. Let 77 be the constant claimed by Lemma 10.1 and let ~ be the constant of the Harnack estimate (1.3). Set
and construct the cylinder
214 VII. Harnack estimates and extinction profile for singular equations
QSr(Xo, to) == {Ix - xol < 8r} x {to - c [u(xo, t o)]2- p 8rP, to
+ c [u(xo, t o)]2- p 8r P} .
Wthout loss of generality, we assume that QSr(X o, to) C [}T. First fix the time level and choose R> 0 from to - t.
= c[u(x.,t.)]2- p RP,
The defmitions of t. and R give
c [u(xo, t o)]2- p r P = to - t. By Corollary 1.1, u (x., t.)
~
= c [u (x., t.)]2- p RP.
')'u(xo, to). Therefore
R
~
')'
2..=l p
rP
== p/T/.
Applying Proposition 10.1 with such a choice of the point (x., t.) and radius R proves the theorem.
11. Bibliographical notes Theorem 1.1 and its proof is taken from [44]. The form of Theorem 1.2 was conceived by Nash [84], who believed it to be true for solutions of the heat equation. Moser [83] pointed out that (1.7) is not dilation invariant for solutions of the heat equation. It becomes scalar invariant in a specific intrinsic geometry. The results adapt to equations of porous medium -type and its generalisations (see [44n. In the context of the plasma equations estimates of the rate of extinctions were derived by Berryman-Holland [13,14]. Proposition 3.1 is due to &nilan and Crandall [9]. The estimates of §§ 4 and 5 are taken from [42]. The subsolution tV of §6 appears in [44]. The subsolution ~ of §7 is a modification of a subsolution introduced in [4]. It is natural to ask whether an intrinsic Harnack estimate continues to hold for non-negative solutions of p.d.e.'s with full quasilinear structure. This is the case if p = 2 and it remains an open issue for degenerate (p> 2) and singular (1 < p < 2) equations. A step in this direction is in [29]. It is shown that Theorem 1.1 holds true for non-negative weak solutions of Vt -
(IDvIP-2aij(X,t)u:J:JI:', = 0,
where (x, t) --+ aij (x, t) are only bounded and measurable and the matrix (aij) is positive definite.
VIII Degenerate and singular parabolic systems
1. Introduction We turn now to quasilinear systems whose principal part becomes either degenerate or singular at points where IDul =0. To present a streamlined cross section of the theory. we refer to the model system
u == (1.1)
{
Ui
(UI,
E C'oc
Ut -
U2, ... , Urn), mEN,
(0, TjL~oc(n))nLP (0, Tj w,!,:{n)) , i=l, 2, ... ,m,
div IDul p - 2Du
=0
in nT'
The solutions are meant in the weak sense
(1.2)
I
t2
tl
for all intervals satisfying (1.3)
+
Uirpi{X,T)dXI
n
II t2
{-Uic,oi,t
+ IDuIP-2Dui·Dc,oi} dxdT=O,
tin
[tl, t2] C {O, T] and all testing functions cP == (cpt. CP2, ••. , rpm)
CPi E W,!;; (0, T; L2{n)) n Lfoc (0, T; wJ,p{n)),
i
= 1,2, ... , m.
For these we derive local sup-bounds on the modulus of the solution space gradient IDul and establish the estimate (1.4)
Ui, zj EC1:,c{nT),
i=I,2, ... ,m, j=1,2, ... ,N,
lui and its
216 VIn. Degenerate and singular parabolic systems
for some a E (0, 1). This is the focal point of the theory. Weak solutions of elliptic systems in general are not continuous everywhere within their domain of definition. We refer to [48J for counterexamples and an account of the theory. Solutions of (1.1) are regular everywhere in nT because of the special nature of the system. If u solves (1.1), then the function IDul 2 is a non-negative subsolution of a parabolic p.d.e. (1) It is precisely such a property, which for elliptic systems is called 'quasi-subharmonicity' ,(2) that permits one to prove (1.4) everywhere in
nT. These estimates can be extended up to t = 0 if the system in (1.1) is associated with a smooth initial datum 110. They also carry over to the lateral boundary of nT if (1.1) is associated with homogeneous either Dirichlet or Neumann data on ST == an x (0, T). If the data are not homogeneous, the theory is fragmented and incomplete. In the case of non-homogeneous Dirichlet data, we will show that Ui
EC6 (nx (e, T» for arbitrary 6 E (0,1), 'rIe E (0, T),
provided p > max {I; J~2}' However the key estimate (1.4) is not known to hold in such a case, and it is a major open problem in the theory. The C 1 ,o regularity (1.4) requires a preliminary estimation ofthe type (1.5)
IIDulloo,K: :5 const,
IC a compact subset of
nT.
1be degenerate case p> 2 and the singular case pE (1, 2) are rather different with
respect to such an estimate. The function class in (1.1) implies that(3) (1.6)
If P > 2, such integrability suffices to establish (1.5). If 1 < p < 2, the sup-bound lui. Precisely,
(1.5) can be derived only if further 'integrability' is assumed on (1.7)
lui E L1oc(nT ),
where r ~ 2 satisfies Ar==N(p - 2) + rp
> O.
This is analogous to the condition imposed in Theorem 5.1 of Chap. V. It implies (1.4) and in addition (1.8)
l-(i). Aboutthe singular case l 2, the behaviour of the solutions of (1.1) is entirely a local fact. In particular the sup-bound (1.5) and the estimate (1.4) are a sole consequence of u being a weak solution of (1.1). If 1 < p < 2 due to the singular (1) See (3.3) in the Preface or (1.8) of Chap. IX. (2) We refer to Meier [77] for some sufficient conditions for an elliptic system to be quasi-subharmonic. (3) See Proposition 3.1 of Chap. I.
1. Introduction 217
nature of the p.d.e. some global infonnation is needed. This is not related to systems. Indeed it occurs also in Theorem 5.1 of Chap. V to establish a sup-bound for solutions of a single equation. Since our estimates involve u and DU the global infonnation needed regards both the solution and its space gradient. Let r ~ 2 satisfy (1.7) and let U be a local weak solution of (1.1) for p E (1,2). We assume that t
U
(1.9)
{
can be constructed as the weak limit in L[oc(f1T) ofa
sequence of bounded subsolutions {Un her of (1.1) satisfying in addition
IDunl E L~oc(f1T)'
We stress however that all our estimates will depend only upon the quantities
lIull r ,K;, IIDull"K;, x:;
a compact subset of f1T .
Such an assumption is not restrictive in view of the available existence theory(l) and the special fonn of (1.1).
1-(ii). General structures We will develop the theory for the homogeneous system (1.1). The same results however continue to hold for the following general class of quasilinear systems (1.10)
~1J,' at I
div A (i) (x , t , Du) = B(i) (x , t , u , Du) in
nT,
i = 1,2, ... ,m,
where the functions
A(i)=(A(i) A(i) A(i»).. l ' 2 , ... , N B(i) :
f1T xRxR Nm
satisfy the structure condition
(S3)
(1) See Lions [73).
--+
R,
rl
uT
i
xR Nm --+RN ,
= 1,2, ... ,m,
218
vm. Degenerate and singular parabolic systems
m
2: IB(i)1 ~ CllDul
(85 )
p-
l
+ !P2,
i=l
where Ci , i =0,1. are given positive constants and !Pi, i = 0,1,2, are given nonnegative functions satisfying !Po
) + !PI#r + !P22 E L q,oc (n uT,
N +2 q > -2-'
Remark 1.1. The structure condition (82 ) is somewhat fonnal since there is no stipulation that Ut.",,,,,,; have meaning at all. More correctly it should be written with Ut,,,,u:; and DUi''''i replaced by tensors ~t,k,j. Neverthless we prefer the formal but suggestive fonn of (82 ). We will develop the main points of the theory for the model system (1.1) and indicate later how to modify the arguments to include (1.10).
2. Boundedness of weak solutions We will use the notation of§3 of Chap. II. Thus Q (0, p) is the cylinder with 'vertex' at the origin. Its cross sections are the cubes Kp and its height is O. The cylinder [(xo, to) + Q (0, p)] has the 'vertex' at (xo, to) and is congruentto Q (0, pl. With ( we denote a piecewise smooth non-negative cutoff function in Q (0, p) vanishing on the parabolic boundary of Q (0, p). THEOREM 2.1 (THE CASEp>2). Letubealocalweaksolutionof(l.l),and let p > 2. Then for all e E (0, 21 there exists a constant 'Y depending only upon
N,p, mand e, such tlultforeverycylinder [(xo, to) + Q (O,p)] c nTandforever CTE(O,I),
(2.1)
sup [(so,t o )+Q(0'9,O'p»)
lui < -
'Y
(1 -
(Ojpp)lIE CT)(N+p)/E
(f!
lu IP - 2+£dXdT)
liE
(so,to)+Q(9,p»)
A
(~);!J .
THEOREM 2.1 (THE CASE l
(2.2)
lui E L,oc (nT ) , r
~ 1
Ar -=N(P - 2)
+ rp > 0,
2. Boundedness of weak solutions 219
and that (1.9) holds. There exists a constant 'Y depending only upon N, p, m and r such thatforevery cylinder [(x o, to) + Q (9, p)] c nT andfor every C1E (0, I),
(2.3)
2-(i). An auxiliary proposition The arguments are similar to the proof of local boundedness of solutions of a single equation and are based on local energy inequalities which we derive next. We set
(2.4)
lul=w.
2.1. Let u be a local weak solution of the system (1.1) in nT, and let f(·) be a non-negative, bounded, Lipschitz function in R+. There exists a constant 'Y='Y(N,p, m), such that PROPOSITION
V(x o, to) E
(2.5)
sup
to-9~t~O
nT Vp, 9> 0 such that [(xo, to) + Q (9, p)] c nT I (1~f(8)d8) ("(x, t)dx 0
(zo+Kp)
+
IIIDw l" f(w)("dxdr
(zo,to)+Q(9,p»)
+
IIIDUI,,-2IDwI2wf'(w)("dxdr
[(zo,to)+Q(9,p»)
~ 'Y I I w" f(w)ID(I"dxdr + 'YI I (fo~f(8)d8) (,,-l(t dxdr. (zo,to)+Q(9,p»)
PROOF:
(zo,t o)+Q(9,p»)
The weak fonnulation (1.2) can be rewritten in tenns of Steklov aver-
ages,as (2.6)
I { ! Ui,htpi
+ [I Dul,,-2 DUi]h·Dtpi} dxdr = 0, Vh E (O,T),
n VO< t~T - h,
VCPi E W~'''(n) n L2(n) i
= 1,2, ... ,m
Since ItUi,h E L?oc( nT ), this implies (2.6)'
!
Ui,h - div [lDul,,-2 DUi] h = 0
a.e. in nT·
220 VID. Degenerate and singular parabolic systems Without loss of generality we may assume that (x o, to) coincides with the origin. In (2.6), take the testing function
We add over i obtain
= 1,2, ... , m and integrate in dt, over the interval -() $ t $ 0, to
j! f(t;/(3)ds) ,'dxdT -8
Kp
t
+ J J [lDulp-2Dudh·Dui,h!(luhl) (PdxdT -8K p
t
+ J J [IDU 1P- 2 a~l Ui,h] h Ui'hr~~IIUhl) U;,h a~l u;,h(PdxdT -8K p
t
= -p J
J [lDulp- 2DUi] h Ui,h! (luhD (p-l D( dxdT.
-8K p
We perform an integration by parts in the rust integral and then let h - t O. The various limits are justified since IDul E Lfoc(lh) and lui EC,oc (0, T; L~oc(l1T». This gives (2.7)
sup -8
J( Kp
10f~J(S)dS) (P(x, t)dx
+ J JIDulP !(w)(PdxdT + JJIDuIP-2IDwI2W!'(w)(PdxdT Q(8,p)
Q(8,p)
'$ PJfiDuIP-IW!(W)(P-IID(ldxdT + p !!(1wS!(S)dS) (P-l(t dxdT. Q(8,p)
Q(8,p)
By Young's inequality for every 1] > 0
! !IDuIP-1w!(w)(P-1ID(1 dxdT Q(8,p)
$ 1] !!IDuIP !(w)(PdxdT Q(8,p)
+ -Y(1]) J J wP !(w)ID(IPdxdT. Q(8,p)
Next by Schwartz inequality
2. Boundedness of weak solutions 221 N
(Ut Ut.z;)2 $;
w- 2
;=1
Therefore sition.
N
m
IDwl2 = w- 2 L
m
L u~ L L U~.z; == IDuI2. ;=1 t=1
l=1
IDulP ~ IDwI P • Combining these estimates in (2.7) proves the propo-
COROLLARY 2.1. The integral inequality (2.5) continues to hold/or non-negative. non-decreasing functions / in R +. satisfying
/orall k > 0,
sup /'(8) <00, O~s~k
provided (2.8) PROOF:
°
Fix k> and write (2.5) for the truncated functions
/(8) fk(S) == { f(k)
for for
°
$; 8 $; k S
~
k.
Letting k -+ 00 gives (2.5) for such an f. The limit of the various terms on the left hand side follows from Fatou's Lemma and the limit of the terms on the right hand side is justified by virtue of (2.8).
2-(ii). Proof of Theorems 2.1 The starting point is the energy estimate (2.5) where we assume, up to a translation, that (xo, to) coincides with the origin. Fix uE (0, 1) and consider the family of nested cylinders Qn ==Q (6n , Pn), where Pn (2.9)
= up + (1;: u) p,
{
n
= 0, 1,2 ... ,
(1 _ u) 2n 6.
6n = u6 +
It follows from the definition that (2.10)
Qo = Q (6, p)
and
Qoo = Q (u6 up) .
Consider also the family of boxes (2.11)
where forn=O, 1,2, ... (2.12)
_ Pn {
=
Pn
+ Pn+l 2
= up +
8 = 6n + 6n+l = n
2
u
6
+
3(1 - u) 2n+2 p, 3(1 - u) 6 2n +2 •
222 VIll. Degenerate and singular parabolic systems
For these boxes we have the inclusion Qn+l C
Qn
n =
C Qn
0,1,2, ....
Introduce the sequence of increasing levels k kn = k -2n
(2.13)
where k is a positive number to be chosen. We will work with the inequalities (2.5) written for the functions (u - kn+l) +, over the boxes Qn. The cutoff function (n is taken to satisfy (n vanis~es ~n the parabolic boundary of Qn { (n
(2.14)
== 1 m Qn
2n +2 2n +2 ID(nl ~ (1 _ (1)p' 0 ~ (n,t ~ (1 - (1)6.
Set if (8 - kn+l) ~ e if 0 < (8 - kn+l) < e if (8 - kn+d ~ 0,
(2.15)
and as a function few) take f~ [(w - kn+l)+]. We put these choices in (2.5) and neglect the non-negative term involving IDul p - 2 since f;(8) ~ O. Letting e - 0 we obtain
We estimate the two integrals on the right hand side as in (7.2)-(7.5) of Chap. V. This gives the inequalities (2.16)
sup
j(W-kn+d!(x,t)dx+ ffID(w-kn+1)+IPdxdT
JJ
-8..
~ (1'Y~n;)p (PPkI6 _
Q.. p
+ 6k!-2) ff(w -
kn)~ dxdT,
Q..
valid for all 6 ~ max {Pi 2}. If P > 2, the proof is now concluded as in the proof of Theorem 4.1 in §12 of Chap. V. If 1 < P < 2, we may take 6 = r in (2.16) and
3. Weak differentiability of IDulEy! Du and energy estimates for IDul 223
obtain the analog of the recursive integral inequalities (10.3) of Chap. V. The proof of Theorem 2.1 for the singular case 1 < p < 2 is now concluded as in the proof of Theorem 5.1 in §16 of Chap. V.
3. Weak differentiability of IDuI P ;2 Du and energy estimates for IDul The main tool in investigating the local behaviour of the of the space-gradient of the solutions of (1.1) are certain local energy estimates for Ui,zj' These are derived by first differentiating' (1.1) and then by taking testing functions roughly speaking of the type ipi = Ui,zj f(lDul), I
up to some localising cutoff function. Here f(·) is a non-negative Lipschitz function in R+. In this section we discuss a rigorous way of carrying the indicated calculations. PROPOSITION 3.1 (THE DEGENERATE CASE p> 2). Let u be a local weak solution in fiT of the degenerate system (1.1). Then IDul2j!Ui,z;
EL1oc(o, Tj W1!;;(fi») , i=I,2, ... ,m, j=l, 2, ... , N,
and there exists a constant 'Y='Y(N,p), such that
j jIDuIP-2ID2UI2dxdT
(3.1)
[(zo,to)+Q(119,l1p)]
$ (I..? 0")2 [p-2 + 0-1J j
J
(1+IDulfI) dxdT
[(zo,t o)+Q(9,p)]
where
m
ID 2 ul 2
==
N
LL
u~,z;z.·
i=1 j,k=1
Moreover (3.2)
Ui,zj
ECloc(O, Tj L1oc(n)) , i=l, 2, ... , m, j= 1, 2, ... , N.
PROPOSITION 3.1 (THE SINGULAR CASE 1 < p < 2). Let u be a local weak solution of the singular system (1.1) in fiT and let the approximation assumption (1.9) hold. Then
224
vm. Degenerate and singular parabolic systems 2 r. 12 ~ . . IDul E.jl Ui.x;ELloc\O,TjW,~(il»), '=1,2, ... ,m, J=1,2, ... ,N,
and there exists a constant 'Y='Y(N,p), such that
IIIDuIP-2ID2uI2dxdr
(3.3)
[(xo ,to )+Q( 1711,17 p»)
5 (1 :
q
)2
[p-2 + 6- 2] (1 + M~)
11(1 + IDuIP) dxdr,
[(xo,to)+Q(II,p»)
where
Moreover Ui,x, E Lfoc
(0, Tj w,!;:(n)) ,
and there exists a constant 'Y = 'Y( N, p) such that
(3.4)
IIID
2 u 1P dxdr
[(Xo ,to )+Q( 0'11,0' p l)
5 (1:q)p [p-P + 6- P] (1 + M:)
11(1 + IDuIP) dxdr.
[(xo,to)+Q(II,p»)
Finally (3.5)
Ui,x, EC,oc(O, T; L?oc
This local regularity pennits to derive local energy estimates for Du. To simplify the symbolism we set (3.6)
v=IDul·
Given a cylinder [(x o, to) + Q (6, p)] c ilT we let' denote a non-negative piecewise smooth cutoff function in [(x o, to) + Q (9, p)] that vanishes on the boundary ofthe cube [xo + K pl. In particular we are not requiring in general that , vanishes for t=to-6.
3. Weak differentiability of IDul ~ Du and energy estimates for IDul 225 PROPOSITION 3.2 (LOCAL ENERGY ESTIMATES). Let u be a local weak solution of (1.1) for p > 1. In the singular case 1 < p < 2 assume in addition that the approximation assumption (1.9) be in force. Let also / (.) denote anon-negative, nOR-decreasing Lipschitz function in R +. There exists a constant 'Y ='Y( N, p) such that
(3.7)
'v' (xo, to) E nT, 'v' [(Xo, to) + Q (6, p)]
f(
sup
ff
r:/(S)dS) (2 (X, t)dx t
vP- 2 1D2 u1 2 /(v)(2dxdr
[(zo,to)+Q(9,p»)
+ (p - 2) $ 'Y
ff
t
nT
t
to-9~t~O [zo+K~)Jo
+
c
ff
+
9 0-
vP- 1 1Dv 12 /,(v)('2dxdr
[(zo,t o )+Q(9,p»)
ff
vP-3IDv.DuiI2/'(v)('2dxdr
1=1 [(zo,t o )+Q(9,p»)
vI' /(v)ID(1 2 dxdr + 'Y
[(zo,to)+Q(9,p»)
ff
(l:/(S)dS)
"t
dxdr .
[(zo,to)+Q(9,p»)
COROLLARY 3.1. The integral inequalities (3.7) continue to holdfor non-negative, non-decreasing functions / in R + , satisfying
sup /,(s)
<00,
forall k > 0,
O~.~k
provided
(3.8) PROOF: Analogous to that of Corollary 2.1.
3-(i). Taking discrete derivatives of (1.1) For a function FE Lfoc(nT) and TJER\{O}. we introduce the discrete derivative with respect to the Xj variable
CjF(x,t)==TJ-l{F (Xl, ... ,X;
+ 11, .. "xN)-F(Xb'"
,x;, .. "XN)}'
This is defined for X
E
nl'7l == {x E nl dist(x,an) > ITJI} ,
where we let 1111 be so small that n l'7l is not empty. We also let discrete gradient of F ,i.e.,
cF denote the
226
vm. Degenerate and singular parabolic systems
The discrete derivative of (2.6)', with respect to Xj, takes the fonn
~6 at '·U· h -
(3.9)
div [6, ·IDuI P- 2Du·] I h
I,
i
= 1,2, ... , m,
In transforming the term
= 0'
a.e. nl'll x (0, T - h).
[6j !Du!P-2Dui] , we only specify the Xj
variable for
simplicity of symbolism. We have (3.10)
6j !DuIP-2 DUi 1
=~! d~ {luDu(x; +,,) + (1 - U)DU(Xj)r-
2
o x (UDUi(X;
+,,) + (1- U)DUi(Xj»)}du
!
+,,) + (1 - u)DU(Xj)I P - 2 du
1
= D6jUi
luDu(Xj
o 1
+(p - 2)6jUl,Z.!luDu(x;
+,,) + (1- u)Du(x;)r- 4
o x (UUl,Z/c(Xj
+ 11) + (1- U)UI,z/c(Xj»)
x (UDUi(X;
+ 11) + (1- U)Dui(Xj) )00.
To simplify the symbolism we let ~P) (u) denote the N-dimensional vector
~P)(u) = UDUi(Xj + 11) + (1- U)DUi(X;) and let ~ (j) (u) be the N x m matrix
~(j)(u)
=uDu(x; +,,) +
(1 - u)Du(xj).
Having fixed the point (xo, to) E nT, if!(xo, to) + Q (9, p)] c nTwe may assume, up to a translation, that (xo, to) coincides with the origin, and then by choosing 111! and h sufficiently small we may assume that Q (9, p) c nl'll x (0, T - h). We multiply (3.9) by the testing function
where, is a standard non-negative cutoff function that vanishes on the boundary of K p' We integrate over ( -9, t) for arbitrary -9 < t ~ 0, and add over i = 1, 2, ... , m and j = 1, 2, ... , N. This gives
3. Weak differentiability of IDul~ Du and energy estimates for IDul 227
/.(J."";/(B)da) ,'(%,t)
+I
I [DjIDul,,-2 DUi] h ·DDjUi,h! (lc5uhl) (2dxdr
-9Kp t
+ II
[c5jIDul,,-2Dui]h·c5jUi,hD! (lc5uhl) (2dxdr
-9Kp t
= -2 I
I [c5j IDul,,-2 DUi] h'DjUi,h! (lc5uhl) (D( dxdr
-9Kp
+2
jJ(f.""';/(B)da)
(C. dztlr.
-9Kp
In this equality we first let h '..... 0, while I'll > 0 remains fixed. The various limits are justified since IDul eLfoc(nT) and ueC, oc (0, T;Lfoc(n»). Making use also of (3.10) we obtain 16 1
)
sup I ( rus!(s)ds (2(x,t)dx
(3.11)
-9
+
Jo
t -
9
l
I I (foiLl(j) (u)I,,-2dtr ) IDDj U 2 !(lc5ul)(2dxdr Q(9,p)
+(p - 2) I I (follil(j)(U)I,,-4ILl(j)(U).Dc5jUI2dtr) ! (IDuD(2dxdr Q(9,p)
+
I I (foiil(j) (u)I,,-2du ) IDIDuflc5ull' (l6ul) (2dxdr Q(9,p)
+(p - 2) I I (foiil(j)(U)I"-4Ll(j)(U)'D6jUil~j)(U)6jUidU) Q(9,p)
x DIc5ul!' (lc5ul) (2dxdr
~ 2(P -
1) I I (foiil(j) (u)I,,-2dtr ) ID6j U116ul! (16ul) (ID(ldxdr Q(9,p)
+2
II(foI6~1!(S)dS) "tdxdr. Q(9,p)
228 VID. Degenerate and singular parabolic systems
First we observe that the sum of the fmt two integrals over Q «(J, p) on the left hand side, is bounded below by
ff
min{Ij (P -I)}
(foiJ1(;) (U)I P- 2d,q) ID6;u1 2f (16ul) (2dxdr.
Q(9,p)
If p > 2, this is obtained by discarding the coefficient (p - 2). If 1< P < 2, we estimate below
(p - 2)
ff
(111J1(;)(U)IP-41J1(;)(U).D6;uI2 d,q ) f (16ul) (2dxdr
Q(9,p)
2! (p - 2)
ff (Li
J1 (j) (u)IP-2d,q ) ID6;ur f (l6ul) (2dxdr.
Q(8,p)
Next by Young's inequality, for all e > 0,
ff (l ff + ff
iJ1 (j) (u)IP-2d,q ) ID6;u116ul f (l6ul) (ID(I dxdr
Q(8,p)
(liJ1(;) (u) IP-2d,q ) ID6;u1 2f (l6ul) (2dxdr
$ e
Q(9,p)
'YE
( liJ1(;) (u)I P- 2d,q) 16ul 2f (l6ul) ID(1 2dxdr.
Q(9,p)
These remarks in (3.11) give the integral inequality involving discrete derivatives
3. Weak differentiability of IDul1j! Du and energy estimates for IDul 229
sup
(3.12)
I(10fI6~f(s)
dS) (2(x, t) dx
-9
II (liil
+ [min{I; (p -I)} - e)
t
-
9
(j) (a)IP-2d,q)
Q(9,p)
x ID6jUl2 f (16ul) (2dxd-r
II (liil II (liil
+
(i) (a)IP-2d,q )
IDl6ufl6ulf' (l6ul) (2dxd-r
Q(9,p)
+(p - 2)
(j) (a)IP-4
(il(i)(a).D6j
u) .1~j)(a)6jUid,q)
Q(9,p)
5')'
II (li JJ (l"'1
x DI6ull' (16ul) (2dxd-r
.1(i) (a)IP-2d,q)
16ul 2f (16ul) ID(1 2dxd-r
Q(9,p)
+7
f (8)d}C, dxd7,
Q(9,p)
for a constant ,),=,),(p, e).
3-(ii). Weak differentiability oflDullj!ui,zi In (3.12) take f == 1 and select a cutoff function that vanishes on the parabolic boundary of Q(0, pl. In particular, (., -0) = O. We discard the first tenn and observe that the integrand in the remaining integral on the right hand side is nonnegative. Therefore letting" - 0 with the aid of Patou's Lemma gives
(3.13)
II
vP-2ID2UI2,2dxd-r 5 ')'
Q~~
II
(vPID(/2
+ v2((t) dxd-r
~~~
for a constant')' = ')'(P). If p > 2, the inequality (3.1) follows from (3.13) by choosing (, a cutoff function that equals one on Q (aO, a p) and such that
1
ID(/5 (1 -
alp'
1
05 (t 5 (1- a)O·
To prove (3.3) for the singular case, we transfonn the last integral in (3.13) by means of an integration by parts as follows.
230
vm. Degenerate and singular parabolic systems IIv 2CdxdT = II Du.Duv1!j! v!TCdxdT Q(9,p)
Q(9,p)
= II UiD [v2:f1 DUi] v!T CdxdT Q(9,p)
+
II UiVEj! DUiDv!jR CdxdT Q(9,p)
+
II UiVEj! DUiv!jR DC dxdT Q(9,p)
~'Yllulloo,Q(9,p) II (vP-2ID2uI2C2) t v!jR dxdT Q(9,p)
+'Yll u ll oo,Q(9,p)
I I (I + IDuI P) IDCI dxdT. Q(9,p)
Finally (3.4) follows from (3.3) and RUder inequality, since
lfiD2ulPdxdT = II(vP-2ID2uI2)P/2v~dxdT. Q(9,p)
Q(9,p)
Since v1!j! ID 2uI E L~oc(nT). the energy inequality (3.7) follows from (3.12) by letting '1--+0.
3-(iv). Continuity of Ui,:I:;(t) in L~oc(n) and energy estimates By virtue of (3.1) and (3.3) the system in (1.1) can be written in the differentiated fonn (3.14)
!!.U· = 0 in 1J'(flT)' &t 1,:1:; - div (IDuIP-2 Duo) 1 :t&; i = 1,2, ... ,N, j = 1,2, ... ,m.
Moreover (3.12) implies that (3.15) These two facts imply that t --+ Ui,:t&; (t) is weakly continuous in L~oc(n). Indeed let cP E L2 (K p) and let {'Pn} be a sequence of functions in C~ (Kp) such that
IIcp -
'Pnll2,Kp
--+
0
as n
--+ 00.
Taking CPn as a testing function in (3.14) and integrating over Kp x (tl, t2) gives
4. Boundedness of IDul. Qualitative estimates 231
!
!! t3
[Ui,z; (t2) - Ui,z; (td] CPndx =
~
(IDul,,-2 DUi) Dcpn,z;dxdr
~~
for almost all -0 < tl
< t2 $ O. Therefore
limsup ![Ui'Z;(t 2) - ui,z;(td] CPndx It 3- t d-..o
= O.
Kp
From this and (3.15)
I
lim sup f[Ui'z; (t2) - Ui,zj (td] CPdxl It 3- t d-..o
Kp
$ limsup I t 3- t lf-
!
[Ui,z;(t2) - ui,z;(td] CPn dx
Kp + 2 sup IIUi,z; 112,Kp(t)lIcp - CPn1l2,Kp. -8$t$O
To prove that Ui,z; is strongly continuous in L~oc(n} it suffices to prove that (3.16)
limsup (lIui,z;(1I2,Kp(t2) -lIui,z;(1I2,Kp(tl») f-
--+
0,
It 2- t l
where, is a piecewise smooth cutoff functions in Kp vanishing on oKp. In (3.14) take the testing function and integrate over Kp x (tl' t2). By calculations similar to those leading to (3.12) we obtain
If
[.'(!,) -
.'(',)J "dxl
Kp
4. Boundedness of IDul. Qualitative estimates Using the weak differentiability of IDul2j1ui,z; we first prove that IDul is in Lloc(nT ) for all q ~ 1. If K-o C K-l are compact subsets of n T • we will show that the norm IIDullq,K:o is bounded only in terms of q, dist {K-o; K- l } and the norm
232
vm. Degenerate and singular parabolic systems
IIDullp,K:l. We will do this in a qualitative way and with no precise specification of the functional dependence. We will use such qualitative infonnation to prove still qualitatively that /Du/ E Ll:c {!1T ), with bounds only dependent on local V-nonns of /Du/. Finally, in the next section, we will tum such qualitative infonnation into precise quantitative estimates of IIDulloo,K: o over compact subsets lCoc{h. LEMMA 4.1. Let u be a local weak solution of(1.1). Moreover in the singular case 1 < p < 2 let the approximation assumption (1.9) be in force. Then
/Du/
E Lloc{{h),
forevery q E [1,00).
PROOF: Consider first the degenerate case p > 2. Let Q (6, p) c {h and let ( be a standard non-negative cutoff function vanishing on the parabolic boundary of Q (6, p). Thus, in particular, (., -6) =0. In (3.7), take J(v) =vP, where P?O is to be chosen. Proceeding fonnally we obtain
(4.1)
sup Iv +pe (x, t)dx $ 'Y jrJf (1 + vP+ 2
-9
Q(9,p)
Kp .
o
(4.2)
lt ) dxdr
IIIDv£¥r
II (1 + vP+
dxdr $ 'Y
-9K p
P) dxdT,
Q(9,p)
where 'Y = 'Y (N,p, p, (t, D(). These are rigorous if the right hand side is finite. We apply the embedding Theorem 2.1 of Chap. I to the functions
x
-+
(v£¥()
(x, t),
a.e. t E (-6,0),
over the cubes Kp. It suffices to consider the case N > 2. Indeed if N = 1, 2, we may consider u as a vector field defmed in RN N ? 3, up to a localisation, and deduce inequalities (4.1 )-(4.2) for it. Let 6 be a positive number to be chosen. Then by Corollary 2.1 of Chap. I and HOlder's inequality
We integrate over (-6,0), to obtain
4. Boundedness of IDuI. Qualitative estimates 233
JJIv.'¥ 'I'dzdT C:~r
1N
JJ....•..
"dzdT $
Q(8.p)
Q(8.p)
- - Kp
Choosing 6 = 2¥1 and combining this with (4.1)-(4.2) gives the recursive inequalities
I I tr.8~+i (2dxdr :5 'Y
(4.3)
Q(8.p)
11(1 +
vJ'+.8) dxdr,
Q(8.p)
o.
for a constant 'Y = 'Y(N,p,/3,(t,D(). The right hand side is finite for /3 = Therefore IDul E Lr;4/N ({IT). We may now again apply (4.3) with /3 =4/N and proceed in this fashion to prove the lemma. We now tum to the singular case 1 < p < 2. In (3.7) assume that (x o, to) == (0,0) and choose a cutoff function ( that vanishes on the parabolic boundary of Q «(J, p). Take also f (v) = v.8, where /3 ~ 0 is to be chosen. By working with the approximations claimed by (1.9) we will use the qualitative infonnation that IDul E L~oc(nT). Our estimates however will be only in tenns of IIDull".Q(8.p). Proceeding fonnally we obtain from (3.7)
(4.4)
Illvl!.±f=.! D
2U(r dxdr :5 -y {
Q(8.p)
II tr.8dxdr + II V2+.8(dxdr} , Q(8.p)
Q(8.p)
where -y=-y (N,p, {3, (t, D(). Also by a fonnal integration by parts
/lv.8+ 2 (dxdr = //V~+f-2 Du.DuvP+~-P(dxdr Q(8.p)
Q(8.p)
= IluD(V~DU) v~(dxdr Q(8.p)
+ //
u vP.±f=! Du Dv~ ( dxdr
Q(8.p)
+ II u vP.±f=! Du v~ D( dxdr Q(8.p)
:5 'Yllulloo.Q(8.p) / /l v l!.±f=3 D2ul( v~ dxdr Q(8.p)
+ 'Yll u lloo.Q(8.p) II v.8+1 dxdr. Q(8.p)
We combine this with (4.4) and make use of the Schwartz inequality to arrive at
vm. Degenerate and singular parabolic systems
234
JJ
v P+2Cdxd'T $ 'Y
(4.5)
Q(6,p)
!J( + 1
v P+(2-p) + vP+l) dxd'T,
Q(6,p)
for a constant 'Y='Y (N,p, (j, Ct, DC, lI u ll oo ,Q(6,p») .
This inequality is indeed rigorous as long as the right hand side is fmite. We apply it fIrSt with (j =p-l to deduce that IDul E L~l (nT ). with bounds only dependent on II Dullp,Q(fI,p) . Then we apply it again with {j =p to deduce that IDul E L~2 (nT ). Proceeding this way proves the lemma. , LEMMA 4.2. Let u be a local weak solution of(1.1). Moreover in the singular case 1 < p < 2 let the approximation assumption (1.9) be in force. Then
PROOF: Consider first the degenerate case p> 2. Let Q (6, p) c nT and let Qn and Qn be the family of cylinders introduced in (2.9)-(2.12). Let also k,. and Cn
be respectively the increasing levels defined in (2.13) and the cutoff functions in Qn introduced in (2.14). We put these choices in the energy estimates (3.7) and as a function f (v) take
f(v) :: (v - kn+l)r 2 • By virtue of Lemma 4.1 and Corollary 3.1. such a choice is admissible. The teon involving D 2 u is estimated below by
~ (~) P-jIID (v -
2 22
I I v"- ID uI f(v)C2dxd'T Qft
12
kn+l)! C!dxd'T.
Qft
1bese choices yield the inequalities (4.6)
-fl:{:~O 1[(v-kn+1)lCn]\x,t)dx K pft
!!ID
+ kP - 2
[(v - kn+1)l Cnr dxd'T
Qft $
(12!;2p2 JJV2(P-l)X[(V -
kn+l)+> 0] dxd'T
Qft
for a constant 'Y='Y(N,p). To simplify the symbolism let us set (4.7)
Yn ::
!!(V-kn)~dXd'T. Qft
4. Bc..undedness of IDul. Qualitative estimates 235
Then we have(l) (4.8)
/ / X[(v - kn+l)+> 0] dxdr
~ 'Y 2k7
Qn
/ / (v -
kn)~ dxdr
Qn
== 'Y2np k -p Yn. By Proposition 3.1 of Chap. I with m=p - 2 and q= 2(N + 2)/N. (4.9)
Yn+1
~/
/[(V - kn+d,:/2
(nr
dxdr
Qn
$
(tfi(-- (.]'''# dxdT) (if>IJ- - oJ dxdT) k.H
w'h
)'."
"-+1)+>
x
~ (1'Y~:)2 k-~Yn~ {p-2 / /V2(P-I)x[(V -
.to
kn+l)+>O] dxdr
Qn
+ (J-I //vPX[(v -
kn+l)+> 0] dxdr} I
Qn
where.\2 == N(P - 2) + 2p. These are the key recursive inequalities needed to derive a quantitative sup-bound for IDul. We will use them first in a qualitative way as follows. First let A denote a lump constant depending upon (J I (I, P and the quantities IIDuIl2,Q(B,p)
II Dullq,Q(B,p)
q
= (N + 2)(P -
Then we estimate
/ / v 2(p-I)X[(v - kn+l)+>O] dxdr Qn
= / / vwhvP~x[(v -
kn+d+> 0] dxdr
Qn
(I) See §7-(i) of Chap. V and, in particular, estimate (7.2).
2) + p.
236 VIll. Degenerate and singular parabolic systems
We have also(1)
II
(4.10)
vPX[(v - kn+d+>O] dxdT
~ -y2np Yn·
Q..
Therefore
II
v 2(P-1)x[(v - kn+1)+> 0] dxdT
~ A2npYn~.
Q..
1bese remarks in (4.9) give the recursive inequalities
Yn+1
~ A k-m bny;+wh ,
n
= 1,2, ... ,
b = 4"+1
where we have also used the choice k ~ 1 and the inequality y;+wh ~ A y1+wh.
It follows from Lemma 4.1 of Chap. I that Yn -+ 0 as n -+ 00 if Yo = (Ak- m ) -(N+2) b-(N+2)2.
1berefore IIDulloo,Q(a9,ap) ~ max{l; k}
~ 1 + A¥b(N+2)2/p IIDullp,Q(9,p). We now tum to the singular case 1 < p < 2. The starting point is still the energy estimate (3.7) where we choose Qn, Qn, (n and the levels kn as before. As a function 1(·) we take if r > 2 if r where
= 2,
Ie (.) is the Lipschitz approximation to the Heaviside graph, introduced in
(2.15). After we let e -+ 0, the first term on the left hand side is bounded below for all t E ( -fJ, 0) by the quantity
f(l
K
p"
tJ
s2- p s (s - kn +1r- 2 dS) (!(x, t)dx +
k"+!
~ r~1 (~r-Pf[(V-kn+1)12(nr dx. K p"
(1) See for example estimate (7.5) of Chap. V.
4. Boundedness of IDul. Qualitative estimates 237 The term involving D 2 u is estimated below by
IIID2U l2 (v - kn+1)~-2 x[(v - kn +1)+>O] (~dxdr Qn
~ r~ IIID (v -
kn+d12
Qn
r(~dxdr.
Combining these estimates in (3.7) we arrive at (4.11)
k 2- p sup -B
-
I [(v - kn+1>12 (n(x, t)] 2 dx
KPn
+ IIID (v - kn+1)1 2 (n1 2dxdr Qn
where 'Y='Y(N,p, r). To simplify the symbolism we set (4.12)
Sn=
JI(v-kn)~dxdr, Qn
and combine (4.11) with the embedding of Proposition 3.1 of Chap. I, with m = p=2 and q=2(N + 2)/N. This gives (4.13)
'Y 2(r+2)n 2 (2N"~+r wh Sn+1 ~ (1-0')2 k+ Sn + X
{p-2 Ilvrx[(v - kn+d+> 0] dxdr Qn
+ 6- 1 II v r+(2- p )x[(v - kn +1)+>O] dxdr} , Qn
where we have used a version of (4.8). These are the key inequalities needed to derive a quantitative bound of IDul in the singular case 1 < p < 2. As before, we will use them fIrst in a qualitative way. We choose k ~ 1 and let A denote a lump constant depending upon p, 6, 0' and the norms
II vIl2,Q(B,p) , IIvll q,Q(B,p) , Then we estimate
q=(N + 2)(2 - p)
+ r.
238
vm. Degenente and singular parabolic systems
ff
l1r+{2- p)X[(v - kn+d+>0] dxdr
Q..
= f/11(2-p)+m l1r~x[(V -
kn+l)+>O] dxdr
Q..
~
$
(IfoftdxdT) (If
v',f.(v -
~,)+>ol dxttr)
~
In deriving the last inequality we have used a version of (4.10). These estimates in (4.13) yield the recursive inequalities Sn+l
S!+~ ~ A S!+rtr . The proof is now concluded as in the degenerate case.
5. Quantitative sup-bounds of \Du\ THEOREM 5.1 (THE CASEp>2). Letubealocalweaksolutionofthedegenerate system (1.1). There exists a constant 'Y='Y(N,p) such that
V(xo,to)eflr, V[(x o,to)+Q(9,p)]cflr, Vo-E(O,l),
(5.1)
.
sup
[(zo.to)+Q{...8 ....p»)
IDul
~
(
'YV(9/p2) 1-
0-
)(N+2)/2
()~ ff IDul dxdr I'
(zo.to )+Q(8.p»)
(p2)i6 .
"9
THEOREM 5.2 (THE SINGULAR CASE 1 < p < 2). Let u be a local weak solution of the singular system (1.1) and let the approximation assumption (1.9) be inforce. Moreover let r ~ 2 satisfy
(5.2)
Vr
== N(P -
2)
+ 2r > o.
Then there exists a constant 'Y='Y(N,p, r) such that
5. Quantitative sup-bounds of IDul 239
(5.3)
Remark 5.1. The constant -y(N,p, r) in (5.3) tends to infinity as vr-O.
5-(i). Proof of Theorem 5.1 We start from the recursive inequalities (4.9) and estimate the first integral on the right hand side as follows:
//v 2(P-l)X[(v - kn +1 )+> 0] dxdr Q..
~ (s~~v) 2-'lJv"X[(V - kn +1)+> 0] dxdr Q..
~ 2 (s~~ v) 2-jJ( v - kn)~ dxdr. n"
Q..
Therefore (4.9) yields (5.4)
Yn +1 ~ (1 ~b:)2 k--ih
{(s~~vr-2 p-2+(r Y~+~, 1}
b=4".
If for some n= 0,1,2, ... we have
there is nothing to prove. Otherwise we rewrite (5.4) as Yn +1 ~
-ybn
[(1 - u)p]
2
(
sup v
)"-2 _~ k
N+2
l+~ .
Yn
Q(9,p)
It follows from Lemma 4.1 of Chap. I that {Yn } neN - 0 as n -
00,
if k is chosen
from
(N+2)/2 b(N+2)/2) k>'2/ 2 == ( 'Y 2 [(1 - u)p]
(N+2~(p-2)
( ) sup v Q(',p)
JrJrv"dxdr.
Q(',p)
240
vrn. Degenerate and singular parabolic systems
We conclude that there exists a constant ,,(=,,(N,p) such that 1-4/>'2
sup
(5.5)
V
Q(a9,ap)
< -
(
"(
[(1 - 0")p]2(N+2)/>'2
sup v ) Q(9,p) 2/>'2
IlvPdxdr
)
( X
Q(9,p)
If O"E (0,1) is fixed. consider the family of boxes
Q(n) :;:: Q
(On, Pn). where
Po:;:: O"p and for n= 1, 2, ... n
n
Pn
= O"P+ (1- O")p LTi
On
= 0"0 + (1- 0")0 L2-i. i=l
i=1
By construction. Q(O) :;::
Q (0"0, O"p)
Set
Mn
and
Q(oo)
== Q (0, p) .
= esssupv Q(n)
and write (5.5) for the pair of boxes Q(n) and Q(n+1). This gives (5.6)
where ,,(>'2/ 4
B :;::
[(1 - 0")p](N+2)/2
(f!vPdxdr)
d 2(N+2)/2. =
1/2
'
Q(9,p)
The proof is now concluded by the interpolation Lemma 4.3 of Chap. I.
5-(ii). Proof of Theorem 5.2 We start from the recursive inequalities (4.13) and estimate
IIvr+(2- )x.[(V - kn+d+> 0] dxdr p
Qn
~ (s~~v) 2-pII vrx[(v - k n +1)+> 0] dxdr
rQn
~ 2 (s~~v nr
p
Sn.
5. Quantitative sup-bounds of IDul 241 We may assume that 2 sup Qn v -
p
> - !!... p2'
tior all n = 0 , 1 , 2 , ....
Otherwise there is nothing to prove. Taking this into account, we rewrite (4.13) as
(sUPQ(8,p) v )2-P Sl+wh
'Ybn k"Nh(r+2-p) (1 _ 0-)2
<
S
n+l -
(J
n
where b = 4r+l. By an argument analogous to that in the degenerate case, this implies
sup
v<
Q(
-
(5.7)
'Y (1 - 0') r~t!p
SUP (
---.l!.±L V2-P) Q(8,p) 2(,+2-p)
(J 1!(r+2-p) )
(
jjvrdXdT
X
Q(8,p)
The proof is now concluded with an interpolation process as in the degenerate case. This is possible if the power of the term sUPQ(8,p) von the right hand side of (5.7) is less than one. Since
(2 - p)(N + 2) 2(r + 2 - p)
=1 _
Vr , 2(r + 2 - p)
this occurs if (5.2) holds. We also remark that the interpolation process applied to (5.7) generates a dependence of the type of l/vr in the constant 'Y(N,p, r) appearing in (5.3).
5-(iii). Interpolation inequalities The inequality of Theorem 5.1 can be interpolated. For example consider (5.1) for (xo, to) == (0, 0) and rewrite it as
sup Q(
v
< -
'Y ..[(iTiJ)
(1 - 0')(N+2)!2
(sup Q(8,p)
v) ~
(n
Vp-2H dXdT)1!2
Q(8,p) ....1....
A( ~y-3
Such an inequality can be interpolated as long as e E (0, 2] and proves the following:
242
vm. Degenerate and singular parabolic systems
THEOREM 5.1' (THE CASE P > 2). Let u be a local weak solution of the degenerate system (1.1). Then for every E E (0,2]. there exists a constant 'Y = 'Y(N,p,E) such that
V(xo, to) E nT,
v [(x o, to) + Q (8, p)] c nT, Vq E (0,1),
(8/,r) l/~
'Y
(5.8)
sup
[(zo,t o)+Q(1J'9,lJ'p»)
IDul $ (
1-
q
)(N+2)/~
(
H
,,-2+~
IDul
dxdT
)
l/~
[(zo,to)+Q(9,p»)
A
Remark 5.1. The constant 'Y = 'Y( N, p, E) /
00
( ,r)~ 8 .
as E '\. O.
Also, (5.3) can be interpolated. We rewrite it for (xo, t o):= (0, 0) and in the fonn
sup
Q(1J'9,lJ'p)
V
<
-
'Y
(,r /8) N/vr
(1- q)2(N+2)/vr
sup
!t!:.=.U "r ( V
(Q(9,P) )
H
vqdxdT
)2/
Vr
Q(9,p)
A
This can be interpolated as long as qE (0, r] satisfies (5.9)
IIq
(!..)"!; p2 .
2(::q) < 1. This occurs if
:=N(p - 2) + 2q > O.
The interpolation process gives THEOREM 5.2' (THE SINGULAR CASE 1 < p < 2). Let u be a local weak solution of the singular system (1.1) and let the approximation assumption (1.9) be in force. Moreover let r ~ 2 satisfy (5.2). Then for every q E (0, r] satisfying (5.9) there exists a constant 'Y='Y(N,p, r, q) such that
(5.10)
Remark 5.3. The constant 'Y(N,p, r, q) in (5.10) tends to infinity as IIq -0.
6. General structures 243
Remark 5.4. Estimate (5.10) is fonnally equivalent to (5.3), the only difference being that q is not required to be larger or equal to 2. The only condition is that (5.9) be verified. In particular, (5.10) holds for q=p provided 2N p> N +2'
(5.11)
ID2Ul eL1oc(nT ). From (3.3), for every [(zo, to) + Q (8, p)] c nT,
COROLLARY PROOF:
5.1.
Let 1
! !I D2U I2dxdT = !!IDuI2-PIDUIP-2ID2UI2dxdT [(zo,to)+Q(8,p»)
:::;
[(zo,t o)+Q(8,p»)
sup [(zo,t o)+Q(8,p»)
IDu1 2-p !!IDUIP-2ID2UI2dxdT < 00. [(zo,t o)+Q(8,p»)
6. General structures Let U be a local weak solution of the non-linear system (1.10) subject to the structure conditions (81 )-(86 ), The local boundedness of U can be established as in the proof of Theorem 2.1. The main modification occurs in the handling of the 'perturbation terms' l(Ji, i = 0, 1, 2. These contribute to the energy inequalities (2.5) with an extra tenn of the type
! !{1(J0 (I(w) + wl'(w» + (1(J1ID(1 + 1(J2) wl(w)} dxdT. [(zo,to)+Q(8,p»)
Given the choice (2.15) of 1(')' these terms are estimated as in the sup-bounds established in Chap.V for general equations. (1) The weak differentiability of the tenn IDulp-2 Du follows from the structure conditions (81 )-(82), We proceed as before by working first with the discrete derivatives. All the tenns involving the 'derivatives' 6j Ui,zc' are dominated by the tenns arising from the right hand side of (82 ), (2) Following the same process of §3 yields local energy estimates similar to (3.7) with constants 'Y ='Y( N, p, Co, C1 ) and with the right hand side augmented by the extra integral
! !{1(J0 (I(v) + vl'(v» + (1(J1ID(1 + 1(J2) vl(v)} dxdT. [(zo,to)+Q(8,p»)
(1) See for example Theorem 3.1 of Chap. V and its proof. (2) See also Remark 1.1.
244
vm. Degenerate and singular parabolic systems
These energy estimates imply that IDul E L~(nT) be the same iterative techniques of §4. The 'perturbation terms' are dealt with as in Chap. V.
7. Bibliographical notes In the case of a single equation the estimate (1.5) up to ST has been established by Lieberman [68]. Estimates in the norm cl,a up to ST for Dirichlet data, are not known even for elliptic systems. Results for a single elliptic equations are due to Lieberman [69] and Lin [72]. The general structures of §1-(l1) have been introduced first by Tolksdorff [95]. The arguments of finite differences to prove that IDul,-2Ui,Xj is weakly differentiable were introduced by Uhlenbeck [99] in the context of elliptic systems. The sup-bound of Theorem 2.1 for the degenerate case p > 2 is new. The same theorem for the singular case 1 < p < 2 is due to Choe [31]. The qualitative Lemmas 4.1 and 4.2 appear in [36] for all p > ~~2' and in Choe [31] for all p > 1 provided (1.7) holds. Even though some quantitative estimates of the gradient appear in a variety of forms in [27,36,37], Chen [25] and Choe [30], the precise form of Theorems 5.1 and 5.2 as well as their interpolated version in §5-(lII}, seems to be new.
IX Parabolic p-systems: Holder continuity
of Du
1. The main theorem The space gradient Du of local weak solutions of the quasilinear system (1.10) of Chap. VIn are locally HOlder continuous in nT provided the structure conditions (St}-(Ss) are in force. We will show this first for the homogeneous system (1.1) and then will indicate how to extend it to the general systems (1.10). The estimates of this chapter hold in the interior of nT and deteriorate near its parabolic boundary r. If /C is a compact subset of nT we let dist(/Cj r) denote the parabolic distance from /C to the parabolic boundary r of nT, i.e., dist (/Cj r) == THEOREM
inf
(.. ,·lelC (1I,·ler
(Ix - yl + ~) .
1.1. Let u be a local weak solution 0/(1.1) o/Chap. VI/l. Moreover
if 1 < p < 2 let the approximation condition (1.9) be in /orce. Then (x, t) -Ui,zj (x, t) E Ct!c(nT), for some a E (0,1), for all i = 1, 2, ... ,m and all j = 1, 2, ... ,N. Moreover/or every compact subset /C OinT. there exist constants a=a(N,p) E (0,1) and-y=-y (N,p, II Dull oo,K:) > 1. such that (1.1)
IUi,Zj (Xl, tt} -
I
Ui,Zj (X2' t2) ~ -y
( IXI-X21+ltl-t211/2)Q dist (/C; r)
,
/or every pair o/points (XI,tl), (X2,t2)E/C.
Remark 1.1. The constants -y and a are independent of dist (/C; r). They however deteriorate as p'\. I, i.e.,
246 IX. Parabolic p-systems: ltilder continuity of Du lim inf 'Y (N, p, II Dull 00 K) , a-I (N,p) p'\,l
'
- + 00.
Remark 1.2. The functional dependence of'Y upon IIDulloo,K will be given in §§3 and 4.
l-(i). Some notation and the two basic propositions The proof of Theorem 1.1 is based on estimating the essential oscillation of [(x o, to) + Q (6, p)] c {IT. After a translation we may assume that (x o, to) coincides with origin. Let IJ and R be positive numbers and consider the cylinders
Ui,%j in cylindrical domains of the type
{
(1.2)
Qn(lJ} == Knx {-1J 2 - P R 2 ,0}, satisfying sup IDul:5 IJ. QR(")
The geometry of Qn(lJ) is intrinsic in that the t-dimension is 'stretched' by a
factor, loosely speaking, of the order of IDuI 2 - p • Let us assume for the moment that such boxes can be constructed. Then Theorem 1.1 is a consequence of the following two propositions. PROPOSITION 1.1. There exist numbers II, It, 6 in (0, 1) that can be determined a priori only in terms of N and p. such that if
there holds //IDU - (Du)"+l12dxdr :5 1t6N +2/ /IDU - (Du},,1 2dxdr,
(1.4)
Q,,, R(")
Q,,,+lR (,,)
for all n= 1,2, ... , where (Du)"
=
ff Dudxdr. Q'''R('')
PROPOSITION
there holds
1.2. There exists numbers j
1. The main theorem 247
IDul(x, t) ~ rn~,
(1.6)
These two facts will be used to establish the following: THEOREM 1.2. Assume that the cylinder Q R (/J) satisfies (1.2) for some /J > 0. There exist constants "Y> 1 and a E (0, 1) that can be determined a priori only in terms of N and p, such that
~~ Ui,z; ~ "Y/J (~r, VO
(1.7)
for all i=I,2, ... ,mandall j=I,2, ... ,N.
1-(;;). Constructing QR(/J)
°
Assume first that p > 2. The number R> being fixed, let /Jo be the smallest value of the parameter /J such that QR(/Jo) C {IT. If IIDulloo,QIlC",o) ~ /Jo, then QR(/Jo) satisfies (1.2). Otherwise we take as /J the largest root of the equation
IIDulloo,QIlC"') = /J. Such an equation has finite roots since IIDulloo,QIlC",o) > /Jo, and
/J-IIDulloo,QIlC",) remains bounded as /J-OO. These arguments are based on the fact that, since p > 2, the 'vertical size' of Q R(/J) decreases as /J increases. In the singular case we consider instead boxes of the type (1.2)'
{
QR(/J)
== {Ixl ~ /J~ R} x {-R2,0}, satisfying
sup IDul ~ /J. QIl("')
As /J increases, the cubes {Ixl < /J ~ which (1.2)' holds.
R} shrink. Therefore there exist some /J for
Remark 1.3. The previous propositions could be stated and proved in the geom-
etry of the boxes (1.2)'. Indeed setting /J~ R=r permits one to recast the 'space scaling' of (1.2)' in terms of the 'time scaling' of (1.2).
248 IX. Parabolic p-systems: HOlder continuity of Du
1-(;;i}. More about the intrinsic geometry Take formally the xj-derivative of (1.1) of Chap. VIII and multiply the ith equation of the system so obtained. by 'Ui.z j ' Adding over i = 1,2, ... , m and j = 1,2, ... ,N. and setting W = IDul 2 we arrive at the formal differential inequality (1.8)
!
W - (al,kw2j!wZk) Zl
~0
in
nT,
where
at,k == { 6t ,k + (p - 2)
(1.9)
U'i;;:j;Zk } .
The matrix (at,k) is positive definite and w is a non-negative weak solution of an equation of the porous medium type. This is a parabolic version of the quasisubharmonicity. The degeneracy of (1.8) is of the order of w 2j! and it is overcome by the choice ofthe parabolic geometry of QR(I-').
2. Estimating the oscillation of Du We assume Propositions 1.1 and 1.2 for the moment and proceed to prove Theorem 1.1. Let QR(I-') be a cylinder satisfying (1.2) and define the two sequences (2.1)
{
1-'0 = 1-', Ro = R and for n I'n+l
= TJl'n,
Rn+l
= 1,2, ... ,
= CoRn,
where TJ and 0' are the numbers claimed by Proposition 1.2 and (2.2) Since TJ E (!, 1). we have Co E (0,1) for all p > 1. Suppose the assumption (1.5) bolds with R = Ro and I' =1'0' Then
IDul ~ TJ 1-'0 == 1-'1'
sup Q.. Ro("'o)
From the definitions (2.1) and (2.2) it follows that Rl < Ro and
R21
a 2..,p-2 'r
R20
I-'f-2 = - 4 - TJP-21-'~
_
2
(0')2 R20
= '2 1JP-2 •
This implies that the cylinder QRl (I-'d is contained in QtTRo (1-'0) and
sup
IDul ~ 1-'1.
QR1(",d
Therefore QRl (I'd satisfies (1.2) an4 if the assumption (1.5) of Proposition 1.2 is verified again for such a box we have
2. Estimating the oscillation of Du 249
sup QR2(~d
IDul ~ 1-'2.
Proceeding in this fashion, suppose the assumption of Proposition 1.2 is verified for the cylinders
QRn (I-'n) , n = 0,1,2, ... ,no - 1 for some positive integer no· Then (2.3)
IDul ~
sup
I-'n
== '1 n l-'o,
n
QRn(~n)
= 0,1,2, ... , no.
From the definitions (2.1) and (2.2) it follows that
One verifies that (2.3) as (2.4)
Co
sup
< '1 for all p > 1, and consequently al IDul
~
1-'0
QRn(~n)
E (0,1). We rewrite
(Rn)Ql ,for n = 0, 1,2, ... , no· D
no
Suppose now that the assumption (l.5) of Proposition l.2 fails for no. We call Rno the switching radius. Then for the box QRno (I-'nJ the assumption (1.3) of Proposition 1.1 holds and we conclude that (2.5)
H
IDu - (DU)i
12 dxdr ~ KiH IDu -
Q ,iRno (~no )
(Du)o 12 dxdr
Q Rno (~no ) i 2 . 1 2 < _KI-'n o ' t=, , ...
Writing (2.6)
I(Du)i+ 1
-
(DU)iI 2
~ 21Du -
(DU)i+l1 2 + 21Du - (DU)ir
and taking the integral average over Q6i+1Rno(l-'no) gives
'Y
= 2 (K + cS-(N+2») .
Therefore {(DU)iheN is a Cauchy sequence whose limit we denote with Du( x o , to). To motivate this terminology we recall that our arguments are carried over an arbitrary cylinder [(xo, to) + QR(I-')] with vertex at (xo, to). Therefore if no is the switching radius of the box [( x o , to) + QR(1-')], the limit ofthe averages,
HDUdxdr, [(zo,tO)+Q'iRno (~no)]
250 IX. Parabolic p-systems: Hl)lder continuity of Du
n
is Du(x o• to) for almost all (x o• to) E T • It follows from (2.6) that 2. 2 ~ 'Y 1t'lJno '
1Du(xo•to) - (Du), 1
i = 1.2•....
Fix 0 < p< Rno and denote with (Du)p the integral average of Du over Qp(lJnJ. Let i be a positive integer such that (2.7) and estimate
I(DU)p - (DU),r
(2.8)
if
~
IDu - (DU),r dxd-r
Qp(""o)
~ 'Y6-(N +2) It'1J!0'
Therefore
IDU(xo• to) - (DU)pr
(2.9)
~ 2IDu(Xo. to) -
(DU),r
+ 21 (Du)p - (DU),r ::; 'Y(6) It'1J!0' It follows from (2.7) and (2.9) that
Let 2Qo =min{Ql; Q2}. Then combining (2.9) and (2.4) we conclude LEMMA 2.1. There exist constants 'Y> 1 and Q o E (0. 1) that can be determined a priori only in terms 0/ Nand p. such that/or almost all (x o•to) E nT such that [(xo. to) + QR(IJ)] c nT. and/orallO
(2.10)
Moreover (2.11)
if
IDu - (DU)pr dxd-r
~ 'Y IJ! (~) 2
Q
o •
Qp(""o)
Remark 2.1. The lemma holds also in the geometry of the boxes [(xo. to) + QR(IJ)] introduced in (1.2)'. Indeed we may set 1J2.j! R=r and work within the cylinder [(xo. to) + Qr(IJ)]. We arrive at a version of (2.10) that reads (2.10)'
3. RUder continuity of Du (thecasep>2) 251
Returning to the geometry of [(%0. to) + QR(P)] proves the assertion. Analogous considerations bold for (2.11).
3. HOlder continuity of Du (the case p > 2) We assume that IDul ELOO(nT ) and set P = II Dull OO,DT •
This is no loss of generality. by possibly working with another compact set
K:,' satisfying K:, C K:,' c nT. and
dist (K:,j K:,') ~
Id.
We will prove the mlder continuity of Du in the time and space variables separately.
3-(i). HOlder continuity in t Fix two points (%0. t.) E K:,. i tl > to and construct the cylinders
[(xo.t.)
+ QR(P)]
= 0.1. with the same 'abscissa' Xo' We let
={IX - xol < pEj! R} x {t. - R2. t.}.
The box [(xo. h) + QR(P)] intersects [(xo. to) + QR(P)] at the point (xo, to). if (tl - to) < R2. Moreover they are contained in nT if (3.1) LEMMA 3.1. Let (3.1) hold. There exist constants 'Y > 1 and Q E (0.1) that can be determined a priori only in terms 0/ Nand p such that/or all pairs (x o • t.) E K:" i=O.l.
PROOF: Let Rn, be the switching radii of the cylinders [(xo. ti) + QR(P)] and introduce the two boxes
Qi
Assume first that (3.3)
= [(xo. t.) + QR", (Pn,)] = {Ix - x.1 < PRy Rn, } x {t. - R~,. ttl .
252 IX. Parabolic p-systems: H6lder continuity of Du
and for i=O,l construct the two cylinders
Ci ==
[(xo, ti) + Q"'2('I-'O) (I'n;)]
= {IX - xol < I'::T J2(tl -
to) } X {ti - 2(tl - to), ti}.
By virtue of (3.3) we have the inclusions Ci C Qi, i = 0, 1. Moreover Co and C1 intersect in a box satisfying meas [Co n Cl ] ~ min {I'no ; Jl.nl} (tr - t o)(N+2)/2.
(3.4) Set
(Du)c; ==
H c;
Dudxdr,
i
= 0,1,
and estimate
IDu(xo, tl) - Du(xo, to)1 ~ IDu(xo, t.) - (DU)Cl I
+ IDu(xo, to) - (Du)C o I + I (Du)c, - (Du)c I· o
By (2.10) and Remark 2.1 we have, for i =0, 1. o IDu(xo, ti) - (Du)c; I ~ 'Y I' ( "'tlR- to)a
To estimate the last term we add and subtract Du(x, t) where (x, t) E Co n Cl , and then take the integral average over such intersection, i.e.,
I (Du)co
-
(Du)co I ~ HI (Du)c, - Du(x, t)ldxdr conel
+
H
IDu(x, t) -. (Du)c o Idt. conel
Without loss of generality we may assume that min {I'no ; I'nl} = I'nl. Then we estimate the first integral by extending the integration over the larger set Cl . Taking into account the definition of C1 , (3.4) and (2.11), we obtain
HI (DU)Cl - Du(x, t)1 dxdr conel
~ 'Y HI (DU)Cl -
Du(x, t)1 dxdr
Cl ~'YI'
( ~)ao R
To estimate the second integral. let fJ be a small positive number to be chosen and assume that
3. .Hl)lder continuity of Du (the case p> 2) 253 (3.5) Then using again (2.11) and (3.4),
H
IDu(x, t) - (Du)co Idt
Co nel
:5 "I ( ~:: )
N(P-2)/2H
IDu(x, t) - (Du)Co Idt Co
(I'no )N(P-2)/2 (~)QO I'nl I' R Qo-{JN(p-2)/2 < ( V(tt - to) ) -"II' R
<
- "I
Therefore if (3.3) and (3.5) hold, the assertion (3.2) follows by taking {3 = Qo/ N (p - 2) and then choosing Q =Qo/2. If (3.5) is violated,
IDu(x o , to) - DU(Xl, tdl :5 21'
Q /N(p-2) ( ~) R o
0
,
and the assertion follows by suitably modifying the defmition of Q. We consider next the case when (3.3) is violated, i.e.,
2(tl - to) > min {R!o j If
R!, :5 2(tl -
IDu(x
R!1}'
to) for i=O, 1, then by (2.4) o, tt) - Du(xo, to)
I:5
I'no
+ I'nl :5 "II'
(v'fl=t;;) R
Q
Therefore we may assume that, say,
R!1 :5 2(tt -
to) < R!o'
We conclude the proof by reducing this case to the situation (3.3). Let n. :5 nl be a positive integer satisfying
R!.
~ 2(tl - to) ~
R!.+l'
and introduce the cylinders Qo and Q •• where
Q. == (x o, ttl Since we have
+ QR". (I'n.) == {Ix - xol < I':f Rn. } x{tl - R!.,tl}' 2(tt - to) :5 min {R!o j R!.} ,
the box Q .. will now play the same role as the cylinder Ql in the case (3.3). The proof is now concluded as before, observing that for Q.. the two inequalities (2.10) and (2.11) hold true.
2S4 IX. Parabolic p-systems: H6lder continuity of Du LEMMA 3.1'. There exist constants 1> 1 and Q e (0,1) that can be determined a priori only in terms 0/ N andpsuch that/or every pairo/points (x"to)elC, i=
0,1,
PROOF: If JI.'~
1 we take R=dist (lC;r) in (3.1). Otherwise we take IJ¥ R=
dist (/C; r).
3-(U). Holder continuity in x Fix two points (xo, to) and (Xl, to) in /C, at the same time level to. and let
[(Xi, to) + QR{IJ») == {Ix - Xii < R} x {to - 1J2- P ~, to}, i
= 0,1,
be two boxes satisfying (1.2). The box [(xo, to) + QR(P)] will intersect Xl iflxoxII < R. Moreover they are contained in nT if
(3.6) LEMMA 3.2. Let (3.6) hold. Thereexistconstants1> 1 andQE (0, 1) that can be determined a priori only in terms 0/ Nand p. such that/or all (x" to) E /C, i =0, I.
(3.7)
I
IDU(XI, to) - Du(xo, to) $ 11J (Ixo ~ XII) Q
•
Let R,." be the switching radii corresponding to the two boxes [(Xi, to) + QR(IJ»). and construct the two cylinders
PROOF:
. Qi ==
[(x" to) + QR", (IJn.)]
== {Ix - xii < R,.,,} x {to -IJ!;-P R!" to} • Consider separately the following two cases: (3.8) (3.8)' If (3.8) holds. construct the two boxes
C, == [(Xi, to) + Q21%o-%11 (1Jn')] == {Ix - x,1 < 21xo - xII} x {to -1J!;-P4Ixo - xll2, to}.
By construction these are contained in Qi and they overlap in a box satisfying
3. ltilder continuity of Du (the case p> 2) 255
Set
(Du)c, ==
if
Dudt,
c,
and estimate
I + IDu(xo, to) - (Du)c I + I(DU)Cl - (Du)c I·
IDu(Xt. to) - Du(xo, to)1 :5; IDu(xI, to) - (DU)Cl o
o
The proof now proceeds as for the HOlder continuity in t with minor cbanges. LEMMA 3.2'. There exist constants "y > 1 and Q E (0, 1) that can be determined a prior; only ;n terms of N and p such that for every pair ofpoints (Xi, to) E /C, i =
0,1.
PROOF: If JJ ~ 1. in (3.6) we take R = dist (/C; r). If JJ JJ2.f! dist (/C; r). Then (3.7) reads
<
1. we take R
(3.7)'
If
2.f! JJ
:5;
0/2 ( IXl - Xo I )
dist (/C; r)
,
there is nothing to prove. Otherwise (3.7)' gives
Thus (3.10) follows by suitably redefining the number Q.
3-(iii). A version of Theorem 1.1 Combining Lemmas 3.1' and 3.2' gives the following form of Theorem 1.1:
=
256 IX. Parabolic p-systerns: Holder continuity of Du THEOREM 1.1' (THE DEGENERATE CASE p> 2). Let u be a weak solution in nT of the degenerate system (1.1) of Chap. VIII, and assume that I-' = IIDulloo.oT < 00. There exist constants 'Y > 1 and Q E (0,1) that can be determined a priori only in terms of Nand p such that, for every compact subset fC of
nT •
(1.1')
IDu(xo, to) - Du(xl. t 1 )1
< - 'YI-'
(Ixo - Xli + max{I;I-'2j!}v'lto- t11)0< dist (fC; r)
,
for every pair of points (Xi, til E fc, i=O, 1. Remark 3.1. The constants 'Y and Q are independent of dist (fc; r) and 1-'. The fonn of (1.1)' suggest we reduce the system (1.1) of Chap. VIII to another for which I-' 1. Introduce the change of variables
=
Vi
Ui == -,
. t
IJ.
Vt -
= 1, 2 , ... ,m,
div IDvlp - 2 Dv
= 0,
and T = tl-'p-2.
in nT
== nx (0, T I-'p-2) ,
and IIDvll oo,~6T ~ = 1. We write (1.1)' for v in the variables (x, T) and return to the original coordinates. This gives
for all pairs (Xi, til E fC, i =0, I, where IJ. - dist (fc; r)
==
inf
( .. ,tIEIC (v,-IET
(Ix - yl + 1J.2j!~)
is the intrinsic parabolic distance from fc to
r.
4. HOlder continuity of Du (the case 1
4. Hl)lder continuity of Du (the case I < p < 2) 257
4-0). Holder continuity in t Fix two points (xo, til E J(" i = 0, I, with the same 'abscissa' Xo. We let tl > to and construct the cylinders
[(Xo, ti) + QR(I-')] == {Ix - xol < I-'Ej!
R}
The box [(xo, t l ) + QR(I-')] intersects [(xo, to) over they are contained in flT if
X {ti
-
R2,ti}'
i
= 0,1.
+ QR(I-')] if (h -to) < R2. More-
max {1-'2jl R; R} :$: dist (J(,j r) .
(4.1)
Proceeding as in the case p> 2 we have
4.1. Let (4.1) hold. There exist constants 'Y > 1 and a E (0,1) that can be determined a priori only in terms of N and p such that
LEMMA
Next if 1-'?I, we take R=d in (4.1), and if 1-'< I, we rewrite (4.2) as (4.2)' Arguing as in the proof of lemma 3.2' and bY.possibly redefining the constants 'Y and a, we obtain LEMMA 4.1'. There exist constants 'Y > 1 and a E (0,1) that can be determined a priori only in terms of Nand p such that for every pair of points (xo, ti) E J(" i = 0, I,
4-0i). HOlder continuity in x Fix two points (xo, to) and
(Xl, to)
in J(" at the same time level to, and let
[(Xi, to) + QR(I-')] == {Ix - Xii < R} x {to -1-'2-p R2, to}, i = 0, 1, be two boxes satisfying (1.2). The box [(xo, to) XII < R. Moreover they are contained in flT if (4.4) We proceed as in the case p> 2 and establish
+ QR(I-')] intersects Xl if Ixo -
258 IX. Parabolic p-systerns: Holder continuity of Du LEMMA 4.2. Let (4.4) hold. Thereexistconstants'Y> 1 andaE (0, I) that can be determined a priori only in terms of N andp,such thatforall (Xi, to)EK:, i=O, 1.
4-(iii). A version of Theorem 1.1 Combining Lemmas 4.1 and 4.2 gives the following fonn of Theorem 1.1 THEOREM 1.1" (THE SINGULAR CASE 1 < p < 2) .. Let u be a weak solution in fh of the degenerate system (1.1) of Chap. Vl/I. and assume that /J = IIDulloo,uT < 00. There exist constants 'Y > 1 and a E (0,1) that can be determined a priori only in terms of N and p such that. for every compact subset K",ofnT.
(1.1")
IDu(Xo,to) - DU(Xlotl)1
~ 'Y/J (
max{l j /J¥}lXo - xII + vito - tll)Q dist (K",j r) ,
for every pair of points (Xi, ti) EK"" i=O, 1.
Remark 4.1. The constants 'Y and a are independent of dist. (K:j r) and /J. Arguing as in §3-(III), the HOlder continuity of Ui.:J:j can be expressed in tenns of the intrinsic parabolic distance /J-dist (K:j r).
5. Some algebraic lemmas We let QR(/J} c nT be a cylinder satisfying (1.2) and consider the system
!
(5.1)
Ui -
div IDulp - 2 DUi = 0, in QR(/J), p > 1,
and the one obtained by taking the derivative with respect to Xj' i.e., (5.2)
!
Ui,:J:; -
div
(IDUIP- 2 DUi,:J:; +
a:; IDuIP- DUi) = 0, 2
in QR(/J), i=I,2, ... ,m, j=I,2, ... ,N. We let V denote a vector in R Nxm satisfying (5.3)
We also let 'Y='Y(N, p) denote a generic positive constant that can be detennined a priori only in tenns of the indicated quantities.
5. Some algebraic lemmas 259 LEMMA 5.1. There exists a constant 'Y = 'Y(N,p). such that for every vector V E RNxm. and/or all p > 1.
{lDul + IVI)Y IDu - VI:5 'YIIDuIY Du -IVIYVI·
(5.4)
LEMMA 5.2. Let 1 < p < 2. There exists a constant 'Y = 'Y(N,p) such that/or every vector V E R Nxm.
IIDuIP-2 Du - IVlp-2VI:5 'YIDu - ViP-I.
(5.5)
Moreover if the vector V satisfies (5.3). then (5.6)
IIDuIP-2 Du -IVlp- 2VI:5 'Y,",p- 2IDu - VI,
(5.7)
IIDuIP-2 Du _IVIP-2vI2IDuI2-p :5 'Y,",p- 2IDu - V12.
Remark 5.1. These lemmas are algebraic in nature and could be stated for any pair of vectors U and V, provided (5.3) is replaced by (5.3)' Also in (5.4) the number (1'-2)/2 could be replaced by any number and in (5.5)(5.7) the number (1'-2) could be replaced by any negative number. PROOF OF LEMMA 5.1: By calculation,
IIDulY Du -lvIYVIIDu - VI 2: 1(IDuI'i'Du-IVI'i'V, DU-V) / f1d = \10 ds ISDu + (1 =
10f~IsDu + (1 -
+
p; 21~ 0
s)VI
s)VI
Y
Y (sDu + (1 - slY) ds,
IDu -
IsDu + (1- s)VI
I Du -
V
)
VI 2ds
y I{sDu + (1- s)V, Du - V)I 2 ds
~ min{l; (p - 1)}IDu - VI 210f~IsDu + (I - s)VI ~ ds. If1
1o~IsDu + (I -
s)VI
Y ds ~ {lDul + IVI) Y
,
and the lemma follows in this case. If p > 2, assume for example that Then
IDul > IVI.
260 IX. Parabolic p-systems: Hi)lder continuity of Du
1o~IsDu + {I PROOF OF LEMMA
s)V/ ~ ds
~
111/2
~
-p IDul--'-- .
{sIDul- (I -
1
s)IVI) ~ ds
1!=.!
5.2: For 1
/IDul,,-2 Du - IVI,,-2V/ ~ IVI,,-2IDu - VI
+ /IDul,,-2 -IVI,,-2/IDul ~ IVI,,-2IDu - VI IDu~-1 ) + {2 - p)IVI,,-2IDu - VI ( eIVI,,-1 + {I _ e)IDul,,-l '
e
for some E [0,1]. Interchanging the role of Du and V gives (5.8)
/IDul,,-2 Du - IVI,,-2vl
~ IVI,,-2IDu - VI { 1 + (2 -
p) eIVI,,-1
~~71~-;)IDUI"-1 } •
e
for some E [0, 1]. and (5.8')
IIDul,,-2 Du - IVI,,-2vl IVI,,-I}
~ IDul,,-2IDu - VI { 1 + (2 - p) '1I VI,,-1 + (1 _ '1)IDul,,-1 ' for some 'IE [0,1]. To prove (5.5) assume rust that 1
IVI> "2 IDu - VI·
(5.9)
This implies
These inequalities in (5.8) prove (5.5). If (5.9) is false. its converse gives the two inequalities
These in (5.8)' imply that the term in braces on the right hand side is bounded above by an absolute constant. Moreover
IVI,,-2IDu - VI ~ IDul,,-2IDu - VI,,-IIDu - VI 2-". The two inequalities (5.6) and (5.7) are an immediate consequence of (5.8) and the assumption (5.3).
5. Some algebraic lemmas 261
Let H be the vector in R Nxm defmed by
Hi == IDul p- 2DUi -IVlp- 2Vi -IVl p- 2 (DtI.i - Vi) - (p - 2)IVlp- 4 Vt,k (Ul,:/:. - Vt,k) Vi, i= 1, 2, ... , m.
(5.12)
We will estimate IHI for all p> 1. For this we first set Wet) == tDu + (1 - t)V,
(5.13)
for t
E
[0,1],
and rewrite (5.12) in the form 1
Hi = j ~ {ltDu + (1 - t)VIP-2 (tDui + (1 - t)Vi)} dt o -IVl p- 2 (DUi - Vi) - (p - 2)IVlp-4 Vt,k (Ul,:/:. - Vt,k) Vi 1
= (DUi -
Vi) j{IWIP-2 -IVI P- 2} dt o 1
+ (p -
2) (DUl - Vt) j{IWIP-4WlWi -IVI P- 4 VtVi }dt o
== HP) + H~2), I I where we have dropped the t-dependence from W. From (5.13) W - V
(5.14)
= t (Du -
V) ,
and for every sE [0,1]. sW + (1- s)V = V + st(Du - V).
(5.15)
5.3. There exists a constant "'( = "'(N,p) such that/or every constant vector V ERNxm satisfying (5.3), and/or all p> 1,
LEMMA
(5.16)
Remark S.2. The lemma holds for every pair of vectors U and V satisfying (5.3)'. 5.3 (p>2): Assume first thatlVI ~ 21Du - VI. Then
PROOF OF LEMMA
1
IHI
~ ",(p)IDu -
VI j(IW(t)I P- 2 + IVIP-2) dt o
~ "'(IDu - VIP-l ~ IZI (lDul + IVj)p-2IDu - V12. Therefore (5.16) follows in this case since V satisfies (5.3). If
262 IX. Parabolic p-systems: Hl)lder continuity of Du
IVI > 21Du - VI,
(5.17)
then by the mean value theorem and (5.14)-(5.15),
IHI ~ 'Y(P)IDu - VI 2
1
flsW + (1 - s)VIP- dt, 3
o for some s E [0,1]. By (5.15) and (5.17) we have
and this implies the lemma. PROOF OF LEMMA
5.3 (1
(5.18) Assume first that
IVI ~ IDu - VI,
(5.19) and let t* E [0, 1] be defined by
t*
=
IVI IDu-VI
Then
IHI ~ 'Y(P)
1
fltlDu - VI_IVII,-2IDu - VI dt + 'YIDu - VIIVI,-2 o
"~ {l~
l'IDu - VHVIIP - 1 dt +
i~
IIIDu - VHVII P - 1dt }
+ 'YIDu - VIIVI,-2 ~ 'Y (lVIP-l + IDu - VIP-l) . Therefore taking into account (5.19) and (5.3), the lemma follows in this case. Consider now the case when (5.19) is violated, i.e., (5.19)' Then fori=l,2, ... ,m,
IVI > IDu - VI·
6. Linear parabolic systems with constant coefficients 263 1 1
Hi
= (DUi -
Vi) I I :alsW + {1 - s)VIP- 2 dsdt o0 1 1
+ (Dut -
Vi) I l:a {lsW + {1 - S)VIP-4 {sWt + (1- s)Vi) o0 X {sWi
+ (1 - s)Vi) }ds dt.
Therefore 1 1
(5.20)
IHI :5 'Y{p)IDu - VI 2 I It IsW + {1- s)VIP- 3 dsdt. o0
Next, by (5.15) and (5.19)'
IsW + (1 - s)VI = Iv + st(Du - V)I
~
IIVI- stlDu - VII
~
IVI{1 - st).
This in (5.20) gives 1 1
IHI :5 'YIDu - V1 21V lp- 3 Ilt{1 - st)P- 3 ds dt. o0 Since 1 < p < 2. the last integral is finite and the Lemma follows.
6. Linear parabolic systems with constant coefficients Let V be any vector in RNxm satisfying (5.3). To the system (5.1) we associate its linearised version (6.1)
a
at Vi -
(IVIP-2 vi ,xI
in QR{/J),
+ (p -
2)IVlp-4Vj,k Vj,xt Vi,l) XI
'
i = 1,2, ... , m.
Let v
==
(Vl,V2, •••
,vm )
and for 0 < p:5 R we let (Dv) p denote the integral average of Dv over Qp (/J).
264 IX. Parabolic p-systems: mlder continuity of Do THEOREM 6.1. There exists a constant'Y='Y(N,p). such thatforall O
(6.2)
H
IDv - (Dv)p 12 dxdT :S 'Y (~) 2
~w
H
IDv -
W1 2 dxdT.
~w
To prove the theorem we introduce the change of variables
v(x, t) -
V
(x, tp.2-,,) .
This transfonns Qp(p.) into Qp(l) == Qp for all 0 < p:S R. and tranfonns (6.1) into a system for which In a precise way, the transfonned vector v is a solution of
8
(6.3)
at Vi
-
( ..
a~:~ Vi,z.
)
Zt
= 0,
where the coefficients
= IVI,,-2 {flijt,lI: + (p _ 2) \-j,ll: Vi,t } IVI2
a i,j t,ll: -
satisfy the ellipticity condition
Co (N,p)leI 2 :S a~~lI:eiltej,ll: :S C1 (N,p)lel 2 ,
(6.4)
Ve E RNxm,
for two given constants Co < C 1 depending only upon N and p. Therefore it will suffice to prove Theorem 6.1 for p. = 1. In the remainder of the section we let v be a solution of (6.3) in QR and let (6.4) hold. Let a denote a multiindex of size lal. i.e., N
a==(al,a2, ... ,aN), ajENU{0},j=l,2, ... ,Nj lal=Laj, j=1
and for f E Coo (Q R) let
For non-negative integers m and n we also set
ID;'fl ==
L
ID:II,
D~f== ~f,
ID:fl
= ID~II = III·
lal=m
6.1. There exists a constant 'Y ='Y(N,p) such that for all non-negative integers m, n and all 0 < p:S R, LEMMA
6. Linear parabolic systems with constant coefficients 265
II ID,:+l D~VI2 dxd1' ~ 'Yp-211ID': D~VI2 dxd1',
(6.5)
Qp/2
Qp
1!ID~+lD':VI2dxd1' ~ 'Yp- 4 !!ID':D~VI2dxd1'.
(6.6)
Qp/2
Qp
The system (6.3) is also solved by the vectors W == D';Div. Let (be a non-negative smooth cutoff function in Qp vanishing on the parabolic boundary of Qp and such that PROOF:
W(2
Multiply the system (6.3), written for w, by the testing function and integrate and over Qp. to arrive at (6.5). To prove (6.6). mUltiply the same system by integrate over Qp. This gives (6.7)
Wt(2
!!lwtI2(2dXdT+ !! (a~,,{Wj,x/c! Wi,X t ) (2dxdT Qp
Qp
= -2 I!
a~:{ Wj,x/c Wi,t( (Xt dxdT
Qp
+~ !!IWtl2(2dXd1' +;!!IDwI 2dxd1'. Qp
Qp
The integral involving a~:{ on the left hand side of (6.7) equals
These remarks in (6.7) give
!!IWtI2dxdT ~ 'Yp- 2!!IDw I2 dXdT. Qp/2
Qp
The lemma now follows by applying (6.5) and suitably modifying the scale of the radii P and p/2.
266 IX. Parabolic p-systems: H61der continuity of Du LEMMA
6.2. There exists a constant 'Y='Y(N,p) such that/or all 0 < p~ R
(6.8)
It suffices to prove the lemma for 0 < p ~ R/2 N +2. Let ( be the standard cutoff function in QR/2N +1 that equals one on QR/2N +2 and such that PROOF:
ID(I ~ 2N+2/R
and
0
< (e ~ (2N+2/R)2.
IfO
!!lvI2 dxdT
~ 'YpN+2"v"~,Qp ~ 'YpN+2"v("~,QIl/2N+1.
Qp On the other hand for all (x, t) eQR/2N+1, t
Iv(l(x, t)
=
I!
De (v() (x, T)dTI
_R2/2 N+1
~ 'Y !ID: Dt(v()1 dxdT QIl /2N+1
Combining this with Lemma 6.1 we obtain the estimate
"V("~,QIl/2N+1 ~ 'YR-CN+2>!!lvI2dxdT. QIl
This in (6.9) proves the lemma.
6.1: Since the vectors vz",z. solve (6.3) for h,s = 1,2, ... , N, we have from Lemma 6.2
PROOF OF THEOREM
(6.10)
!!ID2v I2 dxdT Qp
~ 'Y (~)N+2!!ID2vldxdT' QIl/2
for a constant 'Y='Y(N,p) and for all 0
Let W be any constant vector in R Nxm and multiply (6.11) by the testing function (Wi,z. - Wi,.) (2. where ( is the standard cutoff function iO Q R that equals one in QR/2. This gives
6. Linear parabolic systems with constant coefficients 267
IIID2vI2d3:dr :5 'Y R- 2 IIIDv - WI 2d3:dr. QIl/2
QIl
To estimate the left hand side of (6.10) set
f
(Dv)p(t) = Dv(x, t)dx,
VO
_p2 :5t:50.
K,.
Then x - (Dv,(x, t)-(Dv,)p(t» has zero average over Kp. and by the embedding Theorem 2.1 and Remark 2.1 of Chap. I.
IIIDV - (Dv)p(r)1 2d3:dr:5 'Yp211ID2vI2d3:dr. Qp
Qp
Write
IIIDV - (Dv)pI2 d3:dr :5 'Y (~) NHIIIDV -
(6.12)
Qp
Wl 2 d3:dr
QIl
o
I
+ 'YpN / (Dv)p - (Dv)p(r)j2 dr, -p2
and estimate the last term by
o
pN/I(Dv)p - (Dv)p(r)1 2dr:5'YpN+2 sup
_..2
-p2
I(Dv)p(t) - (Dv)p(r)r·
Next integrate (6.11) over K p x (r, t) and divide by meas{ K p} to obtain
I(Dv)p(t) - (Dv)p(r)j :5 'YP-NIIID 2v 1d3:dr Qp
$. p-N
$.
P"P (£fID'VI'dzdr) 1/'
~p-N/' (£fIDv _WI'dzdr) 1/.
Therefore the last term on the right hand side of (6.12) is estimated by
'Yp2
(~)N+2fIID2vI2d3:dr:5 'Y (~)N+'l/IDV - WI 2 d3:dr. QIl/2
QIl
268 IX. Parabolic: p-systems: mlder continuity of Du
7. The perturbation lemma LEMMA 7.1. There exists a constant 'Y = 'Y(N,p) such that/or every constant vector V in RNxm satisfying (5.3),
PROOF: Let ( be a cutoff function in QR(P.) that equals one on QR/2 (p.). and such that
In the weak formulation of (5.2) we take the testing functions
modulo a Steldov time average. We obtain sup
(7.2)
flDU -
VI2(2(X, t) dx +
-,.2-PR2
f fIDul,,-2ID2uI2(2dxdT
JJ I
QR(")
$;'Y
ffIDU-VI 2((t dxdT + J, QR("')
where
-IVI,,-2Vi,;)z; (Ui,z; - Vi,;)
(D(dxdT.
~ ffIDUI,,-2ID2UI2(2dxdT + 'Yp.;~2 fflDU -
VI 2 dxdT.
J='Y
f fODUI,,-2DUi QR("')
If p > 2. we have
J $;
QR(,.)
QR(,.)
Putting this estimate in (7.2) proves the lemma in the degenerate case. To estimate J in the singular case 1
7. The perturbation lemma 269
~ 'Y ffIIDUIP-2DU -IVIP-2VIID2UI(ID(ldxdT
J
(7.3)
QIl(,.)
ff
+
II DuIP- 2 Du - IVIP- 2 VIIDu - VI(ID 2(ldxdr
QIl(,.)
==Il+h By (5.7) of Lemma 5.2 and Schwartz inequality
~ ~ f fIDUIP-2ID2UI2(2dxdr + i£;~2 f flDU - Vl 2dxdr,
II
~W·
~w
and by (5.6)
12 ~ 'Yp;~2 fflDU - Vl 2dxdr. QIl(")
Combining these estimates in (7.2) proves the Lemma. Let 8pQR/2 (p) denote the parabolic boundary of boundary value problem Vi,t -
(7.4)
(IVIP-2 vi ,x;
{ Vi L')pQIl/2(,.)=Ui,
+~ &=
QR/2 (p).
Consider the
2) IVlp- 4 Vt,k vi,xlo Vj,i) Xj , in Q R/2 (JL)
1,2, ... ,m.
The existence of a unique solution to (7.4) can be established for example by a Galerkin procedureP) The solution v == (vt, V2, ••• , 11m) of (7.4) is 'regular' in the interior of QR/2 (p), in the sense of Theorem 6.1. The next lemma compares U and v. LEMMA 7.2. There exists a constant 'Y='Y(N,p), such thatfor all 0< p< R/2 and for every vector V satisfying (5.3),
fpDU -
(7.5)
Dvl2dxdr
~
'Y
(p_2f! IDu - Vl2dxdr)1I
Qp(")
QIl(")
f flDU - Vl 2dxdr, QIl(")
where a=mint!; i}. PROOF: Write the system (5.1) in the form
! i
Ui -
(IVI P- 2U i,Xj
= 1,2, ... ,m,
(1) See Lions (73).
+ (p -
2)IVl p -
4 Vt,k Ui,:t:1o VjJ) Xj =
div Hi,
270 IX. Parabolic p-systems: H5lder continuity of Du
where the vectors Hi are introduced in (S.12). From this. subtract (7.4). and in the weak fonnulation of the system so obtained, take the testing function Ui -Vi. This is admissible since it vanishes on lJp QR/2 (",). Adding over i= 1, 2, ... , m. gives
",p-1/ IDu - Dvl 2 dxdT
~ ..,/ / IHIIDu -
Q 11/2 (,,)
DvldxdT,
Q11/2 (,,)
where we have taken into account the fact that V satisfies (S.3). Using Schwartz inequality on the right hand side and then Lemma S.3 to estimate IHI2. we arrive at (7.6)
/ / IDu - Dvl 2 dxdT QIl/2(")
~ ..,,,,-2(P-l)//
(IDul + IVI)2(P-2) IDu -
V1 4 dxdT.
QIl/2(")
To estimate the right hand side of (7.6) assume first that N ~ 4 so that
a
= min{!' ~} = ~ 2' N N'
-
To simplify the symbolism we let r =",2-p R2 /4. We have
n.7)
//
(lDul + IVI)2(p-2) IDu -
VI 4 dxdT
QIl/2(")
~ l(;.~~Do' + IVIl2(p-2) IDo _ VI'U) i
7. The perturbation lemma 271
By Lemma 7.1
(7.8)
~*(P-l)
sup -r
(fIDU _V 12dt) i J I KR/2
To estimate the last factor in (7.7) we majorise the integrand by means of Lemma 5.1. It gives
(lDul + IVI)2(p-2) IDu - VI 4 = {(IDuI + IVI) zy! IDu _ VI} 4
~ 'YIIDulZY! Du _IVIZY!vI 4
~ ~P~IIDulZY!DU-IVIZY!vl~· Let x -+ {(x) be a non-negative piecewise smooth cutoff function in KR that equals one on K 3R / 4 and such that ID{I ~ 4/ R. Then for a.e. tE {-r, O}, by the embedding Corollary 2.1 of Chap. I, we have
N-2
~ 'Y~P¥ (![lIDUIZY! Du -IVIZY!VI{] ~ dx)-,;r K3R/4
~ 'Y ~p¥ !
ID
[IDulZY! Du -IVIZY!V] {1 2dx
K3R/4
Here in estimating the last term we have used the algebraic inequality
272 IX. Parabolic p-systcms: H5lder continuity of Du
which follows from (5.6) of Lemma 5.2 with p replaced by (p + 2)/2. Therefore the last factor on the right hand side of (7.7) is estimated by
1V.~~DoI
N-2
+ IVI)2lP-2) IDo - VI'.J") -,,- dT
:5 'Y I'P~ {
jjlDulP-2lD2ul2dxdT Q3R/2C",)
j
+ I'p-2 R- 2 jlDU -
V 12 dxdT}
QRC",)
:5 1'2CP -l)-1f R- 2
j jlDU - V1 dxdT, 2
QRC",)
where we have also used Lemma 7.1. We now combine these calculations in (7.7) and then in (7.6) to obtain
jjlDU - Dvl
2 dxdT
QpC",)
provided N ~ 4. If N = 2, 3, we transform the integral on the right hand side of (7.6) by HOlder's inequality as follows.
7. The perturbation lemma 273
II
(7.9)
dxd'T
ODul + IVn 2(p-2) IDu - Vl 4
QR/2(")
o
=
I
IIDU - VI (IDul
+ IVn 2(p-2) IDu -
Vl 3
dxd'T
-rKR / 2 1
~ 1(l~-V'2dzr x
(/ODU KR/2
~
'
p.2j! sup -r
+ IVI)4(P-2) IDu -
V 16
dx) ! d'T
(fIDU - V12dx) ! J I
KR/2
xl(J.!;IDuI
1
+
IVI)'" IDu -vi]' dz) ·d.
By Lemma 7.1
We estimate the last tenn on the right hand side of (7.9) separately for N
N=2. The case N=3 Let, be defined as before. Then for a.e. t E ( -r, 0).
=3 and
274 IX. Parabolic p-systems: II)lder continuity of Du
(J[(IDul + IVI) E? IDu - Vr
dx )
1
KR./2
~
'Y JL f
Rl
(J[ (lDul + IVI) E? IDu _ Vf
dx)
1
KR./2
,; 7,,1 RI
(L!!IDuI'i'Du-IVI'i'VI,r
~ 'Y JLf Rl
JID [IDulE? Du -IVIE?V] '1 dx. 2
K SR./4
1berefore
l(J.!~,Du' + IVI)'i' IDu -Vi]"
dT
JJIDU - V1 dxdT. 2
QR.(")
Combining these estimates in (7.9) and then in (7.6) proves the lemma for N =3.
TbecaseN=2 We apply the embedding Theorem 2.1 of Chap. I with q = 6,
B
=1. This gives for a.e. t E (-r, 0)
( J[(IDuI + IVI)'i'IDu -Vi]" J I KR./2
,; 7
~ 'Y
)
(f.~~DuI'T' Du -IVI'T'VI'j'
,r
dx
Q
= 2/3 and
8. Proof of Proposition 1.l-(i) 275
1berefore
l(J.!~IDuI + IVI)"'IDu - VI]' ~ 'Y~fR
dz)
! dT
fflD [IDul~ Du-IV\2j!v] (1 dxd.,.. 2
QR(")
We estimate these integrals by means of Lemma 7.1 and combine the calculations in (7.9) and in (7.6) to conclude that (7.5) holds with a=~.
8. Proof of Proposition 1.1-(i) LEMMA 8.1. There exist constants ~, 6, E E (0, 1) that can be determined a priori only in terms of N and p. such that ijVo is a constant vector in R Nxm satisfying (8.1)
H\DU - V l dxd.,. ~ E~2,
(8.2)
o 2
QR(")
then there exists a constant vector V t E R Nxm such that
(8.4)
ffiDU - V l dxd.,. ~ ~6N+2ff\DU - V l dt, t 2
o 2
Q'R(")
(8.5)
QR(")
HIDU-Vt I2 ~E~2. Qu(,,)
PROOF:
Let v be the unique solution of (7.4) and set Vt
==
HDvdt,
Qu(,,)
where 6 E (0,1) is to be chosen. The perturbation Lemma 7.2 with V triangle inequality and (8.2) give
=V o • the
276 IX. Parabolic p-systems: Jl)lder continuity of Du
I PDU - V l l2 dxdT :5 "Yeo IIIDU - V ol2dxdT Qu(,,)
QIl(")
+ I fiDV - V l l2 dxdT. Qu(,,)
By Theorem 6.1
lfiDV - V l l2 dxdT:5 "Y6 N +4IIIDv - V ol2dxdT, Q,Il(")
QIl/2(")
and again by Lemma 7.2 with V =V 0 and (8.2)
II IDv - V ol2dxdT :5 "Y (1 + EO) I IIDU - V ol2dxdT, QR/2(")
QR(")
for a constant "Y="Y(N,p). Combining these inequalities we obtain
I fiDU - Vll2dxdT:5 "Y (6 NH +eO)IIIDU - V ol2dxdT, Q,Il(")
6:5 1/2.
QIl(")
To prove (8.4) choose EO =6NH , and then 6 so small that 2"Y62 :5 ".
Inequality (8.5) follows from (8.4) and the s1I1Illiness assumption (8.2). To prove (8.3) write
Vl
-
Vo
H =H =
(Dv - Vo)dxdT
Q,R(")
{(Dv - Du)
+ (Du -
Vo)}dxdT
Q,R(")
and IVl
-
V o l2 :5 2
H
IDu - Dvl2dxdT + 2
QIIl(")
H
IDu - V ol2dxdT.
Q,R(")
By Lemma 7.2 and the indicated choices of E and 6
H
IDu - Dvl2dxdT :5 "
~RW
Therefore using again (8.2)
H
~RW
IDu - V ol2dxdT.
8. Proof of Proposition 1.l-(i) 277 (8.6)
IV1
-
V ol2 ~ 2 (K, + 6-(N+2»)
H
IDu - V o l2dxdT
Qa(,,)
~ 2 ( K, + 6-(N +2) ) dl 2(NH) p.2 ~ 2K,p.2. By choosing K, sufficiently small we may insure that
and
LEMMA 8.2. There exist constants K" 6, EE (0,1) that can be determined a priori only in terms of N and p. such that if V 0 is a constant vector in R Nxm satisfying (8.1) and (8.2). then there exists a sequence of constant vectors {Vi} el in RNxm. satisfying
H
IDu - Vil2dxdT
(8.8)
~ Ep.2,
Q6I a(")
(8.9)
! !IDU - Vi+!1 2dxdT
~ K,6 N+2!/IDU -
Q6l+ 1 a(")
V i l 2dxdT,
Q6 I a(")
for i = 1, 2, .. " PROOF: The sequence is constructed inductively by using the procedure of the previous lemma. To prove that IVil are in the range (8.7), we refer back to (8.6), i.e.
IVi +! - V i l2 ~ 2 (K, + 6-(N+2»)
H
IDu - V i l2dxdT.
Q6I a(")
We iterate over i and use again the smallness assumption (8.2) to obtain
IVi+l - V i l2 ~ 2 (K, + 6-(N+2») K,i
H
IDu - V o l 2dxdT
Qa(,,)
~ 2p.262(NH) (K, + 6-(N+2») K,i. From this by taking roots and adding over i 00.
IV i+l - Vol
,fK.
~ P.6L.,fit ~ p. 1- ,fK.' i=l
278 IX. Parabolic p-systems: Ifi)lder continuity of Du
where we have used the specific choice of 6 in tenns of It. Choosing now It sufficiently small proves the Lemma.
9. Proof of Proposition 1.1-(ii) The number 11 in the assumption (1.3) can be chosen to insure the existence of a constant vector Vo E RNxm satisfying (8.1) and (8.2). This is the content of this section. Set IDul =v and, for all 0 < p ~ R,
== ((x,t) B; == {(x, t)
(9.1)
E Qp(l-') Iv(x,t)
A;
(9.2)
E Qp(l-') Iv(x, t) < (1 - II)I-'} .
We will choose 11 E (0,
l) and rewrite (1.3) as
(9.3)
IBRI ~ IIIQR(I-')I,
LEMMA
> (1-1I)1-'},
11 E (0,1)·
9.1. There ex;stsa constant."Y ="Y(N, p) such thatforall uE (0,1)
jr fIDul,,-2ID2uI2 dxdT <
(9.4)
P
A:
-
"Y 1-'211 RN. (1- u)2
1t
PROOF: Consider the differentiated equation (S.2) and in its weak fonnulation take the testing function Ui,z;
(v 2 - k2)+ (2,
k = (1- 211)1-',
modulo a Stelclov averaging process. Here ( is a non-negative piecewise smooth cutoff function in QR(I-') that equals one on QtT R (I-') and such that 1 1-',,-2 ID(I ~ (1- u)R' 0 ~ (t ~ (1- u)W' After we add over i = 1, 2, ... m and j I
(9.S)
sup,
/(,:,2 -
=1, 2, ... , N, we arrive at
k2): (2(x, t) dx
-",3-PR2
+ / /v'P-2IDv212(2x. [v> k] dxdT QIt(,,)
m
+
N
?:?: //IDul,,-2IDUi
,z.:/
12 (V 2 -
k2)+ (2dxdT
1=1 .1=1 Q"It(")
~ "Y / /v'P-2IDv21 (v2 -
k2)+ (ID(I dxdT
QIt(,,)
+ "Y //(v 2 QIt("')
k2): ((tdxdT
9. Proof of Proposition l.l-(ii) 279
for a constant "Y="Y(N,p). By the Schwartz inequality
"Y jjvP- 2IDv2I (v 2 - k2)+(ID(ldx.dT QR(p)
~ jjvP-2IDV212(2X[V > k]dxdT QR(p)
+"Y2 j j vP- 2 (v 2 - k2): ID(1 2dxdT. QR(P)
We put this in (9.S) and in the resulting inequality we discard all the non-negative terms on the left hand side except the integral containing DUi,Zi' This gives
fjlDulP-2lD2ul2 (v 2 - k2)+ (2clxdT
(9.6)
QR(P)
~ "Y j j(vP-2ID(12 + (t)
(v 2 - k2): dxdT.
QR(p)
Since (v 2 - k2)+ ~ 411J.1.2,
jj(v 2 - k2): (t clxdT
~ (1"Y~:;~~2 J.l.P- 2IQR(J.I.)1
QR(P)
where we have used the structure of (and the intrinsic geometry of QR(J.I.). Also
f
r fvP-2 (v 2 _ k2)2 ID(1 2dxdT < "Y 1I2 J.1.4. RN.
j'
- (1-0')2
+
QR(P)
This is obvious if p > 2. If 1 < p < 2, we observe that the integral is extended over the set v > (I - 211)J.I.. We estimate below the integral on the left hand side of (9.6) by extending the integration over the smaller set [v> (I - II )J.I.]. On such a set, (v 2 - k2)+ ~ IIJ.1.2. These remarks in (9.6) prove (9.4). Set for all O
==
f
Du(x, t) dx.
Kp
LEMMA
9.2. There ex;sts'(l constant "Y="Y(N,p). such that/or 'all O'E (l,l)
280 IX. Parabolic p-systems: H5lder continuity of Du
Fix UE(O, 1). and foraH tE [-1£2- P (uR)2,O]. set
PROOF:
Vet)
==
f IDulEj!
Du(x, t) dx.
K"R
We apply the multiplicative embedding of Theorem 2.1 of Chap. I to the functions x
-+
IDulEj! Du(x,t) -
Vet),
'Vt E [-1£2-p(uR)2,O],
which have zero average over KtrR. For the choice of the parameters N Q = N + 1' q = 2, s = 1, we obtain
IIIDulEj! Du -
V(t)1 2dx
~ "YI(lDuIP-2ID2uI2) wtr dx
~R
~R
x The last integral is majorised by ''/#£ ityover [-1l 2- P (uR)2,O] gives
V!IDuI'i'Du-V
rn R-Af:r . Therefore integrating this inequal-
(I£rn RHr) -jII IDulEj! Du -
V(T)1 2dxdT
Q"R(/J)
~ "Y II (lDuIP-2ID2uI2) Jfh dxdT Q..RC/.')
="YI!(IDUIP-2ID2uI2) wtr dxdT A: R
+II(lDuIP-2ID2uI2) wtr dxdT B;R
9. Proof of Proposition l.l-(ii) 281 where A~ and B: are defined in (9.1)-(9.2). We estimate the first integral by Lemma 9.1 and the second by using the 'smallness condition' (9.3) and Lemma 7.1. We conclude that there exists a constant 'Y='Y(N,p) such that (9.8)
f1
r [I
IDul
¥
12 'Y 1J2 Vl/(N+l) N+2 Du - VCr) dxdr ~ (1 _ u)2N/(N+l) R .
Q..R(p)
Introduce the vectors wet) by
Vet) == Iw(t)l¥w(t), and observe that
Iw(t)1 ~ IJ,
'
(-1J 2 - P (uR?, 0] .
By the algebraic Lemma 5.1. (9.9)
IIIIDu l¥ Du - v(r)1 2dxd7' Q"It(p)
~
II(lDu l + Iw(r)I),,-2IDu - w(7')1 2dxd7'. Q"It(p)
We treat separately the cases p> 2 and 1 < p < 2.
The degenerate case p
>2
We minorise the left hand side of (9.9) by extending the integration over the smaller set A~R' On such a set.
(lDul + Iw(t)l)P-2 ~ IDul,,-2 ~ 22 -"IJP -
2•
This with (9.8) yields [ [
(9.10)
11 IDu -
2
'Y1J 2 v 1/(N+l)
W(7')1 dxd7' ~ (1- u)2N/(N+l) IQR(IJ)I·
A~1t
Next write
IIIDU - w(t)1 dxdr = IIIDU - w(7')1 2dxd7' + IIIDU - w(7')1 2dxd7'. 2
Q"It(p)
A~1t
B~1t
vi
The first integral is estimated in (9.10) and the second is majorised by 21J2 Q R (IJ) in view of the 'smallness' condition (9.3). We conclude that
I.
282 IX. Parabolic p-systems: HOlder continuity of Du
for a constant "( = "((N,p). The minimum on the left hand side is achieved for V == (Du)aR (t). This proves the lemma if p> 2.
The singular case 1 < p < 2 Since Iw(t)1 $ JJ, we have (IDul
+ Iw(t)l)P-2
~ 2P- 2JJP- 2. Putting this in
(9.9) and combining it with (9.8) gives
jJr[ IDu - w(t)1 2dxdT
"( JJ2 vl/(N+l) $ (1 _ u)2N/(N+l) IQaR (JJ) I·
Q"R(/J)
The proof is now concluded by a minimization procedure.
10. Proof of Proposition 1.1-(iii) Let (Du) p denote the integral average of Du over Qp(JJ), i.e.,
(Du)p == I f DudxdT. Qp(/J)
LEMMA 10.1. There exists positive constants "(, a, b that can be determined a priori only in terms of N and P. such that for all u E (i, 1).
(10.1)
I f IDu - (Du)aR 12dxdT $ "( JJ2 {(1 :au)b
+ (1- u)}.
Q"R(/J)
PROOF: By Lemma 9.2 2 "( JJ 2 v1/(N+l) I f IDu - (Du)aR! dxdT $ (1 _ u)2N/(N+l) Q"R(/J)
+ If! (Du)aR - (Du)aR (T)!2 dxdT Q"R(/J)
and (10.2)
H! (Du)aR - (Du)aR (T)!2 dxdT Q"R(/J)
sup
$
-"l-P(aR)l
-
1
-
lf
(Du(x,t) - DU(X,8»)
dx12.
KtlR
Let (f= (1 +u)/2 and denote with x--+((x) anon-negative smooth cutoff function in K&R that equals one on KaR and such that
10. Proof of Proposition l.1-(iii) 283
-
2
4
ID(I ~ (1-i1)R == (l-u)R'
1D2-1( ~ (l-u)R' 16
Write
!
(Du(x, t) - Du(x,s») dx
!
=
K"R
(Du(x,t) - Du(X,7'»)(2dx
aKR
- !(Du(x,t) - Du(x,s»)(2dx. K.R\K"R
The last integral is estimated above by 'Y( 1 - u )p.RN . To estimate the flJ'St integral we integrate the differentiated system (5.2) over (7', t). multiply by (and integrate over KaR. This gives
!
(10.3)
(Ui,Zj (t)
K.R
=
-
Ui,zj (s)
Iif ('
)(2 dx
div ( .,..-' Du;..,
+ 0:;' Do;)
""dBl·
aK.R
Thecasep>2 The right hand side of (10.3) is estimated by
To estimate the last integral write
! !IDul~ ID2Uldxd7' = ! !IDUI~ ID2uldxd7' ~RW
~R
+! !IDul~ ID2uldxd7' S:;R
~ IQR(P.)!! (f!IDuIP-'ID'U1'' ' tt.! ! Au
+ IBill!
)
(!!IDuIp-2ID'U1'''''tt.!! Qu(,,)
)
284 IX. Parabolic p-systems: mlder continuity of Du The frrst integral is estimated by Lemma 9.1 and the second tenn is estimated by the 'smallness' condition (9.3) and Lemma 7.1. Combining these estimates in (10.2) proves the lemma.
The case l
II( ~u
Ui,:J:; (t) - Ui,:J:; (8)
=
)(2 dx
V/
D(-' {IDol""" Du - I(Du)'R (s)l.-2 (Du)'R
(sn., dzdTl
sK.R
t
~ I IID2(IIiDuI P- 2Du -I (DU)uR (8)11'-2 (DU)uR (8)1 dxdr. sK;;R
By the sttucture of the cutoff function ( and (5.5) of Lemma 5.2, this is majorised by
(1 _
!)2
t
R2 IllDu - (Du)uR (s)l"-1 dxds SK.R
We estimate the last integral by Lemma 9.2 and combine it with (10.3) to prove the lemma.
11. Proof of Proposition 1.1 concluded LEMMA 11.1. Let e E (0, 1) be the number claimed by Lemma B.l. There exists a number v E (0, i) such that if (9.3) holds, then
HIDU -
(11.1)
QR(")
(11.2) PROOF:
Write
(DU)RI 2dxdr
~ ep.2,
II. Proof of Proposition 1.1 concluded 285
/Du - (Du)RI dxdT = O'N+2 H
IDu - (DU)aRI 2dxdT
2
H Q RC,,)
Q.RC,,)
+ IQR(/J)1-1jjIDU -
(Du)Rr dxdT
QR(,,)\Q.RC,,)
+ O'N+2H I(Du)R -
(DU)aRI 2dxdT.
Q.R(")
The flrst integral is estimated by Lemma 10.1 and the second is bounded above by ")'(1 - q)/J2. To estimate the last integral write
(DU)R-(Du)aR
= IQaR (/J) 1-1 {O'N+2jjDudxdT - j j QRC,,)
= IQaR (/J) 1- 1{
(O'N+2 -
DUdxdT}
Q.RC,,)
lifj DudxdT + j j QR(,,)
DUdxdT} .
QRC,,)\Q.RC,,)
This implies that
and
H IDu - (DU)RI 2 dxdT
:s ")' /J2 { (1 :°O')b + (1 -
0') } .
QR(")
:s
To prove (11.1) choose 0' so close to one that ")'(1- 0') ~e. and then v so small that ")'vO(1 - 0')-" ~e. To prove (11.2) we flrst observe that the deflnitions (9.1)-(9.2) imply
:s
Then by the 'smallness' assumption (9.3)
jjlDu I2 dxdT
~ jjlDU I2 dXdT ~ /J2(1- v)3IQR(/J)I.
QRC,,)
Ail
Using now (11.1)
H IDuI 2 dxdT - H QR(")
QRC,,)
(Du)~dxdT =
H IDu - (DU)Rr dxdT
QRC,,)
:s e/J2. From this
286 IX. Parabolic p-systems: fR)lder continuity of Du
I(Du)RI 2 ~
H
IDul 2dxdr -
E",2
~ {(I -
11)3 - E}
",2.
QR(")
1.1: Let E, 6 Ie E (0,1) be fixed as in Lemma 8.1. We start the iteration process of Lemma 8.2 with Vo == (Du)R, and let {Vi} ~ be the corresponding sequence of constant vectors in RNxm satisfying (8.7). It is apparent that, by an application of the triangle inequality, the vectors Vi can be replaced by (DU)i' by possibly modifying the number Ie. PROOF OF PROPOSITION
12. Proof of Proposition 1.2-(i) We assume that the smallness condition (9.3) does not hold, i.e., (12.1) LEMMA
12.1. Let (12.1) hold. There existssome t ••
(12.2)
such tMt
I v(x, t.) > (1 -
(12.3) mess {x E KR PROOF:
1-11
II)",} < 1 _ 11121KR1, v = IDul·
Indeed if not, _,,2-P(~/2)R2
IAR/ ~
f
mess {x E KR / v(x, r)
> (1 - II)",} dr
_,,2-PR2 ~ (1 - II)/QR("')/,
contradicting (12.1). We will work with the function w == IDuI 2, which satisfies (1.8) within the cylinder KR x (t., 0). Introduce the change of variables
r = -tit., (= xlR,
w«(,r) = w(Re,-t.r)
and the convex function of w %
w
I}
== max { ",2 i '2 .
Then KR x (t., 0) is mapped into Ql == K 1 X (-1,0) and, denoting again with (x, t) the transformed variables, % satisfies (12.4)
%t-(At,k%Zt)o;,,:50 in
Ql
and
0<%:51,
12. Proof of Proposition 1.2-(i) 287
where the matrix (At,Al) is uniformly elliptic with eigenvalues bounded above and below independent of p. Indeed it follows from (1.9) and the range (12.2) of t. that
for two constants co(N,p, v) $ Co(N,p, v). The information of Lemma 12.1 in terms of z implies (12.6)
meas {x E Kl
I z(x, -1) > (1 -
I-v
v)} $ 1 _ 1I/2IK11.
Without loss of generality we may assume that z satisfies (12.4) in a slightly larger box, say Q2. This can be achieved by starting for example with Q2R (p). Proposition 1.2 is a consequence of the following: THEOREM 12.1. Let z E C (-2, OJ L2(K2» nL2 (-2,0; W 1 ,2(K2») be a subsolution off12.4)-(12.5), and let (12.6) hold. There exists 11 =lI(N,p, II) E (0, 1), such that
meas {(x,t) E
Q! I z(x,t) > (1-1I)}
= O.
In view of (12.4), the proof of the theorem uses techniques typical of a single equation. Even though these methods have been presented in various forms in Chapters n and m, we reproduce here the main points, to render the theory selfcontained.
12-(i). Some energy estimates/or z LEMMA
12.2. Let 0 < 110 < v and consider the function !li(z) = In+ { II II - (z - (1 - 11»+
+ 110
}
•
There exists a constant"( = ,,(N,p, v) such that for all t E (-1,0) andfor all 0<0'< 1, (12.7)
J
J
K"x{t}
KIX{ -I}
!li2(z) dx $
!li2(z) dx + (I!0")2
JJ
!li(z) dxdT.
Ql
PROOF: Let x - (x) be a cutoff function in Kl that equals one on Ka. and in the weak formulation of (12.4) take the testing function !lilli' (2 , modulo a Steklov averaging process. Then (12.7) follows by estimates analogous to those in Proposition 3.2 of Chap. II.
288 IX. Parabolic p-systems: fR)lder continuity of Du
LEMMA 12.3. ForO
Also
13. Proof of Proposition 1.2 concluded 289
I z(x, t) > (1 - '10)} ~ meas {x E Ku I z(x, t) > (1 -
meas {x E KI
'10)}
+ (1 -
O')IKII
< 1 - v IKllln2 (v/'1o) - 1 - v/2
+
In 2 (v/2'10) '"t In (v/'1o) (1 _ 0')21Kllln2 (v/2'10)
+ (1 -
O')IKII·
Choose 0' so that (1 - 0') ~ v 2 /8 and then '10 so that
'"t In (11/'10) < 112. (1 - 0')2ln2 (11/2'10) - 8 By choosing '10 even smaller if necessary. we may insure that
In 2 (11/'10) < 1 _ 112. 1- 11/2 ln 2 (11/2'10) 2 1-
II
Having determined '10. let So be the largest positive integer such that2- So ~ '10' For s ~ So. set
o A.(t)
== {x E KI
I z(x, t) > (1 -
2- S )
}
,
As
==
jIAs(T)ldT. -I
Then Lemma 13.1 implies that
'It E (-1,0).
(13.2) LEMMA
13.2. For every v.
E
(0, 1) there exists a positive integer s.
> So such
that (13.3)
PROOF: Apply Lemma 2.2 of Chap. I to the functions x-+z(x, t) fortE (-1,0). and for the levels
l
= 1- 2-(·+1),
Taking into account (13.2). we obtain
2- S lAsH
I ~ IKI \~s(t) I
j IDzl dx
A.(t)\A.+1(t)
"~(N,p,
v)
(j ID
(z - (1- 2-'))+
x (lA.(t)I-IA s+1(t)l) i .
I'dx) I
290 IX. Parabolic p-systems: Ht;lder continuity of Du
We square both sides of this inequality, integrate in dt over ( -I, 0) and estimate the resulting integral on the right hand side by the energy inequalities (12.8) written over the pair of cylinders Ql and Q2. This gives
4- s A~H :5 "Y4- s (As - AsH) . Divide through by 4- s and add these inequalities for 8=80'So + 1, ... ,S. obtain
-
1 to
s.
(s. -
So -
I)As. :5 "Y
E (As - AsH) :5 "YIQ11·
Therefore
PROOF OF THEOREM
12.1: Consider the family of nested boxes
and the increasing levels
n=O,I, ... , and set Yn
== meas{(x,t) E Qn I z(x,t) > k n }.
Write the energy inequality (12.8) over the boxes Qn for the functions (z - k n )+, where ( is the standard cutoff function in Qn that equals one on Qn+l' By the embedding Proposition 3.1 of Chap. I, with m =p = 2,
II
(z - kn
)! dxd-r:5 II [(z -
kn )+ (]2 dxd-r
Q"
Q"+1
x
(if
[(z - k,.)+ <) if<
dzd_,y-It. Ynwb
:5 II (z - kn)+ (II~2.2(Q")Ynwh :5 4S • Y~+wh . On the other hand YnH :5 "Y4n+s •
II
(z - kn
)! dxd-r.
Q"+1
Therefore ny'1+1i1h y.n+l < _ "Y4 n ,
n=O,I,2, ....
It follows from Lemma 4.1 of Chap. I that {Yn } - 0 as n -
00
provided
15. Bibliographical notes 291
(13.4) To prove the theorem we have only to pick 8. by the procedure of Lemma 13.2 so that (13.4) is satisfied and then set 'I = 2-(8.+1).
14. General structures Consider the general non-linear system (1.10) of Chap. VIII subject to the structure conditions (81)-(~). The proof of Propositions 1.1 and 1.2 for these systems is analogous to that in §§6-11. The corresponding 'linear' system about a point (x o, to) E flT is
For this, the linear analysis of §6 can be carried with minor changes. The analog of the 'algebraic' lemmas of §5 are a direct consequence of the structure conditions (81 )-(~). In the proof of Propositions 1.1 and 1.2, when working within cylinders [(x o, to) + Q (6, p)], the 'perturbation terms'
15. Bibliographical notes The content of this Chapter is essentially taken from [36,37]. The estimation of the oscillation of Du in §§2 and 3 builds on [37] but it is essentially new. The algebraic Lemmas of §5 are scattered in the literature mainly without proofs. We have attempted to rephrase them in the context of p-systems. The theory of linear parabolic systems of §6 is taken from Campanato [23]. The rest of the Chapter follows [36,37].
X Parabolic p-systems: boundary regularity
1. Introduction We will establish everywhere regularity up the boundary for weak solutions of the parabolic system U::(UlIU2, ..•
(1.1)
{
,Um), meN,
UieC (e,T; L2(O»nV(e,T; Wl,P(O» ,
=Bi(X, t, U, Du),
in 0 x (e, T), ee(O,T), i=I,2,,,.,m, p>max{liJ~2}, Ui,t -div IDulp-2 DUi
associated with Dirichlet boundary data (1.2)
in the sense of the traces on ao, of functions in Wl,p(n). The basic assumptions on ao, the boundary data g and the forcing term B
g:: (91t!I2, .. · ,9m),
are the following: (A l )
(A2)
ao is of class Cl,~
for some oX e (0,1), in the sense of (1.2) of Chap. I. Thus the norm IlaolhH is finite.
The functions gi, i = 1,2, ... , m, are restrictions to ao of functions 9i' dermed in the whole OT, and satisfying
1. Introduction 293
-
(1.3)
.\-
9i,z; E C (nT), i
= 1,2, ... , m ,
j
9i,t E LOO(nT), = 1,2, ... , N.
We set(l) m
/11/1 ==
(1.4)
N
L L {/l9i /loo,nT + /l9i,tlloo,nT + [9i'Z;].\,nT}· i=1 j=1
IB(x,t, u,Du)1 ::; Bo
(A3 )
(1 + IDuIP-1),
a.e. in nT,
for some given constant Bo. We say that a constant 'Y = 'Y (data) depends only upon the data if it can be determined a priori only in terms of
(data) == (N, p, B o, IIOnll1+.\, /11/1) . Let u be a weak solution of (1.1 )-(1.2) in (A I )-(A3) hold. Then
THEOREM 1.1.
fl.i
n x (e, T), and let
E C l - OI (nx (e, Tj) , for every aE (0, 1), i = 1,2, ... , m.
Moreover for every aE (0,1) and every eE (0, T), there exists a constant
'Y
= 'Y (a,e, /lDullp,nx(~,T),data),
such that (1.5)
°
The constant 'Y tends to infinity as either e '\. or as a'\. 0.
Remark 1.1. The constant 'Y is •stable' as p -+ 2. THEOREM 1.2 (HOMOGENEOUS BOUNDARY DATA). Letubeaweaksolution of (1.1 )-(1.2) with 1== and let (At) - (A2 ) hold. For every e E (0, T) there exist constants
°
'Y = 'Y (e, /lDullp,nx(~,T),data)
>1
and a=a(data) E (0, 1)
such that
[fl.i'Zi]OI,nx[~,T) ::; 'Y, The constant 'Y /
00
i
= 1,2, ... , m, j = 1,2, ... ,N.
as e '\. 0.
We will only carry the proof of Theorem 1.1. The proof of Theorem 1.2 follows exactly the same arguments, where in the various estimates the contributions coming from /11/1 are discarded.
(I) For a smooth function r/J, the norm [r/Jh,K is defined in (1.3) of Chap. I.
294 X. Parabolic p-systems: boundaJy regularity
2. Flattening the boundary Let e E (0, T) be fixed. We will estimate the oscillation of Ui about each point (x o• to) E an x (e, T). For this we first introduce a change of coordinates that maps a small portion of an about (xo, to) into a portion of an hyperplane. After a translation we may assume that (xo, to) coincides with the origin. We will work within the cylinder
Q'R, == K'R, x {-'R., O} ,
2'R. = min{po; e},
where Po is the number that determines the structure of an as in (1.2) of Chap. I. The portion of the boundary annK'R, is represented by XN
= 4>(x),
x == (Xl, X2, ••• , XN-l)
,
where 4> is a function of class C l ,>. in the (N -1) -dimensional ball8'R,. satisfying (2.1)
The last condition can be realised by taking a smaller Po if necessary. With respect to the new variables
Xi
= Xi,
i
= 1,2, ... ,N-I;
the portion annK'R, coincides with the portion of the hyperplane XN =0 within
K'R,. We orient XN so that, say, nnK'R, C {XN >O} and set Q~
== Q'R,n{XN > O}.
Denoting again by X the transformed variables x and with Ui, B i , 4>, etc., the transformed functions. the system (1.1) takes the form
(2.2)
(2.3)
(2.4)
A
( ) _ ( X
=
IN-l
-DcI(x)
-D4>(X)) (1 + 1D4>12(x)) ,
where IN-l is the (N -1) x (N - 1) identity matrix. To reduce (2.2) to a system with homogeneous boundary data on Q'R,n{ X N = O} set Wi=U;-.9i,
i=I,2, ... ,m,
and rewrite (2.2) in the form (2.5)
!
Wi -
div Ai (x, t, Dw)
= Bi + a~l A"
in
Q:k,
2. Flattening the boundary 295
Figure 2.1
= at,k (x, Dw + Di) Wi,x" ,
(2.6)
Ai,t (x, t, Dw)
(2.7)
Bi=Bdx,t,w+g,Dw+Di)-
(2.8)
At = at,k (x, Dw + Di) 9i,x".
!9i,
Using the assumptions (Al)-(A3) we find the following structure conditions and regularity properties on the various tenns of (2.5): (2.9)
{
Ai,t(x, t, DW)Wi,XI ~ "YolDw + Dil,,-2IDwI2 Ai,t(x,t,Dw)wi,xt $ "YllDw + Dil,,-2IDwI2,
for two positive constants "Yo $ "Yl depending only upon the data. Moreover for all i=I,2, .. . ,m and k=I,2, ... ,N, (2.10)
IAi,k(x, t,e) - Ai,k(y,T,e)1 $ "Y (1 + lel,,-l) (Ix -
Ve E R Nxm ,
and for a.e. (x,t), (y,T) E Q*,
yl + It - TI)~
296 X. Parabolic p-systems: boundary regularity (2.11)
IBi (x, t, Dw) I :::; 'Y (1 + IDw l,,-I) ,
(2.12)
Ili.l(X,t,w,Dw) I:::; 'YIDul,,-2.
From (2.1) and the definitions (2.3)-(2.4) and (2.6), it follows that
. = I~ + b 1,,-2 ~i,1c 61c,l,
Ai,l (0, O,~)
(2.13)
V ~ E R N xm,
b == (Di) (0,0).
(2.14)
2 -(i). Comparison Junctions Consider cylindrical domains of the type
+ Q (R2+'I,R)] ,
[(X o, to)
where fJ E ( -1, 1) is to be chosen, Xo E K x n { x N =O} and the faces of the cubes (xo + KR] are parallel to the coordinate'axes. These boxes are contained in Qx if (2.15)
which from now on we assume. The proof of Theorem 1.1 is based on comparing w in a neighborhood of each point (xo, to) E QX/2 n {x N ~ O}, with the solution of == (111.112"'" 11m) , mEN, l1i,t-div IDvl,,-2 Dl1i =0, in [(x o, to) + Q (R2+'I, R)] nQ~,
V
(2.16)
{
l1i
= Wi
on 8" [(xo, to)
+ Q (R2+'I,R)]nQ~,
where 8"Q denotes the parabolic boundary of a cylindrical domain Q. The existence of a unique weak solution of (2.16) can be established by a Galerkin procedure. (1) Denote by (f,XN) ,
f==(Xl,X2, ... ,XN-l),
the coordinates in K x. Then since w vanishes for x N =0, we also have v(x, 0, t) = O. We let v and wdenote the odd extensions of v and w in the cylinder [(x o, to)+Q(R2+'I,R)ln{zN~o}, i.e.,
+ Q (R2+'I, R)] n{XN ~ O} [(x o, to) + Q (R2+'I, R)] n{XN :::; O}, [(xo, to) + Q (R2+'I, R)] n{XN ~ O}
in [(x o, to) in
w ==
{
W(:,XN' t), -W(X,-xN,t),
in
in [(Xo, to) +Q (R2+'I,R)]n{xN :::;O}.
Then, by the reflexion principle v is the unique solution of (1) See, for example, (73).
3. An iteration lemma 297
V == (iit,V2, ... , Vm), mEN, { vi.t-divIDvl,,-2Dvi=O, in [(X o ,to )+Q(R2+'7,R)] , Vi = Wi on 8" [(xo, to) + Q (R 2+'7, R)] .
(2.17)
It follows from the interior estimates of Theorems 5.1 and 5.2' of Chap. VIII that IDvI is bounded in the interior of [( x o, to) + Q (R2+'7, R)] and it satisfies the sup-bounds (5.1) and (5.3). We restate these bounds for the special geometry of
[(x o , to) + Q (R 2+'7, R)]. THEOREM
2.1 (THE DEGENERATE CASE p> 2). Let v be the weak solution
0/(2.17). There exists a constant 'Y='Y(N,p) such that/or all O
(2.18)
IDvl '5. 'Y (R'7
sup [(zo.to )+Q(p2+'I.p»)
ff IDvl" dxd1")
[(zo.to )+Q(R2+'I.R)
1/2
+ R-
)
~.
THEOREM 2.2 (THE SINGULAR CASE max { Ii A~2} < p < 2). Let v be a weak solution 0/(2.17). There exists a constant'Y = 'Y(N,p) such that/or all O
(2.19)
IDvl '5.'Y
sup [(zo.t o )+Q(p2+'I.p»)
(R- '7 ff IDvl" dxd1")l/V + Rr-., P
N
[(zo.to)+Q(R2+'I.R)
where 1I,,=N(p - 2) + 2p. Remark 2.1. Theorem 2.2 is a restatement of Theorem 5.2' of Chap. VIII with q p. By Remark 5.4 of the same Chapter, such a choice is admissible.
=
3. An iteration lemma LEMMA
3.1. Let 8"'" rt'( 8) be a non-negative non-decreasing junction defined in
[0, I] and satisfying (3.1)
rt'(p) '5. A
(1/
rt'(R) + ~A(R.8-V"+"R-"),
VO
for given positive constants A, /3,11, K. satisfying in addition /3 > K. and II E (0, 1). Then for every (3.2)
0'5. 6 < K. (1 ~ ~):): /3) ,
there exists a constant'Y depending only upon A, /3, II and 6, such that
298 X. Parabolic p-systems: boundary regularity
P )(j-tf,+6 cp(p) ~ 'Y ( R' (cp(R) + 1)
(3.3)
VO
where q = 1 +
(1 - v)1\:
f3
PROOF: Choose Ro < 1 and define the sequence Rn+l Then cp(Rn+1)
.
= ~, n = 0, 1,2, ....
~ A~l-II)tf,cp(Rn) + ~A ( n!;'~" + ~+~/,) = A~l-II)"cp(Rn) + A~+~/'.
Iteration of these inequalities gives
Now
and
Therefore if A ~ 2 cp(Rn+1)
~ An+1 ( ~1 ) (j cp(Ro) + 2An+1l(+~/'.
Fix 6 in the range (3.2) and set E
=
I\:
(1 ~ ~):): f3) - 6;
f3 -
~ = f3 -
I\:
+ 6 + E.
Then
n+1
The first coefficient in (3.3) is independent of n if n is so large that AR~m Let no be the smallest integer satisfying
~ 1.
4. Comparing w and v (the case p > 2) 299 qno+l
InA
-->--. no+ 1 - pnR~1
It follows from (3.3) that if n ~ no,
If Ra < 1 is fixed, for every p E (0, Raj there exist some n E N such that Rf+l ~p~ Rtf. Therefore the equation p = R!,q" has a root (J E [I, q]. Starting the process with Ra replaced by R!, gives
Remark 3.1. The lemma continues to hold for 11=0. The constant 'Y on the right hand side of (3.3) is 'stable' as 11'\.0.
4. Comparing w and v (the case p> 2) We start by comparing w solution of (2.S) with the solution v of (2.16). Having fixed (xo, to) E Ql'R n {XN = O}, we may assume, after a translation, that it coincides with the origin. Setting
the vectors w and v satisfy (4.1)
!
(Vi - Wi) -
div (IDvIP-2Dvi -IDwIP-2Dwi)
= - div(Ai(x,t,Dw) -
Ai(O,O,Dw» - div (Ai(O,O,Dw) -IDwIP-2Dwi)
(4.2)
Vi - Wi
-Bi(X,t,Dw)- 1:>8 fil(X,t,w,Dw), in+Q1t, uXl ' = 0 on the parabolic boundary of +Q1t.
From (2.10) it follows that IAi,l(X, t, Dw) - Ai,t(O, 0, Dw)1
~ 'Y R).. (1 + IDwIP-l),
for i=I,2, ... , m and 1.=1,2, ... , N. Moreover from (2.13) and (2.14)
300 X. Parabolic p-systems: boundary regularity
IAi(O, 0, Dw) -IDwIP-2 DWil ~ 'YIIDw + bl p - 2-IDwIP-21IDwl ~ l' (1
+ IDwIP- 2 ) .
In the weak fonnulation of (4.1) take the testing function Vi - Wi modulo a Steklov average, and add over i = I, 2, ... , m. We estimate the tenns on the right hand side by the remarks above and the left hand side by making use of the algebraic Lemma 4.4 of Chap. I to obtain
//IDW - DvlPdxdr
~ 'YR).
+Qlc
//(1 + IDwIP-1) IDw - Dvl
dxdr
+Qlc
+1'//(1 +IDwIP-2) IDw - Dvl +1'//(1 +IDwIP-1) Iw - vldxdr
dxdr
+Qlc
+Qlc = [(I)
+ [(2) + [(3) .
In the estimates below we integrate over the boxes Q'k, rather than +Q'k. In doing 50 we think of v and w as defined in the whole Q'k through an odd extension as indicated in §2. By the Schwartz inequality [(I)
~ ~//IDW - Dvl Pdxdr+'YR).P!Y Q~
[(2)
//(1 + IDwIP) Qlc
~ ~//IDW-DvIPdxdr+'Y//(I+IDwIP)~ Q~
dxdr,
dxdr.
Q~.
Since v - w vanishes on the lateral boundary of Q'k, by the Sobolev embedding, (1) z.::.!
1(') S
~R (If (1+ JDwJP) kdT) , (I/,Dw -DvIP
~ ~//IDW-DvIPdxdr+'YRP!Y Qlc
Combining these estimates gives (1) Corollary 2.1 of Chap. I.
//(1 + IDwIP) Q~
r .1
dzdT
dt.
4. Comparing w and v (the case p > 2) 301 (4.3)
//IDW - DvlP dxdT ~ 'YR).t=r,
//(1 + IDwIP)
Q~
Q~
+"'1//(1 +
dxdT
IDwIP)~ dxdT.
Q~
From this we deduce two inequalities. First, since
p- 2 p-l we have
<1
If
(4.4)
and
(1 +
IDvlPdxdT :5 "'1
Q~
IDwIP)~ ~ (1 + IDwIP),
If
(1 + IDwI P) dxdT.
Q~
Second, for 0: > 0 set
:F (0:, "I, R) == Rap
(4.5)
If
(1
+ IDwIP) dxdT,
Q~
and observe that
//(1 + IDwIP)~ dxdT ~ 'YRN+2+,,-ap~ [:F(O:'''I,R)l~. Q~
This implies that VO
//(1 + IDwIP)
dxdT
~ 'YR).t=r,
//(1 + IDwlP)
dxdT
Q~
Q:
+//IDvIPdXdT+'YRN+2+,,-ap~ [:F(O:'''I,R)l~.
Q: By taking R sufficiently small and by interpolation, the first term on the right hand side of (4.6) can be eliminated. This is the content of the following lemma: LEMMA 4.1.
In
(4.6')
There exists a constant "'1 ='Y(data) such that for a/IV 0 < p ~ ! R ~
//(1 + Q:
IDwIP)dxdT
~ 'Y!!IDvIPdxdT Q;p
+'YRN+2+,,-ap~ [:F(O:'''I,R)l~. PROOF:
It suffices to prove the lemma for R ~ R" where Ro is so small that
302 X. Parabolic p-systems: boundary regularity
'Y (2Ro)'\;!-r
~ ~.
Let 0 < P ~ ~ Ro and consider the sequence of radii
Pn ==p+2-"p,
n=O,I,2, ....
Write (4.6) for R= Pn-l and P= Pn,n~h and set
Yn
==/
PI +
IDwIP) dt
Q:n
z == 'Y/ /IDvIPdXdT + 'YRN+2+"-QP~ [F (a, 1], R)l~ . Q~"
Then by iteration from (4.6),
Yoo
==//(1 + IDwI P) dxdr
Q:
We return to (4.6)' and estimate the integral involving Dv in terms of Du. Let 0 < P~ R. Then by Theorem 2.1 and (4.4)
!
/ /IDvlPdxdr
Q:
~ 'YpN+2+"II Dv ll:a,Q: ,;
~pN+2+'
{ Jl!lPI'
(U IDvIPdzdT) (U IDvIPdzdT)
,;
~pN+""'{ Jl!lPI'
x
(UIDvIPdzdT) + R-'-,!.}'
Next choose 1] = a(p -
Then
2),
for some
a> O.
pl'+ R"-,!. } ~
4. Comparing w and v (the case p
Ir',/2
(u
~
IDvl'do:dr )
"
> 2)
303
~ [.1" (Q, ., RlI'" ,
and by suitably modifying the constant 'Y we deduce from (4.6)' that for all 0 < p~R~!,R., (4.7)
//(1 + IDwI Q:
P)
dxdr
~ 'Y [.1' (a, 71, R)l
~
(p)N+2+'I f f R )} (1
+ IDwlP)dxdr
Q1r +'Y[F(a'71,R)l~ RN+2+'I-a~ +'YpN+2+'I R-ap, for a constant 'Y = 'Y (data). Set
F(a) ==
(4.8)
sup
{pap
(:Co,to )EQR/2
H + IDwI (1
P)
dt}
[(zo,t o )+Q:l
O
and
(4.9) We summarise: PROPOSITION 4.1. Let a > 0 and 71 = a(p - 2). There exists a constant'Y 'Y (data), independent 0/ a, 71, p, R, such that
=
/orall (xo,to)EQ:k/2 and/orall O
//(1 +
IDwlP)dxdr
[(zo ,to)+Q;J
~ 'Yg(a)(~)N+2+'I
/f
[( Zo ,to
+ 'Yg(a)RN+2+'I-ap~
(1
+ IDwlP)dxdr
)+Q7tJ
+ 'YpN+2+'I R-aP.
PROOF: The previous arguments prove the proposition for those points
(xo, to) EQ!Rn{XN =O}. The estimate is obvious for boxes [(x o, to) + Q7tl c Q~, by interior estimates. If [( xo, to) + Qkl intersects {x N =O}, then either (4.11)
[(Xo,to)+Q1R]CQR or [(Xo,to) + Q1R]n{x N =O}#0.
304 X. Parabolic p-systems: boundary regularity
!
In the first case we may establish (4.10) with R replaced by R. The general case follows by suitably modifying the constant 'Y. If the second of (4.11) holds, we let (4.12)
X. == (Xo,l' Xo,2, . .. ,Xo,(N-l), 0)
and observe that
[(X., to)
+ QiR]
C
I(xo, to) + Qkl·
We carry on the process leading to (4.10) for such a new box, for all 2X o ,N :5 p < R. This implies that (4.10) holds for all Xo,N < p:5 R. If p:5 Xo,N, we consider
!
!
the cylinder [(Xo, to)
+ Qt.N]' which satisfies the inclusion [(Xo, to) +
Ql z o,N] c Q:k.
Then by interior estimates, (4.10) holds with R replaced by Xo,N. Combining the two cases and suitably modifying the constant 'Y we conclude that (4.10) holds for -+ 1 all (XO, to) E Q!"R and all O
5. Estimating the local average of IDwl (the case p> 2) LEMMA 5.1. For every 0: E (0,1) there exists a constant 'Y = 'Y (0:, data), such thot jorall (xo,to) E Q:k/2 and/orall 0
H
(I, + IDwI P ) dxdr:5 'Y(o:,data) p- oP ,
(5.1)
'1
= o:(p- 2).
[(zo,to)+Q:l
Define the sequences 0: 0 = (N + 2)/2 and for n= 1, 2, ... ,
PROOF:
We will prove inductively that
(5.2) Since IDwIELP(nT),
H
(1
+ IDwI P )
[(z.,to)+Q~O 1
Therefore
dxdr
:5 'YP-¥ P (1 + IIDwll:,nT) .
6. Estimating the local averages of w (the case p > 2) 305 Suppose the lemma holds for Q n and let us show that it continues to hold for Q n +1.
If F(Qn):'5 1'(Qn), the quantity Q(Q n ) introduced in (4.9) is bounded and we may use (4.10) with Q = Q n and TI = TIn. We apply the iterative Lemma 3.1 to the function
tp(p) =
JJ(1 + IDwI
P)
dxdr,
[(zo,to)+Q~l
with the choice of parameters (5.3)
v
We obtain
If
(1
+ IDwI P )
[(zo,to)+Q~n
p-2
= --, p-l
6 = 6n Q np.
dxdr :'5 1'(Qn+d p-On+lP (tp(R)
+ 1).
1
Let 0 < P:'5 'R/2 be fixed and consider the point (xo, to) == (0, 0). Without loss of generality assume that p'r/n+l / p'r/n is an integer, and partition the cylinder [(x o, to) + Q~n+l] into s = p'r/n+l-'r/n adjacent boxes with 'vertices', say (0, td, (0, t2), ... , (0, t s ). Then
If
(1 + IDwI P ) dxdr:'5
n p'r/ p'r/n+l
Q:n+l
~ ~ 3=1
If
(1
+ IDwIP )
dxdr
[(o,tj)+Q~nl
:'5 1'(Qn+l) p-On+lP. We may treat analogously the other points of Q'R./2 and the inductive inequality (5.2) follows. To prove the lemma it suffices to prove that {Q n } -. 0 as n -. 00. The sequence {Q n } is deacreasing. We claim that {Qn} -. O. Indeed if not, lim
n--+oo
Q
n
=
Q
o
> 0,
and the definitions of {Q n } and {6n } would imply
Therefore Qn+l :'5 Qn(I-6o). This in tum implies {Q n } -.0. Remark 5.1. The constant l' on the right hand side of (5.1) is 'stable' as p This follows from the choice (5.3) of the parameter v and Remark 3.1.
~
2.
6. Estimating the local averages of w (the case p > 2) We return to cylinders bearing the natural parabolic geometry, i.e., Q p == Q and will work within the boxes
(p2, p)
306 X. Parabolic p-systems: boundary regularity
Since w vanishes for XN =0, we regard it as defined in the whole QR, by an odd extension across {x N =o}. Let
(w)o,p == HW(X,t)dxdr [(zo,to)+QpJ
denote the integral average of w over I(xo , to) the origin, we let (w)o,p==(w)p' Also let
f
(w)o,p (t) ==
w(x, t)dx,
+ Qp]. If (x o, to) coincides with
t
E
(to
-l, to).
{zo+KpJ
We observe that if Xo E {XN =O}, we have (w)o,p (t) =0 for all t E (to - p2, to). since w is odd across {XN =O}, and in particular (w)o,p =0. LEMMA
6.1. For every a E (0,1) there exists a constant "( = "( (a, data), such
lhat (6.2)
H
Iw - (w)o,p IPdxdr :5 "((a)pp(l-OI),
[(zo,to)+QpJ
for all cylinders satisfying (6.1 ). PROOF: We first observe that from Lemma 5.1 and its proof it follows that for every cylinder satisfying (6.1)
H
(6.3)
(1
+ IDwIP) dxdr :5 "((a) p-OIP.
[(zo,to)+Qp]
If Xo E {XN =O} by the Poincare inequality and (6.3).
H
Iw - (w)o,p IPdxdr :5 "((a) pp(l-OI).
{(zo,to)+QpJ
Consider next the case (6.4)
[(xo, to) + Qp+ap]
C Q~/2'
U E (0, l) to be chosen.
By a translation we may assume that (xo, to) coincides with the origin. We have
6. Estimating the local averages of w (the case p
Iw - {w)pIPdxdT :5 H
H Qp
> 2) 3C17
Iw - {w)p {T)I PdxdT
Qp
+ HI {W)p (T) - (w)p IPdxdr Qp
== ](1) + ](2). By the Poincare inequality and (6.3) ](1)
:5 ')'{a) {I'(I-a).
Next (6.5)
](2)
=H
/H
Qp
Qp
[w{x, t) - w{x, r)] dxdrr dxdt.
We estimate the integrand on the right hand side of (6.5) by making use of the equation (2.5), over the cylinder Q P+tT p. Let x -+ ({ x) be a non-negative piecewise smooth cutoff function in K p+tTP that equals one on Kp and such that ID(I :51/up. In the weak formulation of (2.5), take ( as a testing function and integrate over K p+tTP x [T, t] to obtain
/!
([w{x, t) - w{x, r)] dx/
Kp+"p
:5
t
!!IAi{X,w, Dw).D(+At(xt+Bi(ldxdT
1=1 Qp+"p
/(1 +
:5 u'Yp!
IDwIP-1) dxdT
Qp+"p
By the properties of ( and (6.3) with a suitable choice of a we conclude that
1/[w{x, t) - w{x, T)] dxl :5 ~pN+(I-a) l(p
+ / /[w{x,t) - w{x,T)]dxl. Kp+"p\Kp
Let {w)P+tTP denote the integral average ofw over the Qp+tTP, i.e.,
308 X. Parabolic p-systems: boundary regularity
H
(W)P+ITP ==
w(x, r)dxdr.
Qp+"p
Then
Ij[W(X,t) - w(x,r)] dxl .1
~ IKp+ITP\Kpl~ { (
jIW(X,t) - (w)P+ITP 1PdX) Kp+"p
+
L!.~W(X' (W)..,., I'
r}
p
.1
do:
T) -
Combining these estimates in (6.5) gives [(2)
~
;P
pp(l-a)
+ 'YUp- 1
HIW -
(W)p+ITPI Pdxdr.
Qp+"p
We conclude that for every a E (0, 1) there exists a constant 'Y = 'Y( a) such that for every uE (0,1) and for every pE (0, fR) (6.6)
H
Iw - (w)pI P dxdr
~ 'Y~~) p(l-a)p
Qp
+ 'Y(a)uP- 1
H
Iw - (w)p+ITPI Pdxdr.
Qp+ .. p
This implies the lemma, in the case (6.4) holds, by the interpolation process of Lemma 4.3 of Chap. I. This process yields the choice of u E (0, Finally, having fixed u E (0, !), consider the case when
!).
[(Xo, to)
+ Qp+lTp]n{XN = O} i: 0.
Letx. E {XN =O} be defined as in (4.12) and observe that the box [(x., to) + Q2p] 'centered' at (x., to) contains [(xo, to) + Qp]. Therefore, by the Poincare inequality, since the average ofw over [(x., to) + Q2p] is zero,
H
Iw - (w)o,p IPdxdr
[(zo,to)+Qp)
~ 'Y
H
IwlPdxdr
[(z.,to)+Q3p)
~ 'Y(a)pp(l-a).
7. Comparing w and v 309
6-(i). Proof of Theorem 1.1 (the case p>2) The proof is a consequence of Lemma (6.1) and the averaging theory of Campanato-Morrey spaces [22,23,33,79]. It can also be proved directly, starting from (6.2), by arguments similar to those in §§2 and 3 of Chap. IX.
7. Comparing w and v (the case max {I; &~2}
8
at (Vi - Wi) -
div (IDvIP- 2 DVi
= -
-IDwlp- 2 DWi)
8 (al,k(x, DU)Ui,Zk -IDulp-2Ui,Zk Ot,k)
!3
uXl
- div (lDulp-2 DUi -
IDwlp- 2DWi)
- Bi(x, t, w, Dw), in +Q'k Vi - Wi =0, on the parabolic boundary of +Q'k. The boxes Q'k are formally identical to those introduced in the degenerate case p > 2. In the singular case we will take TJ E ( -1, 0). In writing (7.1) we have used the definitions (2.3) and (2.8). From (2.3)-(2.4) we derive the estimate
lal,k(x, DU)Ui,ZI i=1,2, ... ,m,
-IDulp-2 Ui ,ZI! $ -yR)..IDulp-l, l,k=1,2, ... ,N.
Moreover by Lemma 4.4 of Chap. I,
IIDuIP-2Dui -IDwlp-2Dwil
$ -ydD(u - w)I P- 1$-y,
since the boundary data g are regular. In the weak formulation of (7.1) we take the testing functions Vi - Wi modulo a Steklov averaging process, integrate over +Q'k and add over i = 1, 2, ... , m. Using the remarks above to estimate the corresponding terms on the right hand side gives
(7.2)
//(11ID (sv + (1- s)w) IP- 2dS) IDw - Dvl2 dxdr +Qk
$ -yR).. /
/IDuIP-1IDW - nvl dxdr
+Q~
+ -y/ /IDW +Q~
Dvl dxdr + -y //IBIIW +Q~
vi dxdr.
310 X. Parabolic p-systems: boundary regularity
In carrying the estimates below, we think of v and w as defined in the whole Q'k by an odd reflexion across {XN =o}. By the Poincare inequality and (2.7) 2::!
$
~ ([[<1+ IDwIP) dxdT) ·
2::!
$
.1
~R ([[<1 + IDwIP) dxdT) · ([[IDv - DwIPdxdT) •
Introduce the two sets
£1 == {(x, t) E Q'k I IDw - Dvl ~ IDwl} , £2 == {(x,t) E Q'k IIDw - Dvl < IDwl}· 11lD (sv + (1- s)w) IP- 2 ds ~ ~IDw - DvIP- 2. Therefore from (7.2) it follows
!!IDw -
DvlPdxdr+ !!IDWIP-2IDw - Dvl2dxdr
£1
£2
:5
'YR'>'{!! IDuIP-IIDw - Dvldxdr £1
! ! IDuIP-IIDw -
+
Dvldxdr}
£2
~
+ { [[ IDw - Dvldxd-r + [/ IDw - DvldxdT } 2::!
+
~R ([[<1 + IDwIP) dxdT) ·
"{Jf iDw - Dvl'dxdT+ ff IDw - DvI'dxdT} \ £1
IIp
£2
In t~l~" inequality we absorb the integrals of IDw - vi extended over £1 into the analogous term on the left hand side by means of Young's inequalitj. Using also
the definition of £2 we arrive at
7. Comparing w and v 311
(7.3)
!!IDW -
DvlPdxdr+ !!IDwIP-2IDW - Dvl2dxdr
£1
£2
$
'YR >..JJ(l + IDwIP) dxdr + 'Y!J(l + IDwl) dxdr. Q~
Q~
We estimate the right hand side of (7.3) by
'Y
R + +'1+>"-OP {nap H(1 + IDwIP) dt} N
2
[(Zo.to)+Q~l 1
+ 'Y RN +2+'1-
0
{nap H(1 + IDwIP) dt}
P
[(zo.to)+Q~l
$ 'Y R N +2+'1- o p>"o [.r(a) A
.r1/P(a)] ,
where
~o=max{!; 1-~} E(O,l) P ap and where as before we have set
Therefore
(7.4)
JJ IDw - DvlPdxdr + JJ IDwIP-2IDw - Dvl2dxdr £1
£2
$
RN +2+'1- o p>"o
[.r(a)A.r1/P(a)] .
Rewrite the integrand in the second integral on the left hand side of (7.4) as
Observe also that on the set E2 • IDvl $
21Dw I. so that
IDwIP-2IDvI2 $ 22-PIDvIP. These remarks in (7.4) prove the following:
312 X. Parabolic p-systerns: boundary regularity LEMMA
7.1. There exits a constant 'Y forall (xo,to) E Q:k/2
(7.5)
= 'Y (data), such that
andforall
0
IllDwlP dxdT 5, 'YR N+2+'1- op>,0 max {F(Q); F1/P(Q)} [(xo,to)+Q~1
+ 'Y III Dvl PdxdT, [(xo,to)+Q~1
(7.6)
III Dvl P dxdT 5, 'YRN+2+'1-op>,o max {F(Q); Fl/p(Q) } [( Xo ,to) +Q1t1
+ 'Y IllDwlP dxdT .. [(x o ,t o )+Q1t)
To estimate the last integral on the right hand side of (7 .5) we make use of Theorem 2.2. the defmition of F(Q) and (7.6). Assuming that (xo, to) coincides with the origin, we have for all 0 < p 5, 5, 'R/4
¥l
(7.7)
IllDVIPdXdT 5,
pN+2+'1I1DvIl:O,Q~
Q~
5, 'YpN+2+'1 RP.;!:p
+ 'YPN+2+'1{ R-N'I-op>'o + R-N'I
H
(1 + IDwl
[F(Q) " F1/P(Q)] PIlip
P)
dxdT
}
,
Q1t
where lip = N(p - 2) + 2p > O. Choose (7.8)
'1=Q(p-2),
QE (0; ::2] >0.
Then the first term on the right hand side of (7.7) is estimated above by 'Y pN +2+'1 R-oP.
Setting also Q(Q) '= max {pilip; Fillip; F N (2-p)/lI p ;
I} ,
8. Estimating the local average of IDwl 313
the second term on the right hand side of (7.7) is estimated by -yQ(a)
(~)N+2+'1II(1 + IDwI P )
dxdr
Q1t
+ -yQ(a) pN+2+'1 R-(N'1+ 2a p>.o)p/"p. Using the dermition (7.8) of 1] we have p -(N1] + 2apA o ) lip
ap2
= -a + -(1 lip
Ao).
Since Ao E (0, 1) we estimate R-(N'1+ap>.o)p/"p ~ R-a P, and summarise: 7.1. Let a and 1] be chosen as in (7.8). There exists a constant -y = -y(data), independent of a, 1], p, R, such that LEMMA
forall (xo,to) E Q:k/2
(7.9)
11(1
0
andforall
+ IDwlP)dxdr
[(xo,to)+Q~J
~ -y Q(a) (~) N+2+'1
11(1
+ IDwIP ) dxdr
[(x o ,t o )+Q1tJ
+ -yQ(a) pN+2+'1R- a P.
8. Estimating the local average of IDwl LEMMA
8.1. For every a
E
(0,1) there exists a constant -y = -y (a,data), such
that forall (xo,to) E Q:k/2 (8.1)
H + IDwI (1
P)
andforall dxdr
0
~ -y(a,data)p-ap •
[(xo,to)+Q~J
PROOF: Derme sequences ao=(N + 2)/2 and forn=O, 1,2 ... , 1]n
= an(p -
2),
314 X. Parabolic p-systems: boundary regularity
It is apparent that {on}, {l1n}, {On}
-+
0 as n
-+ 00.
We will prove by induction
that F(on) :5 ')'n (on. data) ,
(S.2)
n
= 0, 1,2, ....
Since IDwleLP(lh),
H+ (1
[(:r:o,to)+Q~o
IDwI P ) dxdr:5
')'p_P(~+3)
(1
+ IIDwllp,SlTt.
I
Therefore F(oo) :5
')'0
== (1 + IIDwllp,SlTt·
Assume now that (S.2) holds for some n and let us show that it continues to hold for n + 1. We apply the iterative Lemma 3.1 with the choice of the parameters (3 to the
== N + 2 + l1n, " = onP, 0 = on,
function
=
II
1/
= 0,
(1 + IDwI P ) dxdr,
1(:r:o,to)+Q~n
I
which satisfies (7.9), to conclude that for all (xo, to) e Q:k/2 and for all 0 < p:5 R:5'R/2,
In particular, p being fixed, (S.3) must hold for radii p. satisfying
Without loss of generality we may assume that "',,-'Intl
p.
2+'In+1
_
D
= (;.
is an integer.
Then we may regard the cube Ixo + K pI as the disjoint union, up to a set of measure zero, of (f..)N cubes Ix; + Kp.1 centered at points x; of [xo + Kpl. Similarly we regard the cylinders [(x o• to) + Q~n+l] as the disjoint union, up to a set of measure zero,of(i.)N cylinders [(x;, to) + Q::]. We write (S.3) for each of these cylinders and add up for j =1, 2, ... ,i. to obtain
II
(1 + IDwI P ) :5 ')' (i.)N p~+2+"n-anp+6n
1(:r:o,to)+Q:n+l1
9. Bibliographical notes 315
Therefore pQn +1P
H + IDwI (1
P) dxdr5,'Y and F(an H,1Jn+t} 5, 'YnH'
[(:l:o,to)+Q:n+1]
8-(1). PROOF OF THEOREM 1.1 (THE CASE max {I;
~~2} < p < 2)
By a cube decomposition technique similar to the one outlined, Lemma 8.1 can be rephrased in tenns of the parabolic cylinders Qp=Q(p2, p). LEMMA 8.1'.
For every a E (0, 1) there exists a constant 'Y = 'Y (a, data), such
that
(8.1)'
H
(1
+ IDwI P )
dxdr 5, 'Y(a,data)p-QP.
[(:l:o,to)+Qp]
With this lemma at hand the proof is now concluded as in the degenerate case. First we may establish a version of Lemma 6.1 and then the HOlder continuity of u follows from the arguments of [22,23,32,79] or by those in §§2 and 3 of Chap. IX.
9. Bibliographical notes The proof of Theorem 1.1 is in [27]. The iteration Lemma 3.1 is in the same spirit of similar results of Campanato [22,23]. The technique is indeed a degenerate version of [22,23]. Techniques of this type near the boundary appear in Giaquinta-Giusti [49]. The boundary behaviour of solutions of (1.1) is essentially not understood. In the case of a single equation some results appear in Liebennan [69] and Lin [72].
XI Non-negative solutions in ETThe case p>2
1. Introduction Non-negative solutions of the heat equation in a strip ET == RN X (0, T) are somewhat special in the sense that they grow no faster than
a < 1/4T,
(1.1)
as Ixl-oo.
Let I' be a 0' - finite Borel measure in R N with no sign restriction. We say that p. bas the growth (1.1) if
/e-~ldlJl < 00,
(1.2)
.
RN
where
IdILI is the variation of 1'. Then the Cauchy problem
(1.3)
{
Llu = 0, u(·,O) = 1',
Ut -
in ET ,
is uniquely solvable within the class of functions satisfying (1.1). The •initial measlUe' is taken in the sense (1.4)
/U(x,t)CPd/-L~ RN
/cpd/-L, as t'\.O, RN
'v'cpEC~(RN).
2. Behaviour of non-negative solutions as lxi- 00 and as t '\. 0 317
Conversely every non-negative solution of the heat equation in ET verifies (1.4) for some u-finite non-negative Borel measure Jl satisfying the growth condition (1.2). The measure Jl is unique and it is called the initial trace of u. In tum the initial trace of u determines u uniquely. These are the basic elements of a classical theory developed by Tychonov [98], Tacklind [94] and Widder [105]. A perhaps rough summary of the theory is that the structure of all non-negative solutions of the heat equation is determined by the heat kernel _~
1
r(x, t) = (411't)N12 e
t,
t
> O.
Consider now non-negative local weak solutions in ET of (1.5)
{
u E C loc (O,T; Ut -
L~oc(RN»)nLroc (0, T; WI!;:(R N)) , p>2,
=0
div (lDulp-2 Du)
in ET.
The analog of r(x, t) for the degenerate p.d.e. (1.5) is the Barenblatt explicit solution ~
(1.6)
_ t -NI>.{ 1 =
B (x, t )
"yp
'Yp
(IXI)P!Y}"-+ ' til>'
t
> 0,
_....l....p-2
== A ,,-1 - - , A = N(p - 2) + p. P
We call this a 1undamental solution' only in the sense that B(x,t)
--+
(411') NI2 r(x,t)
pointwisein ET, asp'\.2.
Solutions of (1.5) cannot be represented as convolutions of initial data with B( x, t). Nevertheless the sup-estimates of Chap. V and the global Harnack estimates of §7 of Chap. VI permit a precise characterisation of the class of non-negative solutions of (1.5) in the whole ET • with no reference to possible initial data. Such a characterisation essentially says that all non-negative solutions of (1.5) behave as t '\. 0 like the 'fundamental' solution B(x, t), and as lxi- 00 they grow no faster than Ixl p/ (p-2). For these solutions we will establish the existence of initial traces and prove their uniqueness when the initial datum is taken in the sense of Lloc
2. Behaviour of non-negative solutions as Ixl-+ 00 and as
t'\.O Let u be a non-negative local weak solution of (1.5) in ET • For e E (0, T) and r > 0 set u(x,r) A=N(p - 2) + p. IIlulllr.T-~ = sup sup >'I( -2) dx,
-
J
Or
Kp
P
I'
318 XI. Non-negative solutions in ~T. The case p>2 THEOREM
2.1. There exists a constant 'Y='Y(N,p) such that/or all eE (0, T),
(2.1)
IIlullr,T-~ $ 'Ye-p!-, { 1 + (~)
;!, p-
u(O,T-e)
}>./P
Moreoverforall tE (0, T-e), and all p~r,
lIu(·, t)lIoo,K
(2.2)
pp/(p-2) p
$ 'Y
tNt>.
pI>'
IIlullr,T-~'
t
(2.3)
j jlDulP-IdxdT $ 'Y tl/>'pl+~ Ilull!~~, OKp
IIDu(', t)lIoo,K
(2.4)
p2/(p-2)
p
2/ >.
~ 'Y t(N+l)/>' Ilullr,T_~'
Moreover (x, t) -+ Du(x, t) is Holder continuous in Kp x (e, T -e) with HOlder constants and exponent depending only upon N, p, 'Y, p, e and IIlullr,~.
Remark 1.1. The functional dependence of these estimates is optimal as it can be verified for the explicit solution 8(x, t).
Remark 1.1. The estimates (2.2)-(2.4) hold for solutions of variable sign. provided we assume (2.1). PROOF OF THEOREM 1.1: The estimate (2.1) is the content of Corollary 7.1 of Chap. VI. whereas (2.2) follows from Theorem 4.5 of Chap. V. The gradient bound (2.3) is Lemma 9.1 of Chap. V and Remade 9.2. Inequalities (2.2)-(2.3) hold for a time interval
(2.5)
for a constant 'Y. ='Y.(N,p). In view of (2.1) they can be considered valid for all tE (0, T-e). Indeed working within ET • we may state them for every substrip
RN x [tl' t2j,
O$t!
t2 - tl ~ 'Y.. llullr,T-~'
In the proof of (2.4) we will work in the time interval (2.5). We begin with a qual-
itative information. LEMMA
2.1. For every e E (0, T) and for every r sup p>r
> 0 the quantities
IIDu(·, t)lIoo,K p P2/(P -2)
arefinitefor all 0< t~ T-e. PROOF:
By the interpolation Theorem 5.1' of Chap. VIII with e= 1,
9=!t we deduce
q
=1/2 and
3. Proof of (2.4) 319
for all r E (it, t). Estimating the right hand side by (2.3) we obtain
IIDu(·,r)lIoo,Kp < p2/(p-2)
-
"Y
(till UnUlp-2 )1/>'111 ullIII r,T-£: +C~ • r,T-£:
Next we will turn such infonnation into the quantitative estimate (2.4).
3. Proof of (2.4) Let t>O and p>r be fixed and consider the box radii {Pn} and time levels {t n } be defined by
Qo==K2pX
[it, tjcE T • Let the
n=0,1,2, ... and introduce the corresponding family of nested shrinking cylinders Qn
== K pft x{tn,t},
with vertex at (0, t). We will estimate the quantity IIDu(·, t)lIoo,K p ' by using the techniques developed in Chap. VIII. The starting point is the the iterative inequality (5.4) in that chapter, which we rewrite here in the context of the cubes Qn as y.n+l
_<
"Y bnk-~'LIy'l+~ + "n ,
where Yn == ! !ODu l -
~2
== N(p - 2) + 2p,
kn)~ dxdr,
12ft
(3.1)
'It ==
{sg: IDul
p - 2 p-2
+ t- 1 },
and k is a positive number to be chosen. By Lemma 4.1 of Chap. I, the sequence {Yn } tends to zero as n-+oo if k is chosen to satisfy
We conclude that there exists a constant "Y = "Y( N, p) such that for all it:5 r:5 t,
(3.2)
IIDu(.,r)lIoo,Kp :5
"Y'It~ (j !IDulpdxdr) t/2 K2p
2/>'2
320 XI. Non-negative solutions in E T • The case p>2
To proceed we introduce the non-decreasing function of t
(3.3)
¥= O
A>( ) _
..... t
sup p>r
IIDu(·, T)lIoo,K p p
2/( -2) P
,
By Lemma 2.1 and (2.2) this quantity is well dermed. In the estimates below we write ~ == ~(t) if the dependence upon t is unambiguous. We estimate the quantity 1t introduced in (3.1) by
and deduce from (3.2) that for alllt $
and
(3.4)
Estimating G1(t) we have
T
$t
3. Proof of (2.4) 321
To estimate G 2 (t) we refer back to the p.d.e. in (1.5). Let ( be a non-negative piecewise smooth cutoff function in K4pX{it, t} that equals one on K2pX{ !t, t} and such that ID(I ~ 2/ p and (t ~ 4/t. Taking u(P in the weak formulation of (1.5) we obtain
In estimating G2 (t) we use the estimation (2.2) and the range (2.5) oft.
Combining these estimates in (3.4) gives for all 0 < t ~ 'Y.lllulll ~,7-E'
322 XI. Non-negative solutions in Er. The case p>2
for a constant 'Y='Y(N,p). It follows that 4>(.) is majorised by the solution of
{
V'(t) ~
V(O)
'Y t- (N+l¥P-2) Vp-l(t),
= 'Ylllull~:;_E' 0 < t ~ 'Y.mull~:;_E·
Solving this explicitly gives
1berefore choosing t so small that {1-
'Y (t l~ull~:T2_E)
2/>.}-1/(P-2)
~ 2,
we will have
t¥
IIDu(·, t)lIoo,K < 2 IlluI1 2/>' p2/(p-2) 'Y r,T-E p
for all such t and all p> r.
4. Initial traces THEOREM 4.1. Let u be a non-negative local weak solution of(1.5) in :E T . There exists a unique Radon measure IL such that
lJ$
(4.1)
f
u(x, t)cpdx
=
RN
Moreover. as
f
cpdIL,
'v'CPEC~(RN).
RN
Ixl- 00, IL 'grows' at most as IxI P/(p-2). Precisely.
(4.2)
sup p>r
f
dIL /( -2)
pP P
< 00,
'v'r > O.
Kp PROOF:
The existence of a Radon measure IL satisfying (4.1 )-(4.2) follows from
the global Harnack estimates of §7 of Chap. VI. Indeed by Corollary 7.1 of that Chapter, for every cube [x o + K pj C RN and all cp E Cr;' (Kp),
If
u(x, t)cp(X)dXI
_ Kp
~ 'Y (N,p, p, T, U(Xo, T-e» IIcplloo,K
p '
5. Estimating lDur- 1 in Er 323
for all 0 < t :$ T -E and all E E (0, T). Therefore {u(·, t)}O
for a Radon measure IJ. The uniqueness of such a measure is a consequence of the following: LEMMA
4.1. Let u be a non-negative local weak solution of (1.5) in ET. Then
'Vp>O, (4.3)
f
'VO'E(O, I},
u(x, t)dx
K(1+")p
~
f
'VO
u(x, r}dx- ;,(t -
r}l/'\p~ Ilull~;'~.
Kp
PROOF: Fix 0 < r < t and 0' E (0, I), and let x ...... (x) be a non-negative piecewise smooth cutoff function in K(1+cr)p that equals one on Kp and such that ID(I :$ 2p/0'. In the weak fonnulation of (I.S) take ( as a testing function. Integrating over (r, t) gives
f
u(x, t) dx
~
K(1+")p
f
t
u(x, r) dx - 0'2p
Kp
f T
PDuI P- 1 dxds. K(1+")p
To prove (4.3) we estimate the right hand side of this inequality by (2.3). We now prove the uniqueness part of Theorem 4.1. Suppose that out of the net {u(x, t)}O
for all 'P E ego (RN) and IJ :/= t ...... 0 along t'. This gives
II.
Then we let r ...... 0 along r' in (4.2) and then let
Interchanging the role of IJ and II proves the Theorem since 0' E (0, I) is arbitrary.
5. Estimating IDu Ip-l in ET Local integral estimates of IDul p - 1 are crucial both in the global Harnack estimate of §7 Chap. VI and in the theory of initial traces. The inequality (2.3) of Theorem
324 XI. Non-negative solutions in Er. The case p>2 2.1 is local but holds for all p > r. Therefore it implies some control on the behaviour of IDul as Ixl -+ 00. This behaviour can be given an integral form, by means of the weights (5.1) where a is a positive number satisfying
.x
(5.2)
ap= --2
p-
+u,
for some u
> o.
THEOREM 5.1. Letu bea non-negative local weak solution of(1.5) in ET. Then for every u > 0, there exists a constant 'Y = 'Y( N, p, u) such that for all r > 0 and all EE (0, T),
(5.3)
sup
O
-
ju(x, t)Aa(x) dx
~ 'Ylllullr,T-E'
aN
t
(5.4)
j jlDulP-l Aa(x) dxdr ~ 'Ytl/>'l~ull~;'~. oaN
Remark 5.1. The constant 'Y(N,p, u) /00 as u '\,0. PROOF OF
(5.3): Without loss of generality we may assume that r
= 1. Then
foralIO
fu(x, t)Aa(x) dx ~ f u(x, t)Aa(x) dx + f: f u(x, t)Aa(x) dx aN
{Izl
n=O
{2n
00
~ mU~lr.T-E + 2~
L 2-
an
mullr,T-E.
n=O PROOF OF (5.4): It will suffice to establish the estimate for t in the interval (2.5). We will use this fact with no further mention. First we observe that the inequality
(5.5) holds for all x E RN and all 0 < t ~ T-E. This is obvious if Ixl ~ r with the constant 'Y depending also upon r. If Ixl > r, we apply (2.2) to the cube K 21zl. Let T/ E (0, T -£ ) and in the weak formulation of (1.5), take the testing function
l/p I_a (Al/p r)P (t - T/ )+ U P a+ 1 .. , P
where x -+ «(x) is the usual cutoff function in Kp. After a Steklov averaging process and standard calculations, we obtain
5. Estimating IDulp - t in
Er 325
t
(5.6)
IP A 1. (Pdxdr f (r - "l)l/pjIDU u 2 / p "'+,. "
Kp
t
~ 'Y j(r -
"l)I/P j u¥up-IID
"
(A~:*()IP dxdr
Kp
t
+ 'Y pr - "l)~-lj u¥ A1/puA",dxdr = J~l) + J~2). "
Kp
As for J~2) we have £=l
t
J(2) < "'f(r_'TI)t- 1jr N <:A 2 ) lu(x,r)I" u(x r)A (x)dxdr p /./ (1 + Ixl p)1/p ,'" , "
Kp
so that by (5.5) and (5.3),
J~2) ~ 'Y(t - "l)!-,AWull!;'~. We estimate J~I): t
J~I) ~ 'Y j(r-"l)* jU¥UP-1A",+;ID(IPdXdr "
Kp
t
+'Y pr - "l)l/pju~uP-IIDA~:*IPdxdr "
Kp
= J~l,l) + J~I,2) . Since
IDA '"I/+P1.1 ,. ~ 'YIA~+:!7+1.IP ,. ,. ,. ~ 'YA", A I/ p At,
by (5.5), (5.3) and the range (2.5) of t, t
J~1,2) ~ 'Y j(r - "l)~ j (u~ A~) "
(u P- 2AI) u(x, r)A", (x) dxdr
Kp
~ 'Y(t - "l)!-,Alllull!;'~. As for J~I,l), since ID(I ~ 2/ p, again by (5.5) and (5.3)
J~l,l) ~ 'Y(t - "l)!-,AI~ull!;'~.
326 XI. Non-negative solutions in Er. The case 1'>2 Combining these estimates in (5.6). (5.7)
where we have changed pinto 2p. Next, for all ,., ~ t ~ T - E
6. Uniqueness for data in Lloc(RN) THEOREM 6.1. Letu and v be two non-negative local weak solutionsof(l.5) in ET. satisfying
Then u==v in ET. PROOF:
Fix r>O and some EE (0. T) and set
Ilullr.T-~
+ Ilvllr.T~ == A.
Then and u and v satisfy all the estimates of Theorem 2.1 with the quantities llu. vllr.T~ replaced by A within the strip ETa' where
0< To = min{T;-y.A-(p-2)}. and -y. is the constant appearing in (2.5). It will suffice to prove uniqueness within the strip ETo' The difference w=u-v satisfies
6. Uniqueness for data in Lloc(RN ) 327 (6.1)
where ai,j(z, t) =
(iID(S' + (1 -
S)V)Ip-2ds) 6,;
1
+ (p -
2) j ID(su + (1 - s)v)IP-4
o X (su
+ (1 -
s)v)x. (su + (1 - s)v)xjds.
The matrix (ai,j) is positive semi-definite and for all eE RN and (x, t) E ETo
ao(x, t)lel 2 S ai,j (x, t)eiej S (p - l)a o(x, t)leI 2 , l { (6.2) ao(x, t) = flD(su + (1 - s)v)IP- 2ds, (x, t) E ETo. o Let Ao(x) be the weight introduced in (5.1) with Q satisfying (5.2). In the arguments below, 'Y denotes a positive constant that can be determined a priori only in terms of N, p, (T and A.
6-0). Auxiliary lemmas LEMMA 6.1. There exists a constant 'Y='Y(N,p, (T, A) such that ifw(·, t) -+0 in Lloc(RN) as f\,O, then
j1w(x, t)IAa(x) dx S 'Ytl/",
0< t < To.
RN
PROOF: The functions w± are both weak subsolutions of (6.1), i.e.,
wf - (ai'i(x, t)w~)x., SO weakly in ETo· By working separately with w+ and w- we may assume that w is a non-negative subsolution of (6.1). In the weak formulation of (6.1) take the test function x-+ Ao(x)«x), where (is the usual cutoff function in Kp. Using the assumptions of the lemma we deduce t
+ IDvD p -
1
IDAo(1 dxdr
S 'Y j jODvl + IDvD p -
1
AoID(1 dxdr
j1w(x, t)IAo(x)( dx S 'Y j jODvl Kp
OKp
t
OKp
t
+'Y j jODvl + OKp
IDvDP-IIDAoldxdr.
328 XI. Non-negative solutions in ET. The case p>2 In the last integral.IDAal $'YAa+l/p and in the first integral. since IDcl =0 on Kp/2 we have AalDCI $'YAa+l/p for p > 1. Therefore letting p-+oo t
jlw(x.t)IAa(X)dX $ 'Y jjODvl RN
+ IDvI)P-l Aa+l/pdxdr,
ORN
and the conclusion follows from Theorem 5.1.
PROOF: Let fiE (0,
*,) be fixed. Then 'v'tE (0, To)
j1w{X, t) I1+'1 A a +fJ /(p-2) {x)dx RN
$ jIW{X, t)l fJ A fJ /(p-2) {x)lw(x, t)IAa{x)dx. RN
By (5.5).lw(x, t)lfJA fJ /(p_2) (x) $ 'Yt-If 'I. so that by Lemma 6.1.
jIW(x, t) I1+'1 A a+fJ /(p-2) {x)dx $ 'Yc IYf j1w(X, t)IAa{x) dx RN
RN
$ 'Yt!
(l-NfJ).
6-0;). Proof of Theorem 6.1 In (6.1) we may assume. by working separately with w+ and W-. that w ~ O. In its weak: formulation we take the testing functions
Integrating over K p x (c, t). 0 < C< t $ To. we obtain
6. Uniqueness for data in L:""(RN ) 329 (6.3)
1 ~ 11 !(W + 6)1+'7AQ(2dx Kp
t
+11
If
IDwI2 (1)2 ao(x,r)(W+6)1-'7 A dxdr
6Kp
~ 1 ~ 11
Q (
!
(w
+ 6)1+'7AQ(2dx
Kpx{6}
t
+'1 !!ao(X,r)(W~~~ (w+6)l:f1 6Kp X
(A!()
ID (A!() Idxdr,
where ao(x, t) has been defined in (6.2). By the Schwartz inequality the last integral is majorized by
t
!
+ '1(11) !ao(x, r)(w + 6)1+'7 (AQID(1 2 + IDA! 12) dxdr. 6K p
We absorb the integral involving IDwl2 on the left-hand side of (6.3) and discard the resulting non-negative term. Finally, we observe that by the definition of AQ and the structure of ( we have
AQID(1 2 + IDA! 12 ~ 'YAQ(X) Alp (x). Carrying these remarks in (6.3) gives (6.4)
!(w + 6)1+'7AQ(2dx Kpx{t}
~
!(w + 6)1+'7AQ(x) dx
Kpx{t}
t
+'Y!!ao(X,r)A;(x)(w + 6)1+'7AQ(x)dxdr. 6Kp
Next by (6.2) and (2.4)
a (x r)A.a(x) < '1 0,
p
-
Ixl2 Ai (p-2)r-(Nr> (p-2). (1 + Ixl p)2/p
Substitute this last estimate in (6.4) and let 6 - 0 for p 2: 1 fixed so that by Lemma 6.2
330 XI. Non-negative solutions in ET. The case p>2
j (w + 6)1+f/Ao(x)dx
--+
°as
6 - 0.
Kpx{6}
Then we let p -
The net result is
00.
j1w(x, t) 11+'1 Ao(x) dx RN
t
:5 'Y j.,.- (Ntl) (p-2) j1w(x, "')11+'1 Ao(x) dxd.,.. o Since.,.- (Nti)
(p-2)
RN
E L1 (0, t), this implies
t - j1w(x,tW+f/Ao(X)dx
== 0,
RN
by Gronwall's lemma, provided t - /lw(x,tW+f/Ao(X)dX E VlO(O,To)' RN
Now the parameter Q in the calculations above is arbitrary and only restricted by (5.2). If Q is replaced by Q+,,/(p-2). then Lemma 6.2 and its proof ensure the Loo(O, To) requirement and the theorem follows. Remark 6.1. For non-negative solutions u and v of (1.5) in ET. the quantities
Ilullr,T-E, IIIvllr,T_
(6.5)
are fmite.
The proof of Theorem 6.1 uses only this information. Indeed by Remark 2.2 such a growth condition implies all the estimates of Theorem 2.1. We conclude that the uniqueness theorem for initial data taken in the sense of Lloc(RN) holds for solutions of variable sign provided (6.5) holds.
7. Solving the Cauchy problem Consider the Cauchy problem u EC
(7.1)
{
Ut -
(0, T; Lloc(RN»nLfoc (0, T; W,!;:(RN») , p>2,
div (lDulp-2Du)
u(·,O) =
Uo
E
=
°
in ET, for some T>O
Lloc(RN).
As indicated in the firstof(7.1) the initial datum is taken in the sense of Lloc(RN). By Theorem 6.1 and Remark 6.1 there is at most one solution to (7.1) within the class of functions u satisfying
7. Solving the Cauchy problem 331
Ilullr,T-£ < 00
(7.2)
for some
EE
(0, T).
Existence of a solution satisfying (7.2) can be established if the initial datum U o satisfies the growth condition
- f
Iluolir = :~~
(7.3)
luo(x)1
P"/(p-2)
dx O.
Kp
Since U o ELloc(RN) if Iluollir is finite for some r >0, it is finite for all r>O. THEOREM 7.1. Let U o satisfy (7.3) for some r > O. There exists a constant 'Y. = 'Y.(N,p) such that defining
(7.4)
there exists a unique solution u to (7.1) in ET. Moreover u satisfies (7.2) for all eE(O, T) and the estimates (2.2)-(2.4) of Theorem 2.1.
Remark 7.1. This is an existence theorem local in time and the largest existence time is estimated by (7.4). The functional dependence in (7.4) is optimal as shown by the following explicit solution.
1'(x,t)= { A ( -TT-t
)~ + (p---2) ..\ _~ ( Ixl )p!r}~ p=-r
P
P
--
T-t
,
where A and T are two positive parameters. By direct calculation we have
~111>(.,O)llr = ~
~
(P;2)P- ("\T)-~,
where W N is the area of the unit sphere in R N. Therefore 1'( x, t) exists up to the blow-up time T
where 'Y.
= 'Y. { ~ 1111>(·,OHlr }
= ..\-~ (~r-2
-
,
(p; 2)P-l
For n= I, 2, ... ,consider the sequence of truncated initial data min{uo(x);n}}, ( ) = {max{-n; 0,
uo,n x -
It is apparent that for all n= 1, 2, ... , (7.5)
Consider also the family of approximating problems
for Ixl < n for Ixl ~ n.
332 XI. Non-negative solutions in Er. The case p>2 {
Un,t -
div IDUnl p - 2 Dun = 0, in RN xR+
un(·,O)
= uo,n·
Since uo,n are compactly supported in RN, (7.1)n can be uniquely solved as indicated in §12 of Chap. VI. By the maximum principle the solutions Un are bounded by n. Therefore the quantities
III unlIII r,t -=
sup sup
!un(X, 'T)
O<.,.<tp>r Kp
P>-/(
P
-2)
dx
are finite for all r, t > o. It follows that the sequence {Un} satisfies (2.2)-(2.4) of 1beorem 2.1. We will tum such n-dependent information into a quantitative supestimate of {un} independent of n. Let x --+ '(x) be the standard cutoff function in K 2p • Then (7.l)n implies
We divide by p>'/ (p- 2) and take the supremum over all p> r. Taking into account (7.5) and (2.3) this gives
for two constants 'Yi ='Yi(N), i=O, 1. Let tn be defined by P-2) 1/>.
'Yl ( tn IIlunllr,t
=
1 2·
Then from (7.6) for all t E (0, t n )
IIlunllr,t ~ 2'Yo Iluolir. This implies that tn ~ Tr for all n = I, 2, ... " where Tr is defined by
We summarise: LEMMA 7.1. Let {Un} be the sequence of the approximating solutions (7.1)n. There exists a constants 'Y = 'Y(N,p) and 'Y. ='Y.(N,p) independent ofn. such
that
(7.7)
where
(7.8)
8. Bibliographical notes 333 Given such an estimate, the Cauchy problem (7.1) can be solved by a standard limiting process. Indeed by Theorem 2.1 the sequences Un } , { -f) f)xi nEN
i
= 1,2, ... ,N,
are locally equibounded and equi-HOlder continuous in RN x (O, Tr). This gives the existence of a unique solution in ETr • The largest time of existence can be calculated from (7.8) by letting r -+ 00. In particular the solution to (7.1) is global in time if
.
lim sup p>r
j
f'-tOO
uo{x)
>./( -2) dx = O.
P
P
Kp
8. Bibliographical notes Theorem 2.1 is taken from [41]. A weaker version of (2.2) in I-space dimension is due to Kalashnikov [58]. It is remarkable that in (2.4) one can also control the behaviour of the space-gradient IDul as Ixl-+ 00. Since IDul 2 is a non-negative subsolution of a porous medium-type equation (see (1.8) of Chap. IX) the same techniques yield a version of (2.2) for such degenerate p.d.e. The analog of (2.2) for the porous medium equation is due to Benilan-Crandall-Pierre [10] in the context of an existence theorem. A rather general version is in [4]. Perhaps the most relevant estimate of Theorem 2.1 is the integral gradient bound (2.3) proved in [41]. A version of such a local bound, for the porous medium equation is in [4] and reads
jlDuml dxdr ~ -yt /"p1+w!=r "lu"I!;.:~l, 1
K.
= N{m - 1) + 2,
Kp
where -y=-y{N, m) and IIIulllr,T-E
- sup sup j = Or
Kp
u(x,t) dx. p ,./(m-l)
The estimate holds for small time intervals and for general non-linearities. We refer to [4] for details. There is no analog of (2.4) for the porous medium equation. Theorems 4.1 is taken from [41]. The analog for the porous medium equations is in [6] and for general non-linearities [4]. It would be desirable to have a version of the uniqueness Theorem 6.1 for initial data measures. This would parallel the analogous theory for the heat equation.
XII Non-negative solutions in E T . The case 1
1. Introduction We will investigate the structure of non-negative solutions in the strip ET of the singular p.d.e. (1.1)
Ut -
div IDulp-2 Du
= 0,
I
A striking feature of these singular equations is that, unlike the degenerate case p>2, non-negative solutions of (1.1) are not restricted by any 'growth condition' as Ixl- 00. Nevertheless they have initial traces that are Radon measures. More-
over they are unique whenever the initial traces are in Lloc{RN ). Accordingly, the Cauchy problem for (1.1) associated with an initial datum(1.2)
U o ~O,
is uniquely solvable, regardless of the behaviour of x-uo(x) as Ixl-oo. The case 1 < p < 2 is noticeably different from the case p > 2, both in terms
of results and techniques. The main difference stems from the fact that, unlike the degenerate case, solutions of (1.1) are not, in general, locally bounded. In a precise way, if (1.3)
and
2N P>-N +r '
1. Inttoduction 335
then the solution '1£ of (1.1)-(1.2) belongs to Lroc{ST) , "It> O. This is the content of Theorem 5.1 of Chap. V. In §13 we will give a counterexample that shows that if '1£0 violates (1.3), then '1£ ¢ L~c{ET). The basic formal energy estimate for (1.1) is VO<8
VKp t
(1.4)
ju 2 {x, r) dx + f flDulPdxdr
sup
11 sKp
s
- Kp
Thus ifu e L~oc{ET), the left hand side of (1.4) is finite and IDul e Lfoc{ET)' However if '1£0 e Ltoc{RN), there is no a priori information to guarantee that (1.5)
We have spoken oholutions of (1.1); however if (1.5) fails, one of the main problems is to make precise what it is meant by solution. Thus the starting point of the theory is to give a precise meaning to Du to make sense out of (1.1). The previous remarks suggest that IDul might fail to be in Lfoc{ET ), roughly speaking at those points where '1£ is unbounded. Motivated by these remarks, we have given a novel formulation of non-negative weak solutions. Such solutions are 'regular' in the sense that the truncations
Vk > 0,
(1.6)
Uk
= min{u, k},
satisfy (1.7)
Then (1.1) can be interpreted weakly against testing functions that vanish 'whenever '1£ is large'. A suitable choice of such testing functions is (~- '1£)+
== max{(~ - u);O},
~
e C~{ET); ET'
The notion is introduced and discussed §§2 and 3. We prove that these solutions coincide with the distributional ones if (1.5) holds and that the truncations Uk are distributional super-SOlutions of (1.1) Vk > O. We derive a spectrum of properties of such local weak solutions, regardless of their initial datum. In particular we investigate the behaviour of DUk as k -+ 00. A relevant fact is the estimate (1.8)
frlDulP-l dxdr
11 BKp
VO<8
~ "Y s
VK2p,
- K2P
336 XII. Non-negative solutions in E r . The case I
where A=N(p- 2) + P and 'Y='Y(N,p). We remark that in Chap. XI an estimate of the local integral nonn of IDuI P- 1 was crucial to establish the existence of initial traces. In the singular case 1 < p < 2 it is precisely (1.8) that pennits one to prove an integral Harnack-type inequality, which in turns implies the existence of initial traces. The estimate (1.8) is essential also for the solvability of the Cauchy problem. A solution to (l.l )-(1.2) is constructed by using the increasing sequence { u o,n} of approximating initial data
Uo,n = min{u o ; n},
(1.9)
n
= 1,2, ... ,
and solving the approximating problems
Un (1.10)
{
E
C (0, T; L~oc(RN»nLP (O,T; WI~:(RN») ,
Un,t - div IDu n IP- 2 DUn = 0, in ET, un(·,O) = uo.n , in the sense of Lloc(RN).
The comparison principle and (1.8) yield the Lloc(ET) convergence of the approximating solutions {un}. A one-sided bound on uo,n and hence on Un is crucial to this process in view of the regularising effect of Proposition 6.1 of Chap. VI. In §5 we show uniqueness of weak solutions if they take their initial datum in the sense of Lloc(RN). Namely, if U and v solve (1.1) weakly and if
then the difference w =
v satisfies
U -
=
'Y( N, p, q). The theorem follows by letting p -+ 00 after we for a constant 'Y choose q so large that N(p - 2) + pq>O. If, in (1.3), r = 1 and p > J~ l' the existence and uniqueness theory remains valid if U o E Lloc (RN) with no sign restriction. Indeed in such a case the sequences
are locally equibounded and equi-HOlder continuous in ET. If 1 < p < J~2' the singular equation (1.1) is not fully understood. For example it would be of interest to investigate questions of existence and uniqueness for the Cauchy problem (1.1 )-( 1.2) if the initial datum is a measure JL. Finally, we notice that all the results of this chapter hold true for equations of the type N
Ut - L(lux;IP-2 ux;)x, i=l
=0
in ET.
2. Weak solutions 337
2. Weak solutions A measurable function U : ET -+ R + is a local weak solution of (1.1) in ET if
uEC(O,T:Lloc(RN )), !DUk!ELfoc(ET), :tUkELloc(ET)
(2.1)
for all k>O and'v'ep E C:'(E T ),
!!{Ut(ep - u)+ + IDuIP- 2 DuD(ep - u)+}dxdr
(2.2)
= o.
ET Introduce the spaces
(2.3)
== {ep E Xloc(ET) !ep(x, t) = 0,
XloC (ET)
(2.4)
'v'lxl
'v't E (0, T), for some p
>0
>
p} .
o
By density, (2.2) holds for all ep E X loc (ET). We denote with S the set of all non-negative local weak solutions of (1.1) in ET. LEMMA
2.1. LetuES. Then 'v'1/JEXloc(ET) and'v'l1EC:'(ET).
(2.5)
!
!{Ut(1/J - U)+l1 + IDul p - 2 DuD[(1/J - U)+l1)} dxdr =
o.
ET PROOF: Let IC c IC' be compact subsets of ET such that dist (8IC, 8IC') = d > 0 and let (E C:'(IC') be such that 0 ~ (:51 and (== Ion IC. Choose 1/J E Xloc(ET) and in (2.2) take
where (2.6)
11 E C;:"(IC)
and
k=
111/Jlloo,K:/.
We have a.e. in IC'\IC (ep - u)+ =
((1/J - U)+l1 + Uk( - u)+
= (Uk(-U)+
=0.
Moreover (ep - u)+ =
((1/J - U)+l1 + Uk - u)+,
a.e. in IC.
This vanishes unless U< 1/J. In such a case, Uk = U and a.e. in IC.
338 XII. Non-negative solutions in ET. The case I
We conclude that this holds a.e. in I:T and (2.5) follows. Let (f E (0, 1) and let x x) denote the standard cutoff function in K p that equals one on K tTp , (f E (0, 1). By density. (2.5) implies
«
'Vt/J E X'oc(I: T ),
(2.7)
'VO<s
t
j j {Ut(t/J - u)+(P + IDul p- 2DuD[(t/J - u)+(PJ} dxdT
= O.
BRN
Conversely. if t/J E C~ (I:T ). we may write (2.7) for s < t such that supp{ t/J} C RN x (s,t). By taking (so that p>2diam(supp{t/J}). we obtain (2.2). We conclude that the fonnulations (2.2), (2.5) and (2.7) are equivalent. 2.2. Let UES satisfy
LEMMA
Then
Ut - div IDul p- 2Du = 0 PROOF:
In (2.5) take t/J=un
in 1>'(I:T).
+ 1 E X'oc(I:T). nEN. We obtain 'VTJE C~(I:T)
j j{UtTJ + IDulp-2 DuDTJ}(un
-
u + l)+dxdT = j j IDuI P'1 dxdT .
I:T
I:Tn[n
Since IDul E Lfoc(I:T ). the right-hand side tends to zero as n- 00. The left-hand side converges to
j j {UtTJ + IDuI P- 2 DuDTJ} dxdT
= O.
I:T LEMMA
2.3. Let U E S. Then/or all k > 0, Uk is a distributional super-solution
0/{1.1) in I: T . PROOF:
Fix k>O and Q,eE(O, I), and in (2.5) take t/J = Uk
+ [(k - u)+ + el o
E X'oc(I:T)
to obtain 'VTJEC~(I:T). TJ~O
jj{UtTJ + IDuI P - 2 DuDTJ}(t/J - u)+dxdT = jjlDulP'1 dXdT I:T
I:Tn(k
+ Q j jIDUkIP[(k - u)+ + elo-l'1dxdT ~ O. I:T
First we let e-O as Q E (0,1) remains fixed. Since
2. Weak solutions 339
(1/J - u)+ - (k - u)+ we deduce
j j{u'TI + IDul,,-2 DuD.,.,}(k - u)+dxdr
~ 0,
Voe(o, 1).
I:T Now letting 0
-
0 gives for every non-negative TI e Or:'(I:T )
{! UkTl + IDUkl,,-2DUk.DTI} dxdr ~
jj
(2.8)
O.
I:T The next proposition pennits a large class of testing functions in (2.5). H ko > 0, let F(ko ) denote the set of all the Lipschitz-continuous functions f : R+ - R such that f(k)=O, Vk>ko, and set
PROPOSITION
2.1. Let u e S. Then 'If e F and VTI e Or:' (I:T ).
j j{Utf(u)TI + IDul,,-2 Du·D(f(u)TI)}dxdr
= o.
I:T Assume first that f e 0 2 (0,00). Write (2.5) for 1/J = k, multiply it by - f" (k) and integrate in dk over (0, 00). By interchanging the order of integration with the aid of Fubini's theorem we obtain
PROOF:
00
j j {U'Tlj!"(k)(k - u)dk
I:T
U
+ IDuIP-'D,..D [~ P"(k)(k - U)dk] }dzdT = o. Since
00
j!"(k)(k - u)dk = f(u), U
the assertion follows for f tion.
e 0 2(0,00). TIle general case is proved by approxima-
340 XU. Non-negative solutions in Er. The case 1
3. Estimating LEMMA
IDul
3.1. There exists a constant 'Y = 'Y(N, p) such that Yk> 0,
Yp > 0,
YO<s
Yu
t
j jlDUkl PdxdT ::5 'Y kPlKpl (k2-P +
E
S
t; s) .
BRN
PROOF:
Let ( be the standard cutoff function in K2p. Then from (2.7) with 1/J =k t
t
j jIDUk/P(PdxdT::5 p jj/DUk/P-1(P-l(k - u)+ID(ldxdT BRN
SRN t
+~
JJ:T (k - u)!(PdxdT BRN
p; J t
::5
1 jIDUkIP(PdXdT SRN
J t
+ pp-l fik - u)~ID(IPdxdT SRN
+ ~ fik - u)!(Pdx. RNX{t}
For all 0< s < t::5T and all p>O set (3.1)
MB,t(P) = sup fU(X,T)dx. 1'e(B,t) Kp
LEMMA
3.2. LetueS.ThenYo:e(O,p-l)
I
IDuP-~-Q e Lroc(I:T ), tmd there exists a constant 'Y='Y(N,p) such thatYO< sO
PROOF: Fix k > 0 and e e (0,1). and in (2.8) take '1 = (P1/J-Q. where ( is the standard cutoff function in K 2p and
3. Estimating
IDul
341
u>e
u ~e. We obtain t
o /JIDuIPu-Q-l(Px[e
t
~ P //IDuIP-lu-Q(P-IID(lx[e
(3.3)
t
+ p / /IDu~IP-le-Q(P-IID(1 dxdr sRN
By Young's inequality, the first integral on the right-hand side is majorised by t
i //IDuI Pu-Q-1(Px[e < u < kjdxdr sRN
By virtue of Lemma 3.1 the second integral tends to zero as e -+ 0 at the rate of eP - 1 - Q • Combining these calculations we deduce t
o j/IDuIPu-Q-lx[e
~ O(eP-1-Q) +
"(_1 { oP
(sup
ju(x, T)dx) l-Q (2p)QN
'TE(s,t) K2p
(3.4)
+
(~) (sup pi'
ju(x,r)dx)P-I-Q (2P)N(2-P+Q)}
'TE(s,t) K2p
~
0:-1{[M ,t(2P)j1-Q + (t ; s) [MB,t(2P)jP-I-Q} pN
+ O(eP - 1-
s
Q ).
342 XII. Non-negative solutions in ET. The case I
c S)
If
~
~ [Ms ,t(2p)]2-",
the quantity in braces on the rightmost side of (3.4) is majorised by [Ms ,t(2p)j1-0<. Otherwise it is majorised by
t-S)~ . [Ms ,t(2p)]1-0< + ( -;:;In either case t
(3.5)
JJIDul"u-(O<+l)x.[e
,; O(e,-(.+1») + :, pH { M •.• (2p) +
(t;.) ~ }
1-. ,
and the lemma follows by letting first e -+ 0 and then k -+ 00. Estimate (3.2) deteriorates as Q -+ O. TIle next lemma gives some information
for the case Q LEMMA
=O.
3.3. Let uES. There exists 'Y='Y(N,p) such that \t'O<s
J
\t'p > 0,
f IDu¥ I"x.[n
\t'n ~ 1
< u< n + 1] dxdT
sKp
-;:;. ( + n1) [M ,t(2p) + (t-S)~l In (2.8) we take ,,= (1/J, where ( is standard cutoff function 1/J=ln+ Here = {n, if 0 < n > ~ 'Yin
1
s
the
PROOF:
and
(~).
u ~
u(n)
u,
ifu
n.
We get t
(3.6)
JJIDul"u-1x.[n
~ jJ:T un+lln+ (:~)1) ("dxdT SK2p
t
+ ~ JJI Dul,,-lln+ (:~)1) dxdT = I~l) + ~ I~2). SK2p
in K 2p
3. Estimating IDul 343
Setting, for simplicity of notation,
A = K 2p x (s,t), we have
I~2) ~ln
(1 +~) ff
IDuIP-1u-(Ot:+l) (P;ll u(Ot:+1) (P;1) dxdT
An[u
Vi
.P=l
$
~ In(,+ ;) ([.rID.."¥' I'dxdT) ·
.1
U
dxdT)'
If Q E (0, P - 1) is so small that (Q + 1) (p - 1) ~ I, both integrals in parentheses are finite. Taking Lemma 3.2 into account in estimating the first integral we have
~ I~2) ~ 1pN In (1 + ~) , [Ms,t(2 P) +
(t ~
8)
,.!;; ] (l-Ot:)¥
(~ j /
.1
U(Ot:+1HP_l)dxdT)
P
SK2p
The last integral above is estimated by
(~
f t
.1
/U(Ot:+lHP-l) (x,
T) dxdT )
P
SK2p
Therefore
~ t.') $ ~pNIn (I+~) [M••• (2P) + ('
;;;:t].
As for I~l) we write
ff In (1 +~) +ff In+ (n: 1) = ff! In (1 +~) +ff In+ (n: 1) ~1pNln(1+~)Ms,t(2P)+ ff :T (jln+ (n;')d{\ An)
I~l) =
(PdxdT
Ut
An[u
Un
A
Ut
(PdxdT
An[u>n]
(PdxdT
:T u(n)
(Pdxdr
A
+
("dxdT.
344 XII. Non-negative solutions in ~T' The case 1
3.1. Let u E S and define u{x,t)
(x,t)
->
z(x,t) =
/
(~lnl+E~)-;dx,
E
E (O,p - 1).
e
Then IDzl E Lfoc(ET) and there exists "Y="Y(N,p) such that'v'O < s< t ~ T and 'v'p>O,
PROOF: Divide both sides of the inequality of Lemma 3.3 by lnl+E n, and add over all n=2,3, ...
The estimate (3.7) deteriorates as E-O. The following corollary gives some information in the case E = O. COROLLARY
3.2. Let uES. Then 'v'O<s 1. t
ffIDuIP-I-lX[k
lim
k .....
s Kp
PROOF: Without loss of generality we may assume that k and Ck are positive integers. Divide both sides of the inequality of Lemma 3.3 by In n and add for n=k, k + 1, ... ,Ck. This gives
t
//IDunUln u)-lX[k
Ii'S)~l
~"Y{lnlnCk-Inlnk) [ M s ,t(2p)+ ( t -
= "YIn ( 1 + InC) Ink
[ M s •t (2p)
+
(tIi'S) ~l .
4. The weak Harnack inequality and initial traces In the definition of local weak solutions of (2.1) in ET, no reference has been made to initial data. We will show that each u E S has a unique non-negative u-finite Borel measure J.I. as the initial trace. The existence of such a trace will be a consequence of the following weak Harnack-type estimate.
4. The weak Harnack inequality and initial traces 345
4.1. Let U
THEOREM
E
S. There exists "I = "I(N,p), such that "10 < s
< t $. T
andVp>O
sup
(4.1)
TE(s,t)
!
j u(x, t)dx + "I ( t-S)~ P
u(x, r)dx $. "I
Kp
-.>.-
,
K2p
A = N(p - 2) + p. The uniqueness of the initial trace J.I. relies on the next gradient estimates. LEMMA
4.1. Let uES. There exists a constant "I="I(N,p) such that VO<s
h
Vp> 0,
1 IDulp-1dxdr p}}
(4.2)
$. "I
"10'
E
(0,1),
"Iv> 0,
(t-S)~ 7-
8 Kp
:!iE..=..ll
+ "I
(t -/) *{ P
sup ju(x, r)
S
dx}
P
K2p
Moreover
Ih'lt IDulp-1dxdr $. "I sup
(4.3)
-
p
S
j u(x, r) dx + "I (t-S)~ P -.>.-
.
K2p
PROOF: The proof is the same as that of Propositions 4.1 and 4.2 of Chap. VII. The only difference is that instead of working with the solution u we work with the truncations Uk and use the fact that these are supersolutions. In (2.8) we take the testing functions
t/J = (t -r)*(uk + v)l-:
E
X'oc(E T ),
where v> 0 is arbitrary. We proceed as in Chap. VII and then let k ~ 00. THEOREM
4.2. Every u
E
S has a unique Radon measure J.I. as initial trace at
t=O. PROOF:
From Theorem 4.1 it follows that V'1EC~(RN), the net
{ jU(r)'1dx} RN
TE(O,t)
is equibounded, with bound depending only upon I '1 II oo,RN • A subnet indexed with {r'} converges to a Radon measure 1', in the sense of the measures, i.e.,
346 XII. Non-negative solutions in ET. The case 1
Suppose now that there exist another subnet, indexed with {r"} and a Radon measure jJ., such that
We will prove that J.I. == jJ.. Let u E (0,1) and write (2.8) with 1/J == 1 and ( the standard cutoff function in K(l+u)p. Letting k -+ 00, standard calculations give
VO<s
(4.4)
jU(S)dXS j u(t)dx+ :pjjIDuIP-1dxdr. Kp
K(1+")p
BK2p
We estimate the last tenn by using (4.2) and let s '\. 0 along r' while t> 0 remains fixed. Then we let t '\. 0 along the net r" to get
Since uE (0;1) is arbitrary, interchanging the role of J.I. and jJ. proves the theorem.
5. The uniqueness theorem Let S· denote the subclass of S of those non-negative local weak solutions of (1.1) in ET, satisfying
(5.1)
for some "'( = ",((N,p, t),
(5.2)
lim frJf
k-oo
Vk E R+,
IDuI P.!. dxdr = 0, u
K:n[k
for every compact subset K:. C ET and for all C> 1. In section §§8-12 we will construct solutions of the Cauchy problem (1.1)-(1.2) that satisfy both (5.1) and (5.2); therefore S· is not empty. Corollary 3.2 suggests that (5.2) is almost satisfied
s. The uniqueness theorem
347
by all solutions in S. It would be of interest to know whether the inclusion S* c S is strict. . THEOREM
5.1. Let Ul. U2 E S* satisfy
5-(i). Preliminaries LEMMA
5.1. LetuEs*. Then/oraIlO<sO, VC> 1, t
lim ff1ut/x[k
= O.
BKp
Consider (2.8) written for Uk replaced by UCk. against testing functions
PROOF:
7J = ( In (k/2w k,C) where x-(x) is the standard cutoff function in K2p. and
Wk,C
!k, { u, Ck,
==
o '5, u '5, !k !k < u < Ck u~Ck.
It follows from these definitions that 7J '5, 0 a.e. in ET and 7J = 0 a.e. on the set [O
(5.3)
t
I I ! uC1c7Jdxdr '5, IIIDU1P;X(k/2
SK2p
t
+In2C IIIDUCkIJl-IX(U
> k/211D(ldxdr.
BK2p
The fll'St integral on the right hand side of (5.3) tends to zero as k of (5.2). We estimate the second integral. formally. by
00
by virtue
348 XII. Non-negative solutions in ~T. The case I
ljlD t
In-2C = p
UCk
,
IP-l U-
(<>+I)(p-l)
[ P U(<>+I)(p-l) P XU>
k/2]dxdr
SK2p
~ In;c (p _~ _0) p-l (if,Ducr 1PdXdr)
cl P
SK2p
1
X
(if u(O+!)(P-l)X[U > k/2]dxdr)
P
SK2p
+ l)(p - 1) :5 1, the estimate is rigorous and the last tenn in the right hand side of (5.3) tends to zero as k --+ 00, since UE LJoc(ET)' These remarks in (5.3) give
If we choose 0 E (O,p - 1) so small that (0
t
ffUt In
(2W:,c) 'X[Ut < 0] X [(k/2)
SK2p
t
:5 ff Ut lin
2W:,c I'X[Ut ~ O]X[u > k/2] dxdr + 0 (~) .
SK2p
In view of the definition of Wk,C this gives in turn t
ff1utlX[k
t
~ 'YffUtX[Ut > O]X[u > k/2]dxdr+O (~). SK2p
1be last integral is estimated by means of (5.1) and the lemma follows.
Remark 5.1. The assertion of the lemma is trivial if Ut E LJoc{ET)' We give next a weak fonnulation for the difference of two solutions U1, U2. First we recall that, by Lemma 2.3, the truncated function ifO
u2~k
is a distributional supersolution of (1.1), Vk > O. We write (2.8) for U2,k against the testing functions
S. The uniqueness rheorem 349
where ( is a non-negative piecewise smooth cutoff function in K(1+a)p, such that
( == 1 on
(5.4)
Kp
and
0' E (0,
1),
ID(I:5 1/O'p.
In view of the definition of X 10c (ET) and the regularity properties (2.1) of Ui, i
=
1,2, such a choice of testing function is admissible, modulo a density argument. On the other hand the weak formulation (2.7) of Ul holds against the same testing functions. Therefore setting kER+, we obtain by difference the weak formulation t
(5.5)
{!W(k)(1/J -
/ /
ulh(P + J kD(1/J - Uil+(p} dxdr
SK(l+,,)p
t
:5 -p / /Jk(1/J - ud+(p-l D(dxdr \:/1/J E Xloc(E T ), SK(l+O')P
where
Jk == IDuIlp-2Dul -IDu2,klp-2Du2,k 1
.
= / ~ {ID (~Ul + (1 - ~)U2,kW-2 D (~Ul + (1 - ~)U2,k) } d{ o
~ (iID({Ul + (1 - O....)IP-'d{)
Ow(')
1
+ (p -
2) (
/ID(~Ul + (1 - ~)U2,k)IP-4 o
XD(~Ul + (1 - ~)U2,k)(~Ul + (1- ~)U2,k)zjd{ )W(k),Zj' Set also 1
Ao ==
/ID(~Ul + (1 - ~)U2,k)IP-2d{. o
LEMMA PROOF:
5.2. Ao:5P~1IDw(k)IP-2. If IDu2,kl ~ IDw(k)l, we have
350
xu. Non-negative solutions in ET. The case I
= IDu2,1c + eDW(1c) I ~ IIDu2,1cI- eI DW(1c)11 ~ (1 - e)IDw(1c) I·
1berefore
A. $
(/<1 -(~'d{) IDw(.r'
= ~IIDW(1c)IP-2. p-
where eoE(O, 1) is defmed by
IDu2,1c I ( ) eo =_ ID leo, 1 . W(1c) From the definitions set forth and Lemma 5.2 we have
(5.6)
{
JlcDw(lc) ~
IJlcl S
(p - I)AoIDw(Ic)12,
AoIDw(1c)1
s p~IIDw(1c)IP-I.
In what follows we will use these inequalities without specific mention.
6. An auxiliary proposition PROPOSITION
6.1. Let Ui E S· ,i = 1, 2, satisfy
wet) == (UI - U2)(t) - 0
in Lloc(RN) as t - O.
6. An auxiliary proposition 3S I
Then W E Loo (0, Tj Lfoc(RN ») , Vq E [1,00). Moreover Vq ~ 1 there exists a constant ",(=",(N,p, q), such that (6.1)
jlw(tWdX
~
t
(0';)" j jlwl9+(,,-2)dxdT,
Kp
OK(1+O')p
for all p>Oandforall O'E (0,1).
The proof is based on an iteration procedure and uses recursive inequalities obtained from (5.5) with suitable choices of testing functions 1/1.
6-(;). Testingfunctions in (5.5) For h>O, set
W/i).h" (UI -
(6.2)
", ••
)t -
{:/i)
ifW(k)
~O
ifw(k) < h ifw(k)
~
h
and in (5.5) consider the testing function
1/1 == Ul,l/r: + ~ (w(t),n + a(w(t),m +
e)
(6.3)
e)
b
E X'oc(E T ) ,
where
eE(O,I),
a,b>O,
n,mENj
n>m+1.
We obtain (6.4)
j W(k) (1/1 - ul)+("dx - j W(k)(1/1 - ud+("dx RNX{B}
RNX{t}
t
t
- jjW(k)!(1/1-ud+("dxdT+ jjJkD(1/1- u d+("dXdT BRN
BRN t
::; -p j jJ k (1/1 - Ud+(,,-l D(dxdT. BRN
In using 1/1 as a testing function in (6.4) we keep in mind that the truncated functions Ui,h, i = 1,2, Vh > 0, are regular in the sense of (2.1). In particular the first two integrals on the left hand side of (6:4) are well defined V0 < 8 < t ~ T. We will
eliminate the parameters e,k,8,n,m by letting e-O, k-oo, 8-0, n,m-oo in the indicated order.
352 XII. Non-negative solutions in I:r. The case I
6-(U). The limit as E-+O We multiply both sides of (6.4) by E and let E -+ 0, while k, 8, n, m remain fixed. From the definition (6.3) of t/J it follows that \;IrE (0, T] the net [W(k)(Et/JEud+](" r), is equiboundt:d in Lloc(RN). Moreover it converges to
[W(k)
(w~).nf (w~).m) b] (., r)
a.e. K 2p ,
and it is majorised a.e. in RN by
W(k)
(w~).n + If (w~).m + I)b (·,r) ELtoc(RN).
1berefore for all 0 < r ~ T, as E -+ 0 (6.5)
f
W(k)(t/J - ud+(?dx
RNx{r}
-+
f
(w~).nf (w~).m)b (Pdx.
W(k)
RNx{r}
This determines the limit for the first two terms on the left hand side of (6.4). To examine the remaining terms we let 'iii, i = 1, 2, be arbitrarily selected but fixed representatives out of the equivalence classes Ui, define iii, iii(k) accordingly, and let
Next t
ff = ff w~).n (w~).n +Er- 1(W~).m w~).n(PX(gE)dxdr ff w~).m(w~).n +Er(W~).m +E)b-l ! w~).m(PX(gE)dxdr -ff
LE == -E
W(k) :r (t/J - ud+(Pdxdr
sRN
t
-a
+E)b!
SRN
t
-b
sRN
t
w(k)(1 - EUlhx(FE)(Pdxdr
BRN
==
L~l)
+ L~2) + L~3) .
We claim that L~3) - 0 as
E -+ O.
Indeed
6. An auxiliary proposition 353 t
IL~3)1::; JJeIW(k)II!UIIX(Fe)dxdr. aK2p
On the set
Fe we have 1
-
e
1
::; Ul ::; -
e
1 'Y + -(n + 1)a+b == -, e e
eIW(k)/ ::; 'Y
a.e.
Fe·
Therefore (6.6) and the assertion follows from Lemma 5.1. Since k, n, m are fixed, the integrands in L~i) •i = 1, 2, are in Lloc(ET) unifonnly in e. Moreover they have a.e. limits that are in Lloc(ET) and their absolute value is majorised almost everywhere in ET, unifonnly in e, by functions in Ltoc(ET). Therefore as e-+O (6.7)
:: -a: -b!
1
jJ! (W~),n)a+1 (W~),m)b (,Pdxdr aRN
t
1
!
JJ (w~),n) a (wtk),m) b+l (?dxdr. aRN
Since n>m + 1, .
+ )a Or{) (+ {) (wtk),m )b+l (w(k),n w(k),m )b+l = (wtk),m )a Or {) ( -- a +b+b +1 1 Or w+(k),m )a+b+l '
We obtain from (6.7) (6.7')
c:: -
a:
1
J (wtk),n) a+l (wtk),m) b(Pdx RNX{t}
b - (a + l)(a + b + 1)
+
a:
J( w~),m )a+b+ 1 (Pdx RNX{t}
1
.
J (wtk),n) a+1 (w~),m) b(Pdx RNx{a}
+ (a + 1)(: + b + 1)
J (wtk),mf+b+ 1 (Pdx. RNx{a}
a.e.ET.
354 XII. Non-negative solutions in ET. The case 1
We combine this with (6.5) and conclude that the sum of the first three tenns on the left hand side of (6.4) has a limit. as e - 0, that is minorised by (6.8)
1
a + b+ 1
I(w~).m
)4+/1+1
aNx{t}
1
- a + b+ 1
("dx
I (
w~) w~).n
)4+/1
("dx.
aNx{s}
We tum to estimate below the lim-inf as e - 0 of the last integral on the left hand side of (6.4): t
e IIJIcD(1/J - ud+("dxdr saN t
= a I I JIcDw~).n ( w~).n + e) 4-1 saN
x(w~).m + e) t
+b IIJkDW~).m
/I ("x(g,Jdxdr
(w~).n +e)4
saN X ( w~).m
(6.9)
+ e)
/1-1
("X(g,;}dxdr
t
+ IIJ kD(1- eud("X(F,;}dxdr saN
~ a(p -
t
1) I I
AoIDw~).nI2 (w~).n + e) 4-1
saN
t
- p
~ 1 I I AoIDw(k)IIDull("x(F~)dxdr saN
== H~I) + H~2) . By weak lower semicontinuity (6.10)
lim in! H~I) ~-o
t
~ a(p-1) I IAoIDw~).nI2 (w~).nr-l (w~).mt ("dxdr. saN
6. An auxiliary proposition 355
We claim that H~2) - 0 as
€-
O. Using Lemma 5.2 we have
t
IH~2)1 ::; €C(N,p) jfiDUl -
DU2,IcIP-1IDullx(FE)dxdr
aK2p
t
t
::; C€ j jIDU1IPx(FE)dXdr + C€ j jIDu2, Ic IPx(FE) dxdr aK2p
SK2p
= H(2) +H(2) E,l E,2· Since IDu2,1c1 E LfoAET) the second tenD tends to zero as write
€ -
o. As for H!~l
where "1= 1 + (n + l)G(m + l)b. This implies. since Ul ES· t
H!~l::;c(P,n,m)jjIDu~IPx(~ ::;Ul::; ~)dxdr-o as €-o. aK2p
.
We fmally estimate above the lim-sup as € - 0 of the integral on the right-hand side of (6.4). Using the definition (6.3) of 1/J and (5.6) t
(6.11)
Ip jjJIc(1/J-Ut}+(P- 1D( dxdr l saN
t
j AoIDw(lc) I (w~),n + aaN
::; "1 j
€) a
t
+ 'Y j j AoIDw(lc) IWIX(FE)(P-l ID(ldxdr. saN
The last integral tends to zero as € - O. Indeed it can be majorised by
356 XII. Non-negative solutions in Er. The case 1
(6.12)
'Y !fiDW(k)IP-IX(:Fe)dxdr SK2p
t
:$ 'Y !!IDUIIP-1x[Ul
~ :ldxdr
t
+ 'Y ! !I DU2,kI P- 1x[Ul ~
:ldxdr,
sK2p
for a constant 'Y = 'Y(P, n, m, a, b). The second integral on the right hand side of (6.12) tends to zero as e ..... O. since Ul E Lloc(ET). As for the frrst integral. let oE (O,p - 1) be so small that (0 + 1)(p - 1) < 1. Then t
'Y //IDUI!P-IX[UI > :ldxdr SK2p t {{
=
_ (a+1)(e-l)
'Y JJ IDuI!p-l ul
(a+l)(p-1)
u1
P
P
X[UI > : ldxdr
sK2p
t _
{{
p-1-a
=7 JJ IDu l
P
1
(a+1)(p-1)
IP- U1
P
X[UI > : Jdxdr
sK2p
<~IIDu P-~-Q II P- 1
-
p,K2px(s,t)
1
~
(rJJ(u(O:+l)(P-l)X[Ul > 1
)P
1 1dxdr •
SK2p ---+
0
as e ..... O.
We examine the lim-sup as e ..... 0 of the first integral on the right-hand side of (6.11). The numbers k E R + , n E N being fixed. if e is small enough. we have the inclusion
We write
6. An auxiliary proposition 357 t
ffAoIDW(k)1
(w~),n
+ef (w~),m +e)b (P-1ID(IX(Qe)dxdT
BRN
t
=ffAoIDW~),nl (w~),n
+ef (w~),m +e)b (P-1ID(ldxdT
aRN
(6.13)
t
+f f
AoIDw~)I(n + e)a(m + e)b(P-IID(lx[w~) > nlx(Qe)dxdT
BRN
t
+f f
AOIDw~)lea+b(p-IID(IX(Qe)dxdT = K~l) + K~2) + K~3).
BRN
As for K~l) the integrand tends to AoIDw~),nl(w~),n)a( w~),m)b(P-IID(1
a.e. K2p x (s, t),
in a decreasing way. Therefore t
K~l) -+ f f AoIDw~),nl(w~),nt(w~),m)b(P-lID(ldxdT. BRN
The last integral tends to zero as e -+ O. Indeed
The operation Dw~) coincides with the weak derivative of w~) only on those sets Ai where w~) is bounded by a positive constant i, i.e.,
Dw~)x (At) == Dw~),t. Since Dw~) is not well defined a.e. in the whole strip ET we estimate K~2) as follows: t
K~2) -:; 'Y (m ~ l)b ffiDUl - DU2,kIP-lu~X[Ul > n + U2,k]X(Qe) dxdT aK2p t
-:; 'Y (m
~ l)b ffIDUIIP-IU~X[Ul > n]x(Qe) dxdT aK2p
+ 'Y
(m+l) Up
b
t
a Jff J IDU2,kI P- 1 UIX[UI > n + U2,k]X(Qe) dxdT. aK2p
358 XII. Non-negative solutions in Er. The case I
If 0 E (O,p - 1). write t
JJIDUIIP-IU~X[UI > njx(ge)dxdr aK2p
t
=JJIDUIIP-IU~ (Q+l~p-l) u~Q+1)('P-ll+"P X[UI > njx(ge)dxdr aK2p
c.!
~ 'Y (jjIDu;- :-° 1, /hd) •rjj .\0+1 BK2p
')
~BK2P
!
)(,-1 )+«,
Y
xl'l > n]dzdT
')
Choose 0 and a> 0 so small that (0 + l)(P - 1) + ap ~ 1.
(6.14)
Then u~Q+l)(p-l)+ap E Ltoc(ET) and \fEE (0,1) t
JJIDUIIP-IU~X[UI > njx(ge)dxdr ~ 0
(;).
BK2p
Analogously t
JfiDU2'kIP-IU~X[Ul > n + U2,k]X(ge)dxdr aK2p
x
(jj,:'u!,o+I)(,-I)xl'l
c.!
BK2p
~ (jfiDu~IPdxdr) ~ 'Y
BK2p
I·
> n + ....]dzdT
)
6. An auxiliary proposition 359
We conclude that
provided 0: and a> 0 are chosen so that (6.14) holds. Combining these estimates and limiting processes as parts of (6.4) we obtain
a+ b1 + 1
J(w(k),m +
)a+b+l
(P dx
RNx{t}
t
+ a(p - 1) JJ AoIDw~),~J (w~),n) a-I (w~),m)" (Pdxdr BRN
(6.16)
$
a+ ~ + 1 Jwt (w~),nr+" (Pdx RNx{a} t
+ ~JJAoIDW~),nl
(w~),nr (w~),m)" (P-1dxdr
BRN
+-y(m+ 1)"0 (~).
6-(iii). The limits as k--+oo and 8--+0 If n E N and k > 0 are fixed, we let
iii{k),n and Dw~),n be arbitrarily selected
but fixed representatives out of the equivalence classes w~),n and Dw~),n and introduce the sets
where C is the constant appearing in the last integral on the right hand side of (6.16). This integral is estimated as follows:
360 XII. Non-negative solutions in ~T. The case I
~ I IAoIDw~),nl (w~),nr (w~),m)b (P-
1dxdr
sRN t
$
~/IAOIDw~),nl (w~),nr (w~),m)b (P-l x(tddxdr sRN t
+ (p ~~)crp I fiDW~),nIP-l
(w~),n) (w~),m) b(p-1 X(t2 )dxdr 4
SRN
$
a(p-l) 2
It/
AoIDw~),nl
2(w~),n )4-1 (w~),m )b (Pdxdr
sRN
+
4PC P aP-1(p _ l)p(crp)p
1/( t
w+ (k),n
)P-l+4 (w+
(k),m
)b dxdr
•
SRN
We carry this estimate in (6.16), move the integral involving IDw~),nI2 on the left hand side and discard the resulting non-negative tenn to obtain
(6.17)
-y(p) + (crp)P
It I( w~),n )P-l+4 (+
w(k),m
s
)b dxdr
K(l+")p
+ -y(m + l)bO (~). We let now k - 00 while B > 0, n, mEN remain fixed. Since w~) - w+ in a decreasing way we may pass to the limit under the integrals in (6.17) and obtain the same integral inequality written for w+. In particular the first integral on the right hand side takes the fonn (6.18)
1
a+b+l
jW+(W+)4+b(Pdx. n
RNx{s}
Now letting 8-0, the integral in (6.18) tends to zero since it can be majorised by as B -
0.
6. An auxiliary proposition 361 These limiting processes yield
j(W:at+b+l (Pdx (6.19)
~ 'Y(P)(~;)! +
I)!
RNX{t}
t
j(W:;y-1+ a (W:a)b dxdr
OK(1+")p
+'Y(a+b+ 1)(m+
l)bO(~).
6-(ivJ. Proof of Proposition 6.1 We let n -+ 00 in (6.19), while mEN remains fixed. The integrand in the last integral tends to (w+)p-1+a(w~)b a.e. in K(1+CT)p x (0, t) in an increasing fashion. Moreover if a is so small that p - 1 + a E (0,1),
(6.20)
it is dominated, uniformly in n, by the function
The limit process gives t
j(W~t+b+1 (Pdx ~ 'Y(P)(~;)! + 1) j J(w+)p-l+a(w~)bdXdT.
(6.21)
RNX{t}
OK(l+ .. )p
This inequality holds true "1m E N, Vb ~ 0, "10- E (0,1), Vp > O. The positive number a is fixed, satisfying the restrictions (6.14) and (6.20). The sequence {w~} increases to w+ a.e. in ET . Therefore as m -+ 00, we may pass to the limit under the integrals in (6.21) for those b ~ 0 for which
(w+)p-1+ a+b If bi
~0
E LtoAET).
is one such b, letting m -+ 00 we find that
which implies that
(w+)p-1+a+bi+l E LloAE T ),
bi +l=bi +2-p>bi .
Let bo ~ 0 be defined by p - 1 + a + bo = 1. Then the previous remarks show that
(w+)p-l+a+b o+i(2- p) == (W+)1+i(2- p) E Ltoc(ET), i Interchanging the role of UI and U2 proves the Proposition.
= 0,1,2, ....
362
xu. Non-negative solutions in ET. The case I
7. Proof of the uniqueness theorem From (6.1) by HOlder's inequality, since pE (1, 2)
(7.1)
Let p > 0 be fixed and for n = 1, 2, ... defme
Pn
= (~ ~ ,=0
2-i) K= K p,
n
P.. ,
q
n -- 2-(n+1) ,
Rewrite (7.1) over Kn and Kn+l to obtain (7.2)
By the interpolation Lemma 4.3 of Chap. I we conclude that for every q E [1,00) there exists a constant -y=-y(N,p, q), independent of p, such that for all tE (0, T) (7.3)
To prove the theorem we choose q so large that N(P - 2) q
+ pq > 0
and then, such a q being fixed. we let p-oo in (7.3).
8. Solving the Cauchy problem We will establish the existence of a unique non-negative solution to the Cauchy is non-negative and merely in L}oc(RN). For n = problem (1.1)-(1.2) where 1, 2, ... consider the sequence of approximating problems
"0
I
Un
(8.1)n
9. Compacbless in the space variables 363 E C (O,Tj
L10c(RN»)nV
/:rUn - div IDun l
Un
( )_ x,O -
p - 2 DUn
(0, Tj W,!;:(RN»)
= 0,
_ {min{uojn}
= '0
U on
in ET for Ixl < n II for x ~n.
The initial data are bounded and compactly supported in RN. Therefore the unique solvability of (8.1)n can be established as indicated in §12 of Chap. VI. Since the initial data {uo,n}f1S\l form an increasing sequence of functions in Lloc
't/p>O.
The solution of (1.1 )-( 1.2) will be constructed as the limit of the sequence {u n }f1S\l in a suitable topology. For this we establish flJ'St some basic compactness of {un}f1S\l. LEMMA 8.1. There exists a constant "'( = "'( N, p) independent of n such that for all t,p>O
MoreoverforallaE(O,p-l),
PROOF: The Lloc-estimate follows from (4.1) with s = 0, and the gradient estimate (8.4) is a consequence of (4.2) with s = O. Finally (8.5) is the content of Lemma 3.2.
9. Compactness in the space variables 9.1. Let a E (0, p - 1) be so small that (a + 1)(P - 1) < 1. There exists a constant ",(='Y(N,p, a) such that LEMMA
364 XII. Non-negative solutions in I:T. The case 1
'VO < t
~
T,
'Vk,
'VC
> 1,
'Vn
= 1,2, ... ,
t
/ /IDunIPu;;tx.lk
(9.1)
OKp
< 'Vk _
_(l-
I
No2=.!
P
p
(t )*{/ -
pP
u 0 dx +
(- )~}P_Q~P_l) t
p>'
K3p
+ InC
/ uoX[Uo > k] dx. K2P
The constant 'Y( 0) /00 as either 0'" 0 or 0 / p - l. PROOF: We drop the subscript
n for simplicity of notation. If C > 1 is fixed, let
u~2 == {:
Ck
ifO
< u < Ck u? Ck
if k if
and in the weak formulation of (8.1)n, take the testing function
(k») ( In U~k ((x), where x -+ (( x) is the standard cutoff function in K 2p that equals one on K p. We obtain t
/
t
f!DUIP~X[k
OK p
-j/ (i :T
OKb
2; /
!IDuIP-1X[U
OK3p
In
min{~jCk} de)
((x)dxdT
==
+
k
Let 0 be any positive number satisfying
·OE(O,p-l) Then by virtue of (8.5)
> kJdxdT
and
(0+ l)(P-l)<1.
G~l) + G~2).
9. Compactness in the space variables 36S
jjlD IP-l t
G{l) k
2')'
~-
U
p
U
(o+l)(p-l)
(o+l)(p-l)
pup
j xu>kdxdT [
OK2p
')'(a,p) jilDu =-t
I
.-1-0 P- 1 (0+1)(,-1) PUP
p
kjdxd
[ XU>
T
OK2p
(0+1)(,-1)
X{
jU(X,
sup O
T)dx}
P
K2p 1-(0+1)(,-1)
X {
sup
O
J
X[U > kj
dx}
P
K2p
~')'(a,p)(~);{Juodx+(:~)~}
~
P
K4p 1-(o+l)(p-l)
SUP { O
jX[U> kj
K2p
dx}
P
366 XII. Non-negative solutions in E1. The case 1
The last step follows by use of (8.3). Using it again we obtain
1berefore
As for G~2) it is estimated above by
10. Compactness in the t variable LEMMA
10.1. Let a
E
(O,p - 1). There exists a constant "{ = ,,{(N,p, a) such
that VO<s
VB
~
a
+ 1,
,,{p-aN
+ -s -
Vn
= 1,2, ... ,
{f ( )~}l-a uodx+
t
~
P
K2p
The constant ,,{(N,p,a) /00 as either a'\.O or a/(p - 1). PROOF: Let 0 < s < t ~ T and p> 0 be fixed. Consider the cylinders
QoEKpx(s,t), and let (x,
Ql EBtpX(i,t),
r) -(x, r) be a non-negative pieceWise smooth cutoff function in Ql which equals one on Qo and such that ID(I ~ 2/ p and (t ~ 2/ s. At first we will proceed formally. The calculations below will be made rigorous later. In the weak formulation of (8.1 )n, take the testing function
10. Compactness in the t variable 367 !.&n,t (un
+ 1)-8(2,
and integrate by parts over Ql. Dropping the subscript n, we obtain
II{u+
1)-8u~(2dxdr = - IIIDuI P- 2DUD{Ut (U+ 1)-8(2)dxdT
Ql
Ql
=
-t
I I !IDuIP(u + 1)-8(2dxdr Ql
f
+ (J IIDuIP(u + 1)-8-1Ut(2dxdT Ql
- 2 IIIDu IP- 2Duut{u + 1)-8(D( dxdr Ql
~ ~ (p -
1) IIIDuIP{u + 1)-8-1Ut(2dxdr Ql
+ ~ IIIDuIP(U + 1)-8((Tdxdr Ql
+ ~ IIIDuIP-1(u + 1)-t ({u + 1)-8u~(2) I dxdr Ql =
n(l)
+ nP) + n(3).
In estimating 'R,( 1) we use the regularising effect of Proposition 6.1 of Chap. VI,
1
u
Ut < - - - . - 2-p t
(10.2)
Then,
By Young's inequality 'R,(3)
~ ~ II{u + 1)-8u~(2dxdT + ; Qo
I I IDuI 2(p-l){U + 1)-8dxdr. Qo
Since 1 < p < 2, this last integral is majorised by
Combining these estimates we find that
368 XU. Non-negative solutions in I:T. The case 1
By (8.5). if aE (O,p - 1). this is estimated by
~ ffiDuIPu-(O+1)(u + 1)-[6-(o+1)ldxdr Ql
~ "Y(a,~)pON JJIDUp-~-a IPdxdr Ql
and the lemma follows by fonnal calculations. The calculations are fonnal since Un,t (un + 1) -6 (2. need not be an admissible testing functions in (8.1)n. The arguments would be rigorous if
(10.3)
Indeed. if so. we may take in the weak fonnulation (8.1)n the testing function
Un(t + h) - un(t) ( h un The limit as h - t
+ 1)-6/"2 .. ,
h (0 T _ E,
) Is,
Is~t
°
is justified and we may proceed as before.
lO-(i). Approximating estimates Therefore to prove the lemma it suffices to establish (10.1) for a sequence of approximating solutions satisfying (10.3). The unique solution of (8.l)n. can be approximated by the solutions of
Vn,j E C (0, T; L2 (Bj)}nLP (0, T; W:,p (Bj ») (10.4)
{ j = n + 1, n + 2, ... , /rvn,j - div (lDv n,jlp- 2 Dvn,;)
=
,
°
in B; x (O,T),
vn,;(·,t) 11%1=;= 0, vn,;("O) = uo,n,;, where B j is the ball of radius j about the origin and {uo,n,; };:n+1' is a sequence offunctions in C':' (B n +1). such that .
Uo,n,; and
--+
uo,n in L~oc (Bn+t> ,
10. Compactness in the t ¥ariable 369
/ uo,n,;dx ::; 2/uodx Vp > O. Kp
Kp
As indicated in §12 of Chap. VI. {)
{)
Vn,;, -{) Vn,; -- Un, -{) Un, in C1!c (ET) Xl
Xl
Vl= 1,2, ... ,N, for some Q E (0,1). The unique solvability of (10.4) can be established by a Galerlcin procedure. Such a method also yields
To establish (10.1) for Vn,j is suffices to show that
ID!
(10.6)
vn,;1
E L?oc (B;).
In the remarks below we drop the subscript n, j and write v (10.4) for the time levels t+h and t and set
= vn ,;. We write
w = vet + h~ - vet) , hE (0, T - !s), !s::; t < T - h. By difference
Wt-h-ldivJh=O
(10.7)
where
Jh
inBjx(O,T-h),
=IDv(t + hW- Dv(t + h) -IDv(t)IP- Dv(t). 2
2
(t- V+ which van-
In the weak formulation of (10.7) take the testing function W ishes on Ixl =j and for t::; ~. This gives
T-h (10.8)
/ /(t -
T-h
i) + Ao,;IDwI dxdr ::; 'Y / 2
! Bj where
t+h
/1 f :r
vex, r)drr dxdr,
t
0 Bj
1
Ao,; = /ID(svn,;(t + h) + (1 - s)vn,;(t))IP- 2 ds. o If /C is a compact subset of B; x (s, T). we have A o,; ~ 'YIIDvn ,; lI~i It follows from (10.8) that
.
370 XU. Non-negative solutions in 1::1. The case I
The last integral is finite by virtue of (10.5) and the lemma follows.
11. More on the time-compactness We record a simple consequence of Lemma 10.1. If x - ((x) is the usual cutoff function in K 2p that equals one on Kp. we find from the weak fonnulation (8.i)n.
VO<s
t
t
ffiUt)-(dxdT - ffiUt)+(dXdT aK2p aK2p
=-
ffUt(dxdT aK2p t
= f f1Du1JI-2 DuD( dxdT.
aK2p 1berefore
The fust integral on the right hand side is estimated by (10.2). i.e .•
t-s{f $ 'Yuodx + s-
(t),!p} p>'
•
K4p Estimating the second integral by Proposition 8.1 gives LEMMA
(l1.1)
11.1. There exists a constant 'Y ='Y( N, p), such that
VO<s
if
Vn
= 1,2, ... ,
t-s{f !{~un.tldXdT $ 'Y -sK4puodx +
(t)¢P} p>'
•
12. The limiting process 371
12. The limiting process By construction Un /' U a.e. in ET and by (8.3)
for all O
By Lemma 8.1 the sequence {'Un
J>
}
is equibounded in
V (0, Ti W1,P(Kp ») , 'rip> 0, provided
0
E (O,P -1).
Since the whole sequence {un },1E:N -+u in Lloc(RN). p-l-..
'Un
J>
p-l-..
-+
U
weakly in LP(O, Ti W1,P(K p
J>
», 'rip> 0.
This implies that the sequences Un,A:
are equibounded in £P
Un,k
(12.3)
-+
= Un 1\ k = min{un,k}
(0, Ti W1,P(Kp )) , 'rip> 0. and
U1\ k weakly in LP (0, T; W1,P(Kp ») , 'rip> 0, 'rIk > O.
LEMMA 12.1. DUn,A: -+ DUA: strongly in Lfoc(ET). PROOF: In the weak fonnulation of (8.I)n. take the testing function
to obtain (12.4) jjIDUn,A:IPl()dxdT
ET
= jjIDUnIP-2DUn.DUkl()dxdr ET
+ jjlDunlP-2 D(u + v)(UA: -
Un,k)Dl()dxdr
ET
+j j
Un,t (Uk - Un,k)tpdxdr == 10 + It + 12.
ET We ftrst estimate the integrals Ii, i = 1, 2. Let 0 E (O,p - 1) be so small that (0 + l)(p - 1) ~ 1. Then by Lemma 8.1
372 XU. Non-negative solutions in !:T. The case I
1_
~ riD II I < fJ
Un 11'-1 Un-(01+1)7 Un(a+1)7(Uk - Un,k )cp .l-dT «=
ET I'
~ IIDu:-~-"IIP-1
1u~+1)(p-1)(Uk
(I ET
p,supp{
"~(Q,P' u.,'I') --+
0
(flu
1
- un'k)Pcp"dXdT) ; .1
+ l)(u, - Un"l"'I"'dzdT) ,
as n-oo.
We estimate 1121 by making use of the regularising inequality (10.2). 1121
~ ;~~ Ifun (Un,k -
Uk) cpdxdT
ET --+
2-p
0 as n--oo. p
We return to (12.4) and estimate
10
= IIIDun,kIP-1IDUkl cp dxdT ET
~
p;
1 IllDun,klPCPdxdT
+ ~ IIIDUkIPcpdxdT.
ET
ET
Combining these calculations in (12.4) gives
IllDun,klPcpdxdT ET
~ IllDUklPcpdXdT + o(~) . ET
From this, by lower semicontinuity
IflDUklPcpdxdT ET
~ l~~~ IIIDUn,kIPcpdxdT ET
~ IIIDukIPcpdxdT. ET This proves the lemma. Next, by Lemma 10.1 the sequence
12. The limiting process 373
2-.} nEN { ata (Un + 1)2" is equibounded in L~oAET) for all (J~ 0: + 1 and for all o:E (O,p - 1). Therefore
a (Un + 1)2" 2-' at
-+
a (u + 1)"'2-' at
weakly in L
2{ s, t; L 2(Kp) ) ,
for all 0 < s < t :5 T and all p> O. This implies that
{:t Uk,n}
E
L~ocET
unifonnly in n and
(12.5) o
Choose 1/J EXloc (ET) and in (S.I)n consider the testing function r.p= (1/J - u)+. Fix O<s
(1/J - u)+
= (1/J -
U" k)+
o
EXloc (ET),
so that r.p is an admissible testing function. It gives t
(12.6)
!!{
!Un(1/J - u)+ + IDunIP-2Dun ·D(1/J - U)+} dxdr = O.
SRN
Since Un :5 u, Vn EN, we have
Therefore in view of (12.5) t
n~!! :r un (1/J SRN
t
u)+dxdr = ! j
t
== ! !ut (1/J - u)+dxdr. SRN
Analogously,
374 XII. Non-negative solutions in l::r. The case 1
= ID(Un "
k)IP-2 D(un "k)D('I/1 - U" k)+ = (IDun,kIP- 2DUn,k -IDuklp-2 DUk) .D('I/1 - Uk)
+ IDuIP- 2 Du·D('I/1 - u)+. By a calculation similar to that in Lemma 5.2 and leading to (5.6) we have
Therefore taking into account Lemma 12.1 and letting n-+oo in (12.6) gives t
j j {Ut('I/1 - u)+
+ IDulp-2 Du·D ('1/1 - u)+} dxdr = 0,
·a
N
o
for all '1/1 EXloc (ET)' It remains to prove that u takes the initial datum U o in the sense of Lloc(RN) and that ueS*.
12-(i). Continuity in Lloc(RN) at t=O Fix p > 0 and let uo,e be a net of functions satisfying
{ uo,e == 0, uo,e --+ u o,
for Ixl > 4p in L1 (K2p )'
Such a family can be constructed by first defining a function that coincides with U o in K 3p and zero otherwise and then by mollifying the function so obtained. Let also Ue be the unique solution of (1.1) with initial datum uo,e' We take the difference of (8.1)n and the equation satisfied by Ue. In the p.d.e. so obtained take the testing function tp
= [(un -
u e)+ + 6]",(
where (1,6 e (0,1) and x-((x) is the usual cutoff function in K 2p that equals one on Kp. We perfonn an integration by parts and let 6-+0, 8-0, (1-0. to obtain
j(Un(t) - ue(t»+dx
~
Kp
j(Uo,n - uo,e)+dx K2p
+
2;
t
j j (IDunIP-1
+ IDueIP-1) dxdr.
OK2p
We use (8.4), interchange the role of Un and Ue and, for t This gives
> 0 fixed, let n -+ 00.
12. The limiting process 375
jIU(t} - UE(t}ldx
~ jluo -
Kp
uo,EI dx
K2p
From this
jlu(t} - uoldx ~ Kp
2
j luo - uo,Eldx + jluE(t} - uo,Eldx + O(t;) .
K2p
Kp
Letting t '\. 0 lim-suPt'\,o jlu(t} - uo}ldx
~2j
Kp
luo - uo,Eldx, 'VeE(O, I}.
K2p
12-(ii). uES· By (10.2). 'Vn E N and for all k > 0
o(U I\k } < -1- -Un. n - 2-p t
-
at
As n-+oo
(12.7)
(u 1\ k}t
~
1
u
-- 2-pt
a.e. in ET.
The limit is flISt taken in 1)'(0, T) and then (12.7) holds almost everywhere in ET in view of (12.5). Next from Lemma 9.1 it follows that 'VC> 1 t
jjlDunlP ~ X[k < unlx[u < CkldxdT = O(~). sKp
Here we have used the fact that Un / u implies [un < Ckl ~ Iu < Ckl. Letting n -+ 00 for k > 0 and C> 1 fixed yields by lower semicontinuity t
jjIDuIP~X[k
We conclude by remarking that the requirement u E S· is necessary and sufficient for uniqueness. Indeed. if solutions in S are unique. they can be constructed starting from their traces on t = T E (0, T) to yield u E S·. Vice versa solutions in S· are unique.
376 XII. Non-negative solutions in Er. The case I
13. Bounded solutions. A counterexample Let r 2: 1 satisfy Ar ::= N(p - 2) + rp > O. If U o E L'oc(RN ), then by energy estimates, the sequence of approximating solutions of (8.1)n satisfies
{un} E L'oA~T)
uniformly in n.
Therefore by Theorem 5.1 of Chap. V, {un} E L~(ET) uniformly in n. It follows from the regularity results of Chaps. IV and IX that
{un}, {un.x;} E C;:'c(~T)' j=l, 2, ... ,N, uniformly in n, for some a E (0, 1) depending only upon N and p. This gives a regular solutions to the Cauchy problem (1.1)-(1.2). A similar analysis can be carried if the initial datum is a measure JI. and Al > 0, i.e.,
2N p> N+l'
(13.1)
We show next that if(l3.1) is violated, then initial data in Ltoc(RN) might produce unbounded solutions.
13-(i). A counterexample Let a E (0, 1) be a positive constant and let Be denote the ball of radius a in RN centered at the origin. Consider the functions
z=
(13.2)
where {j, h
(a 2-lxI2)2
+
and
Ixl N lin Ix121f3
v = (1 - ht)+ z,
> 1 are to be chosen. One verifies that
Consider also the Cauchy problem {
(13.3)
Ut -
div IDulp-2 Du = 0, in E1 ::=RN x (0,1),
u(·,O)=z.
The p.d.e. is meant in the sense of (2.1 )-(2.2) and the initial datum is taken in the sense of Ltoc(RN). LEMMA
(j, h
13.1. Assume that N(p - 2)
+ p = O. The constants a
E (0,1) and
> 1 can be determined a priori so that v is a non-negative, weak subsolution
0/(13.3) in E 1 • PROOF:
By calculation on the set 0 < Ixl < a,
Dz =
z
-lxl 2 Fx,
.
13. Bounded solutions. A counterexample 377 where
2/3 41X12} F = { N + In Ixl2 + a2 _ Ix l2 . We choose a =e -k and k> 1 so large that F> O. Compute .
_ IDzlP 2Dz
zp-IFp-1 x Ixl p . zp-l FP-l zp-2 FP-l div(lDzlp-2 Dz) = -(p - 1) Ixl p Dz . x + P Ixl1>+1 Dlxl· X
=-
Zp-l FP-l
-N
Ixl p
zp-l FP-2
-(P-I)
Ixl p
DF·x.
Using the fonnulae
Dz . x
= -zF,
DF . x =
-4/3 +
In21xl 2
Dlxl· x 81xl 2 (a 2
= Ixl +
-lxI 2)
(a 2
81xl 4 -
Ix12)2'
we obtain
We calculate the expression in braces on the right-hand side using the definition of F and the fact that N(p - 2) + p=O, to obtain
Consider the sets
Cil) == {~e-2k
~ Ixl 2< e- 2k } ,
ci2) == {lxl2 <
One verifies that on Cil) we have
1t > -
8(2 _!!.) _Nf3. k 2k
~e-2k }, k > 1.
378 XII. Non-negative solutions in I:T. The case I
Therefore 'H. ~O on £1 if k is sufficiently large. On£~2) we have F~ (N - f3/k) > 1. Therefore div(IDzIP-2 Dz) > zp-1 'Y(N,p). - Ixl P In Ixl 2 Finally we compute in {O< Ixl
.c(v) == Vt - div(IDvI P- 2Dv) = -hz - (1 - ht)~-l div(IDzIP-2 Dz). On £(1) .c(v) -< 0 and on £(2) k' k
zp-2 'Y(N,P)] .c(v) ~ z [-h - Ixl p In Ixl2 . By calculation on £~2) •
zp-2 'Y(N,p) - Ixl p In Ixl 2
(a 2 _lxI2)2(p-2)
~ 'Y(N,p) lin IxI21~(p-2)+l '
where we have used the fact that Al == N (p - 2) + p = O. We select f3 > 1 so that f3(P- 2) +1 > O. This gives
zp-2 'Y(N,p) • -lxlP Inlxl2 ~'Y (N,p,k). Therefore
.c(w) Cltoosing h ='Y. ( k) proves that (13.4)
.c(V)
~
~
z( -h + 'Y·(k».
on {O
0
To prove that indeed v is a weak subsolution in the whole E .. multiply (13.4) by a non-negative function x - cp(x) E C~ (E1 ). and integrate over the cylindrical domain with annular cross section Q~ =={e< Ixl
II{
vtCP+IDvI P- 2Dv.Dcp} dxdr
1:1
= lim
~\,O
~ ~~
jrJ{{vtcp+IDvlp-2 Dv.Dcp}
dxdr
Q. I
II
IDvl p- 2Dv·
1:1 tpdtrdr
o{lzl=G-~}
I
-
lim
~ .... "'O
J"J{IDvIP-2 Dv . -ixix tpdtrdr, I
O{lzl=~}
14. Bibliographical notes 319 where du denotes the surface measure on {Ixl =e} and on on the right hand side are zero. In particular we have
V( E C~(RN),
Vt/J E x'oc (Ed, (13.5)
{Ixl =e}. The limits
( ~ 0,
jj{vt(t/J - v)++IDvI P- 2Dv·D(t/J - v)+} dxdr
~ O.
1:1
One also verifies by direct calculation that v satisfies (5.1) and (5.2) and therefore is a subsolution of (13.3) in the class S·. Next we return to (13.3). This problem has a unique solution U E S·, by the construction of §§8-12 and the uniqueness theorem 7.1. By the comparison principle U ~ v and therefore U is not bounded. The comparison principle here is applied as follows. By the definition of weak solution the truncated functions Uk == mint Ui k} are, for all k > 0 distributional subsolutions of (13.3). Setting
w == v -
U
W(k) == v - Uk
and
and using (13.5) we find
VO<s
Vt/JeX,oc (Ed,
V(EC~(RN), ( ~ 0
+ [lDvlp-2 Dv -IDuklp-2 DUk]·D «t/J - v)+() }dxdr
~ O.
Observe that w(t)-O as t'\,O in Lloc(RN). Therefore we may proceed as in the proof of the uniqueness theorem and establish an analog of Proposition 6.1, i.e. VO
Vq>l,
Vp>O,
~
11
Vue (0, 1)
t
j (w+(t»)q dx Kp
_'Y_
(up)p
(w+)P-2+ q dxdr.
OK(1+a)p
Proceeding as in the proof of Theorem 7.1 we find w+ = O. Remark 13.1. If N(p-2)+p>0 then v satisfies (13.4) but it is not a subsolution of (13.3) in the whole E 1 • In particular it does not satisfy the requirement (5.2) of the class S·. If N(P-2)+p<0 then v does not satisfy (13.4).
14. Bibliographical notes Equations of the type of (1.1) arise in modelling of non-newtonian fluids (see Kalashnikov [57], Martinson-Paplov [74,75], Antonsev [5] and Joseph-NieldPapanicolau [56]). Questions of solvability, even though in a different context,
380 XU. Non-negative solutions in ET. The case 1
were fmt investigated by Btizis and Friedman [18J. The notion of weak solution introduced in §2 is taken from [42J. B6nilan has infonned us of a more general notion of solution. introduced in [II J. that would include solutions of variable sign. The remainder of the chapter is essentially taken from [42J. It would be of interest to investigate questions of existence/Uniqueness for (1.1) in ET when the initial datum is of variable sign or is a measure. Singular equations are little understood. mostly if p violates (13.1). Preliminary investigations seem to indicate questions of limiting Sobolev exponent (see [19]) and differential geometry.
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