The Schwarz Function and Its Generalization to Higher Dimensions

THE SCHWAf\Z FUNC TION AND ITS GENEf\ALIZATION TO HIGHEf\ DIMENSIONS

A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY & SONS, INC.

m �

University of Arkansas Lecture Notes in the Mathematical Sciences WILLIAM H. SUMMERS, Series Editor

VOLUME

9

THE SCHWARZ FUNCTION AND ITS GENERALIZATION TO HIGHER DIMENSIONS

HAROLD S. SHAPIRO Mathematical Institute Royal Institute of Technology, Stockholm, Sweden

A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY AND SONS, INC. New York

Chichester

Brisbane

Toronto

Singapore

In recognition of the importance of preserving what has been written, it is a policy of John Wiley & Sons, Inc., to have books of enduring value published in the United States printed on acid-free paper, and we exert our best efforts to that end. Copyright © 1992 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. Library

of Congress Cataloging-in-Publication Data:

Shapiro, Harold S. The Schwarz function and its generalization to higher dimensions / by Harold S. Shapiro. p. cm.-( University of Arkansas lecture notes in the mathematical sciences: v. 9) An expanded version of a series of lectures delivered at a conference at the University of Arkansas in April 1988. "A Wiley-Interscience publication." Includes bibliographical references. 1. Schwarz function. 2. Functions of complex variables. 3. Geometry. Analytic-Plane. 4. Numerical analysis. I. Title. II. Series. QA331.7.S53 1992 91-41118 515' .9-dc20

CIP ISBN 0-471-57127-X (cloth: alk. paper) Printed and bound in the United States of America by Braun-Brumfield, Inc.

10 9 8 7 6 5 4 3 2 1

Dedicated to the memory of ALLEN LOWELL SHIELDS

1927-1989

CONTENTS Pneface

ix

Chapter 1 The Schwarz principle of reflection 1.1 Introduction, 1 1.2 The Schwarz function, 2 1.3 Anti-conformal reflection, 4 1.4 Schwarzian reflection, generalized, 5 1.5 The Schwarz potential, 7

1

Chapter 2 The logarithmic potential, balayage, and quadrature domains 2.1 Analytic continuation of the potential, 9 2.2 Balayage, 11 2.3 Quadrature domains, 13

9

Chapter 3 Examples of "quadrature identities" 3.1 About the terminology, 16 3.2 Examples, 17 3.3 Examples, continued, 21 3.4 Examples of a q.d. with a singular boundary point, 22 3.5 An example of ''balayage inwards", 23

16

Chapter 4 Quadrature domains; basic properties, I 4.1 Notations, etc., 24 4.2 Quadrature domains and the Schwarz potential, 25 4.3 Some applications, 29 4.4 Subharmonic quadrature domains, 36

24

Chapter 5 Quadrature domains: basic properties, II 5.1 Regularity of the boundary, 39 5.2 Valency of the Schwarz function, 40 5.3 Variational properties of q.d., 41 5.4 Other varieties of quadrature domains, 44 5.5 Existence of q.d., 50 5.6 Conclusion, 51

39

Chapter 6 Schwarzian reflection, revisited 6.1 Reformulation in terms of harmonic functions, 52 6.2 Study's interpretation of Schwarzian reflection, 55 6.3 Failure of Schwarzian reflection in 1R 3 , 60

52

Chapter 7 Projectors from L 2 (a!1) to Jt2(a!1) 7.1 Introduction, 63 7.2 The Hilbert operator of a plane domain, 63 7.3 Relation to the Neumann-Poincare problem, 66 7.4 Proofs of the preceding theorems, 68 7.5 A property of the Szego projector S, 72

63

Chapter 8 The Friedrichs operator 8.1 Introduction, 74 8.2 The Friedrichs operator, 74 8.3 Weak* limits of sequences in L~(!1), 76 8.4 The Friedrichs operator, geometry, and the Schwarz function, 80 8.5 The Friedrichs operator and quadrature domains, 82

74

Chapter 9 Concluding remarks 9.1 The Schwarz potential in en, 84 9.2 The ellipse, revisited, 85 9.3 Propagation of singularities, an example, 87 9.4 The ellipse, concluded, 89 9.5 The sphere, revisited, 91 9.6 Further horizons, 92 9.7 Conclusion, 95

84

Bibliography

97

Index

105

viii

PREFACE This book is a somewhat expanded version of a series of five lectures delivered at a conference at the University of Arkansas, Fayetteville, in April, 1988. The first three chapters are introductory and should provide ample motivation for the study of our subject, so here I will only discuss some features of the presentation. Although the Schwarz function originates in classical complex analysis and potential theory, I have favored a mode of treatment which, as far as possible, unites the subject with the modem theory of distributions and partial differential equations. Since classical function theorists (comprising most of the audience of the lecture series, and for whom this book is principally intended) do not always feel comfortable with this approach, I would like to illustrate, with a familiar example, why I consider it advantageous. Suppose 0 is a bounded open set in IRa and we have a mass distribution with density f, where (say) f is a bounded measurable function vanishing outside o. The "body" n induces a gravitational field on IRa with (Newtonian) potential (a)

u(x)

=

(1I41T)

10 f(y) . Ix -

yl-l dy.

Now, it is easy to check that u satisfies the conditions lim u(x)

(b)

x .... ",

(c)

-flu

= f,

= 0,

in the distributional sense.

Moreover, (b) and (c) imply (a), so (b) and (c) give an equivalent definition of the Newtonian potential of a body. The point is, that Poisson's equation (c) must be understood here in the distributional sense: the u defined by (a) is not, in general, twice differentiable. The exact amount of smoothness of u, which is crucial in many questions, can now be derived either by estimations based on the integral (a) (as in the classical textbooks), or by applying "elliptic regularity theory" to the equation (c); and this well-developed theory immediately yields e.g. that u E C 1 (IRa) and the first partial derivatives of u are Holder-continuous of order A for every A < 1. ix

Most of the results in this book have been published, but the material in Chapter 7 is new, as well as some things in Chapter 8. The book is intended as an introduction, and many relevant topics are left out or only mentioned. The bibliography is not encyclopedic, but probably the references we give, plus the references in those works, comprise essentially all the literature of the subject. I feel that the Schwarz function, and closely related topics, still provide an attractive field of research to the analyst. The subject has deep roots, originating in classical problems of complex analysis and potential theory (Schwarzian reflection, balayage) and of geometry (focal properties of algebraic curves). It relates to modern investigations concerning inverse problems of Newtonian gravitation, free boundaries, Hele-Shaw flows, and propagation of singularities for holomorphic p.d.e. Difficult unsolved problems are not lacking, i.e. the uniqueness problem for simply-connected quadrature domains in ~2 as well as all manner of concrete questions concerning quadrature domains in ~3. The study of the singularities of the Schwarz potential in ~n leads to profound questions concerning the global propagation ofsingularities in en for solutions ofCauchy's problem for the Laplace operator with data on an algebraic hypersurface; although that is not treated in these lectures, there are some remarks and references in our concluding chapter.

Acknowledgements. Without the help and good will of many people (not counting those who created the mathematics and, hopefully, are adequately acknowledged in the text) this book would not have been written. I wish first and foremost to thank Omitry Khavinson who conceived the idea of a lecture series devoted to the Schwarz function and did the organizing, as well as all those in Fayetteville who contributed to making the conference so pleasant. Moreover, OK contributed many valuable mathematical impulses, beyond what is evident from our joint papers, and suggested several improvements in the manuscript. Most of my work on the subject of these lectures has been done during the 17 years I have been at Kungliga Tekniska Hogskolan (KTH) and the support of colleagues who made me feel welcome here and encouraged my research activities has been invaluable. Here I wish especially to mention Matts Essen (who took the initiative for my initial invitation to KTH) , and Bo Kjellberg. My own involvement with the Schwarz function, originating in joint work with Oov Aharonov on an extremal problem in conformal mapping, evolved in collaboration with a series of graduate students at KTH, all of whom contributed to keeping lit the torch of enthux

siasm. One former student and now colleague, Bjorn Gustafsson, a major contributor to the subject, was consulted often by me during the writing of the book, and it is a pleasure to thank him for enlightening me on many points. Gunnar Johnsson discovered the relevance ofLeray's classic work on the holomorphic Cauchy problem to the Schwarz potential, and has helped me to understand this beautiful connection. Also, I want to thank Lavi Karp and Henrik Shahgolian for useful suggestions. The Swedish Natural Science Research Council (NFR) generously funded my travel to the Arkansas meeting. Last but not least, I express my gratitude to the series editor, Bill Summers, for his editorial patience and to Berit Lundberg for her skillful typing of the manuscript. In the final preparation of the manuscript, I have received invaluable help from Yishao and Xinzhu Zhou.

* * * During the writing of this book I was greatly saddened by the death of my dear friend Allen Shields. He had attended the Arkansas lectures, and expressed enthusiasm for the subject. This book is affectionately dedicated to his memory. HAROLD S. SHAPIRO

xi

THE SCHWARZ FUNCTION AND ITS GENERALIZATION TO HIGHER DIMENSIONS

Chapter 1 The Schwarz Principle of Reflection " ... [Riemann hat] von einer Function, welche z. B. die Flache eines ebenen geradlinigen Dreiecks auf die Fliiche eines Kreises conform abbildet, bereits die Existenz nachgewiesen, w8.hrend die wirkliche Bestimmung einer solchen Function ... die Krafte der Analysis zur Zeit noch zu iibersteigen [scheint] ... Zur Losung dieser und vieler anderer Abbildungsaufgaben fiihrt das fruchtbare Theorem: Entspricht bei einer analytischen Function einer stetigen Folge reeller Werthe des complexen Argumenten eine stetige Folge reeller Werthe der Function, so entsprechen je zwei conjugirten Werthen des Argumentes conjugirte Werthe der Function" . H.A. Schwarz [Schwarz, 1869] 1.1 Introduction. Let D be a domain lying in the upper half of the complex z = x + iy plane whose boundary aD contains a nontrivial segment u of the real axis. If J is continuous in D (the closure of D), analytic in D and real-valued on u, it extends analytically across u into the domain '"

D := D u u u D', where D' is {z: zED} (i.e. the "mirror reflection" of D in u) and the extended function (still denoted by f) satisfies the functional equation J(z) = J(z) for zEn. In particular, if w = J(z) is bijective on D and maps D conformally onto a domain E lying in the upper half of the w-plane, with u mapped on a segment T of the real axis in the w-plane, the extended function is a bijective (conformal) map of D on E:= E U T U E'. It is easy to see that the positioning of the rectilinear segments u, T along the real axes is not essential, and a similar principle holds whenever J maps a domain D conformally onto E such that a segment of straight-line boundary of D is carried onto a segment of straight-line boundary of E. By the continued application of this principle, H. A. Schwarz derived a simple ordinary differential equation for the conformal map of the upper half-plane onto a polygon, from which in some cases the mapping could be computed explicitly. He also carried out a similar, but more complicated argument concerning the conformal map of the upper half-plane onto a triangle bounded by

1. THE SCHWARZ PRINCIPLE OF REFLECTION

2

circular arcs, based on a corresponding extension of the reflection principle (this will be further elaborated below). For a good account of this theory the reader may consult, besides the original works [Schwarz, 1869, 1870] the book [Nehari, 1952, Ch. V]. In the above formulation of the reflection principle, straight line boundaries seem to play an essential role because it is not clear how to "reflect" a point in a general curve. However, inversive geometry long ago provided the appropriate notion of reflection (or "inversion") with respect to a circle, and Schwarz proved that the principle of reflection, when "reflection" is interpreted as inversion, remains valid when u and T above are circular arcs. Indeed he went beyond that: There is an appropriate notion of reflection for analytic arcs, and a corresponding reflection principle [Schwarz, 1870, pp. 150-151]. Our first task will be to present this generalization. Although the basic ideas are due to Schwarz (and were elaborated by many authors, see e.g. [Picard, 1893, Tome II, Ch. X] and the references in [Davis, 1974, p. 210]), I prefer to follow [Davis, 1974, Ch. 6] in basing the presentation on the "Schwarz function" . 1.2 The Schwarz function. The designation "Schwarz function" is due to Philip Davis. Actually, the function it designates does not seem to appear explicitly in the works of H.A. Schwarz. The first explicit occurrence known to me is in a paper by D.A. Grave [Grave, 1895] noted by Davis. Later, and apparently independently, it was employed by [Gustav Herglotz, 1914, p. 305]. This remarkable paper is concerned, not with Schwarzian reflection, but a different problem that had been posed as a "Preis aufgabe" by the Jablonowskische Gesellschaft zu Leipzig, concerning the analytic continuation of the logarithmic (or Newtonian) potential of a mass distribution into the domain occupied by the masses. Herglotz' paper was awarded the prize. Concerning the theme of the Preisaufgabe we will have much to say below; for now however, we wish only to recall the definition of the "Schwarz function." Let r be a non-singular, analytic Jordan arc in R2, i.e. a Jordan arc satisfying either of the following two equivalent conditions: (i)

r

°: ;

is parametrically represented as z = f(t), t ::; 1 where is analytic on a neighborhood of the segment [0,1] and is bijective on [0,1]. Moreover (the "non-singular" part): f'(t) "I0, t E [0,1].

f

1.2 THE SCHWARZ FUNCTION

3

(ii) To each point Zo = Xo + iyo E r there is a neighborhood U of (XO,Yo) in C2 and a function cp(X,Y)(X = x+ix',Y = y+iy') holomorphic in that neighborhood and real for real X, Y such that

r nU =

{(X, Y) E Un R2 : cp(X, Y) = O}

and grad cp(xo,yo)

=/: (0,0).

We leave it to the reader to verify the equivalence of these conditions. We can now state PROPOSITION 1.1. (Grave, Herglotz, ... ) Let r be a nonsingular analytic Jordan arc in R2 ~ C. Then there is a neighborhood n of r and a (uniquely determined) analytic function S on n such that

S(z) =z,z E

(1.1)

r.

PROOF: We'll work with definition (ii) above. We'll give one proof now, and another in §1.5. Clearly it is sufficient to construct S on a neighborhood of an arbitrary point Zo E r since (1.1) then ensures that these local functions fit together to one analytic on a neighborhood of r. By assumption, there is a function cp( X, Y) holomorphic on a neighborhood of (xo, YO) in C2 with grad cp(xo, Yo) =/: (0,0) and such that r is, near Zo, defined as {(x, y) E R2 : cp(x, y) = O}. Let F(z,w) =cp«z+w)/2,(z-w)/2i), where z,w vary on neighborhoods of Zo, Zo respectively. Then F( Zo, zo) = 0 and

:~ =

(1/2)cpx (z

-(l/2i)cpy

+ w)/2,

(z - W)/2i)

(Z + w)/2, (z - W)/2i)

where cp x, cpy denote acp / ax, ac.p / ay respectively. Hence ~~ (zo, zo) = (l/2)(CPx(xo, YO) + icpy{xo, YO») =/: o. By the implicit function theorem for analytic functions, there is a unique complex-valued function w = S(z) defined on a neighborhood of z = Zo, satisfying S(zo) = Zo and F(z, S(z» = o. Moreover, S is analytic. Now, for z E r, z near Zo, the equation F(z,w) = 0 is solved by w = z (since F(z,z) = cp(x, y) = 0 for z E r). This implies that S(z) = z for z E r, z near Zo which implies the Proposition (uniqueness is obvious).

4

1. THE SCHWARZ PRINCIPLE OF REFLECTION

Notice also that if r is a closed analytic Jordan curve, S(z) exists and is analytic on a ring-shaped neighborhood n of r. If n conforms sufficiently closely to r, S is necessarily single-valued in n, because of (1.1). Following Davis, we now call the function S = Sr, whose existence is asserted in Prop. 1.1, the Schwarz function of the (non-singular, analytic) open or closed Jordan arc r.

1.3 Anti-conformal reflection. With the Schwarz function in hand, we can now define "reflection", at least for points near r, where r is any nonsingular analytic Jordan arc. The essential facts are contained in PROPOSITION 1.2. (Essentially, H. A. Schwarz)' Let r be a non-singular analytic arc and Zo E

r. Then, there exists a neighborhood V of Zo homeomorphic to a disk, and a bijective map R of V on itself such that (i) R is anti-confonnal, i.e. R(z) is analytic on V (ii) R( z) = z for z E r (iii) Denoting by Vb V2 the components of V\r, R maps VI on V2 and vice-versa. (iv) R is an involution, i.e. R 0 R = identity. PROOF: Denoting by S the Schwarz function of r observe first that S'(zo) I- O. Indeed, if r is parametrized near Zo in terms of arc length s as z = z(s) we have S(z(s)) = z(s), so S'(z(s))(dz/ds) = dz/ds. Now, writing res) = dz/ds, so that res) is a unit tangent vector to r (encoded as a complex number) at the point z( s), we have S' (z( s )) =

r(s)2. Thus, IS'(z)1 = 1 for z E r. Hence S is univalent for {Iz-zol < a} if a > 0 is small enough. Assuming a so small that {Iz - Zo I < a} is divided into two parts by r, denote one of these by VI. Let now R(z) := S(z). We'll show (if a is small enough) that R maps each point in VI to the opposite side of r. Indeed, from Taylor's Theorem S(z)

= S(zo) + (z - zo)S'(zo) + O(lz - zol2)

holds for Iz - Zo I small, i.e. denoting by r the unit tangent vector at Zo

1.4

SCHWARZIAN REFLECTION. GENERALIZED

5

in view of the calculation just made of S'(z). Now, the map z 1-+ Zo + (z - Zo )r2 =: R# (z) is just reflection with respect to the line through Zo in the direction of the unit vector r. Since R(z) = R#(z)+O(lz - Zo 12) and R# is reflection in the tangent line to r at Zo, it is easy to see (for a small enough; we leave the details to the reader) that R maps points in VI to the opposite side of r. If we define Vi = RV1 and V to be the union of VI, V2 and r n aVI we see easily that the pair R, V satisfy all the assertions of Prop. 1.2. We'll call R = S the anti-conformal reflection associated to the nonsingular analytic curve r. Observe that it is only defined near r, but may of course have conjugate-analytic continuation to a larger domain, even (as when r is aline) to the whole complex plane. The reader should verify that if r is a line or circle, R is "mirror reflection" or inversion, respectively. For more examples see [Davis, 1974, Ch.6j as well as works of Gustafsson and Sakai referred to in the bibliography. 1.4 Schwarzian reflection, generalized. In terms of the anti-conformal reflection Rr associated to formulate Schwarz's principle in a general way.

r

we can

PROPOSITION 1.3. Let D be a plane domain part of whose boundary is a non-singular analytic arc u, and suppose that f, continuous on the closure of D, maps it conformally on a domain E part of whose boundary is the non-singular analytic arc r, and f(u) = r. Then, f extends analytically across u, and for points zED sufficiently near u, the extended function maps R/7(z) into Rr(f(z». PROOF: Clearly we can define a function by

f

for points z near u, z ¢ D

This function is easily checked to be analytic and continuously extendable to points of u. For ( E u we have

'" f«() = f«().

By a well-

'"

known elementary theorem this implies that the pair (f, J) fit together to a single analytic function across u, and this proves the proposition. REMARK: The "elementary theorem" referred to doesn't require analyticity of the arc u, cf. [Nehari, 1952, p. 183]. We can outline a less elementary, but instructive approach which is applicable also to some

6

1. THE SCHWARZ PRINCIPLE OF REFLECTION

more general situations we'll encounter later. The essence of what we have to prove is contained in PROPOSITION 1.4. Let f be continuous in the open unit disk 0 and analytic on D\r where r is a "smooth" Jordan arc dividing 0 into two parts. Then f is analytic in D. This is a proposition of "removable singularities" type, variants of which occur in many other contexts. The crux is, how "thin" or, in the present instance "smooth" do we require the set r to be. Actually it suffices for r to be rectifiable; this is a classical result of Painleve. For present purposes we'll take "smooth" to mean: a non-singular Cl arc. To prove the proposition, we require U E C(D) n Cl(D\r). Then the distributional derivative oxu is identified with the (locally integrable) function g = ou/ox on D\r (since r has planar measure 0, g is a well-defined element of Ltoc); and similarly for oyu.

LEMMA 1.5. With 0 and r as in Prop.1A, let

PROOF: Let Db O 2 be the two parts into which 0 is divided by r. Consider first the special case where u = 0 on O 2 • What we have to prove is, for every

where
88:

88:

2 2 , we is identified with ~: (on the set O\r). Since OX';;,2 is get oxU = ~: and the proposition is proved. We can now easily deduce Prop. 1.4: by virtue of Lemma 1.5, the z-derivative of f in the distributional sense (i.e. (1/2)( Ox + iOy)f) is o on D. By a well-known general principle (Weyl's lemma) valid for

1.5 THE SCHWARZ POTENTIAL

elliptic PDE and systems (see e.g. [Treves, 1975]), in D.

7

f

is then analytic

1.5 The Schwarz potential. In view of the fundamental importance of Propositions 1.1 and 1.2 let us give an alternate approach which relates the Schwarz function to Ca1J.chy's problem. Consider again the problem, to exhibit an "anticonformal reflection" with respect to the non-singular analytic curve r. The function R = 1J. + iv must satisfy near r the relations (1.2)

1J. z

+ Vy =

(1.3)

1J. y

-

0

= 0

Vz

and

(1.4)

= x, V = Y

1J.

on

r.

If we confine z to a small neighborhood N of Zo E r, then from (1.3) there is a function W E C2(N) with W z = 1J., Wy = v whence from (1.2), ~w = O. Because of (1.4), W_(x2 +y2)/2 has vanishing gradient along r n N so w is constant there. We may take the constant to be 0 and we then see: w satisfies the Cauchy problem ~w

(1.5)

= 0

nearr

and

(1.6)

w - (x 2 + y2)/2

vanishes with its gradient on r.

Thus, to find R we are led to the search for a solution w of the Cauchy problem (1.5), (1.6). By the Cauchy-Kovalevskaya theorem, there is a solution (and only one) which is real-analytic on a neighborhood of r. We'll call this function w the Schwarz potential of r. In terms of it, the anti-conformal reflection R = 1J. + iv can now be gotten by (1.7)

8w 8x

U=-,

8w 8y

V=-

or, R = 2 C:;;. In turn, the Schwarz function arises from w by

(1.8)

S = 28w. 8z

8

1. THE SCHWARZ PRINCIPLE OF REFLECTION

Conversely, if the existence of the Schwarz function 5 of r is known, the Schwarz potential w can be constructed by

w(z) = Re1z 5(()d( + (1/2)lzoI2.

(1.9)

Zo

Here Zo is an arbitrary point of r, and integration is along an arbitrary path remaining in some sufficiently small neighborhood of r. It's easy to verify that w satisfies (1.8). Moreover, for zIon r we get

1

Z1 5(()d( = 1.91 5(((s») d((s) = 1.91 ((s)d((s) .90

Zo

.90

hence

2Re1Z1 5(()d( =1.9 1 dl((s)12 = 1((sdI 2 -1((soW Zo

.90

=IZ112 -lzol2 (where So, SI are the values of the arc-length parameter s corresponding to Zo, ZI respectively). Thus, w as defined by (1.9) satisfies w(z) = zz/2 on r, so in terms of (x, y) we get (1.6). Hence the concept of the Schwarz function is reducible to that of the Schwarz potential, and vice versa. But, the Schwarz potential has the advantage that its definition extends in an obvious way to nonsingular analytic hypersurfaces in Rn, n > 2. We'll return to this later in the lectures. For some purposes, it is convenient to use a slight variant of w. The function (1.10) is called the modified Schwarz potential of r. It satisfies the Cauchy problem (1.11)

l:!..w# = 2

nearr

and (1.12)

w# and grad w# vanish on r.

Chapter 2 The Logarithmic Potential, Balayage, and Quadrature Domains "Die homogene materielle Ellipsenfia.che ist, was ihre Wirkung auf 8.ussere Punkte anbelangt, ersetzbar durch eine gewisse materielle Linie im Innern der Ellipse... Die von der [Fusspunktkurve einer Ellipse] umgrenzte homogene materielle Fla.che ist, was ihre Einwirkung auf 8.ussere Punkte anbelangt, ersetzbar durch zwei einzelne Masspunkte im Innern (dieser Kurve]". Carl Neumann [Neumann, 1907] "1st n8.mlich das Potential eines Korpers im Aussenraume darstellbar als Potential geeigneter Massenbelegungen im Innern des Korpers, so liisst es sich auch sicher ins Innere desselben bis an jene Massenbelegungen heran analytisch fortsetzen. Liisst sich umgekehrt das Potential ins innere des anziehenden Korpers hinein analytisch fortsetzen und werden die sich herbei ergebenden singularen Stellen des Potentials durch irgendeine Fla.che F umschlossen, so ist dasselbe - weil ausserhalb von F regular - in mannigfacher Weise als Potential von Massenbelegungen der Fla.che F darstellbar." Gustav Herglotz [Herglotz, 1914, p.300] 2.1 Analytic continuation of the potential. Let I denote a (Laurent) Schwartz distribution of compact support in R2 ~ C. It is meaningful to form its convolution with the locally integrable function E, where E(z) := (27r)-110g Iz-(I and the resulting distribution, which shall be denoted by U I, satisfies (in the distributional sense) the Poisson equation t::J.U I = I in R2 (see e.g. [Gilbarg and Trudinger, 1977, p. 17]. It is easy to see that U l is harmonic oft'supp I. This harmonic function on C\ supp I may be analytically continuable across some points of supp I, and encounter singularities at others. The latter case occurs e.g. at any isolated point of supp I. Let us give an example where the former occurs. Suppose n is a domain bounded by a non-singular analytic Jordan curve r and I is the function in LOO(R2) defined by I(z) = g(z), zEn and I(z) = 0 for z ¢ n, where 9 is real-analytic on a neighborhood of n. (We stress

10

2. LOGARITHMIC POTENTIAL

that the notation does not imply g is holomorphic in z!) Let us show that, under these conditions, U f is analytically continuable from C\O to a neighborhood of r. To prove this, we shall require first a "removable singularities" result, analog of Prop. 1.4, which is proved in a similar fashion: If v E G1(D) where D is an open disk, and harmonic in D\r, where r is a nonsingular smooth Jordan arc, then v is harmonic in D. (Later we'll discuss this important result in more detail in Rn, and with weaker hypotheses.) Now, by the Cauchy-Kovalevskaya theorem, the Cauchy problem ~v = g has, in a certain neighborhood N of r, a (unique) real-analytic solution v with vanishing Cauchy data on r (i.e. v and grad v vanish on r). The potential U f of J satisfies ~U f = 0 in R2\0 and ~Uf = g in O. Hence U f - v coincides in 0 n N with a certain harmonic function h. Since U f is of class G 1 (R2) (indeed, this follows from ~Uf E LOO(R2) by "local regularity theory" for elliptic equations, cf. [Gilbarg and Trudinger, 1977] or [Kellogg, 1929]), the Cauchy data of h + v, and so that of h, on r coincides with that of the harmonic function U f in R2\0. Thus, by the aforementioned principle (on removable singularities) U f extends inwards across r to be a harmonic function in (R2\0)UN, and this proves our assertion. (A similar argument also shows, under the above assumptions, that U fin is analytically continuable outwards across r, to a neighborhood of 0.) In particular, the logarithmic potential of a uniform mass distribution (g = const.) on 0 extends inwards, as a harmonic function, from R2 \0 across 00 when this is a non-singular analytic curve (of course, this is also true locally: it extends across any non-singular analytic arc of

00). Suppose now, for a distribution J of compact support, Uf is harmonically extendable from a neighborhood of infinity "inwards" to R2\K where K is a compact set. To obtain an interesting situation, suppose supp J\K is not empty. Let 0 1 be an open set which includes K. Then, there is a function u E Goo(R2) equal to U f on R2\01. Now, u is the potential of ~u. Indeed, letting V denote the potential of ~u, we have (since u - Uf is of compact support): u-Uf=~(u-Uf)*E=(~u)*E-J*E=V-Uf so u=V. This shows that ~u, which is a "charge density" supported in 0 1 (an arbitrary neighborhood of K) generates the same potential as J on a neighborhood of 00. This is the situation Herglotz is referring to in the passage quoted at the beginning of the chapter. Needless to

2.2 BALAYAGE

11

say, all our remarks apply mutatis mutandis to Rn and Newtonian potentials; indeed, it is to make the generalizations obvious that we are using methods based on p.d.e. and distributions, rather than explicit calculations with logarithmic kernels, Cauchy's integral formula etc. as in Herglotz's work. Let now n be a bounded plane domain whose boundary consists of a finite number of non-singular analytic Jordan curves. Let no be an open subset of n (which can be disconnected) such that ano contains an and the modified Schwarz potential of an is real-analytic in no. Then, from the argument given earlier in this Section, the potential of the uniform mass distribution on n extends harmonically from R2\n to (R2\n) u no; i.e. this potential extends inwards harmonically as far as the modified Schwarz potential does (or, what comes to the same) as far as the Schwarz potential, or the Schwarz function, of an. This can be called Herglotz's principle. Herglotz studied laminae of uniform density bounded by irreducible algebraic curves, and showed that the potential of such a lamina is harmonically extendible except where the Schwarz function of the bounding curve has singularities; such singularities he calls "foci of the curve, in the sense of Plucker". (In the case of an elliptic lamina, these are indeed the (ordinary ) foci, as we'll see later). Moreover, as one can check by examining the above proof, the density needn't be constant, it suffices if it is (the restriction to the lamina of) an entire analytic function on C2 (e.g. a polynomial in (x, To prove this, one requires however an additional result concerning analytic continuation of the solution of Cauchy's problem, Theorem 3.1 of [Khavinson and Shapiro, 1989a).

y».

We have now seen that the Schwarz function is remarkable in (at least) two independent connections: (i) Schwarzian reflection with respect to analytic arcs, and (ii) analytic continuation of potentials. Moreover, (ii) is intimately related with (iii) analytic continuation of solutions of Cauchy's problem for the Laplace operator. There is also (iv) the relation to the projective geometry of the underlying curve (focal properties), more about which in Chapter 9. The aspects (ii), (iii) and (iv) (but not (i» extend in a natural way to higher dimensions.

2.2 Balayage. The term "balayage" (sweeping) as a concept of potential theory was coined by Poincare, and has been used in various ways. Here I will use it in a very classical sense going back to Green's original researches,

12

2. LOGARITHMIC POTENTIAL

more precisely "Green's equivalent stratum" in the mathematical theory of electrostatics [Jeans, 1925, §204]. We saw in the last Section that a uniform mass (or charge) distribution in a lamina bounded by a non-singular analytic curve could be replaced by another charge distribution on a proper subset, which generates the same exterior potential. This procedure, to "sweep" the charges inwards in a fashion that does not alter the external field, could be called "balayage inwards"; the possibility to do this is quite exceptional, requiring high regularity of the boundary and the initial density. On the other hand, G. Green showed that balayage outwards is always possible, in the following strong sense: PROPOSITION 2.1. (Green) Let f be a distribution of compact support in Rn and suppose supp fen where n is an open set whose boundary is a smooth non-singular hypersurface. Then, there is a distribution F supported in an whose potential coincides with that of f outside n. (Moreover, F can be chosen as a smooth charge density, or "simple layer" on an; cf. [Kellogg, 1929, p. 10].) PROOF: For definiteness we'll work with n = 3; the potential of a distribution f with compact support is, by definition, f * E where E = En is the fundamental solution for the Laplace operator in Rn; for n = 3 we have E = -(47l"lxl)-1 where x = (Xl, X2, X3), Ixi = (X~ + X~ + xD 1 / 2 • By "Green's formula", assuming sufficient regularity of u and v,

where dx denotes Lebesgue volume measure in R3, dS is area measure on an and denotes differentiation with respect to the outer normal. Now, (2.1) remains valid (with obvious interpretation) if u is a distribution which is smooth outside a compact set Ken and v is, say, in COO(n). Let h be the harmonic function in n with the same boundary values on an as U f , and apply (2.1) with u = U f - hand vex) = E(x - y) where y E R3\n. We get

:n

f Jn

fl.(h - U f ). E(x - y)dx

= f ~(h -

Jan an

U f ). E(x - y)dS(x).

2.3 QUADRATURE DOMAINS

13

The left side is just -UI(y), whereas the right side is the potential evaluated at y of a "simple layer" on an with density (h - UI). This proves Prop. 2.l.

tn

tn

The charge density (UI - h) on an is "Green's equivalent stratum" for the given distribution f. Assuming an sufficiently smooth (so that then h E CI(n)) this charge density varies continuouslyon an. In case an is homeomorphic to a sphere, it is easy to show that this density is unique, i.e. no other continuous charge densit yon an can generate the same potential outside n. (On the other hand, there are plenty of other Schwartz distributions with support in an that generate the same potential outside n). An interesting property of Green's equivalent stratum is: it is nonnegative if f is a non-negative measure. Indeed, in that case U I is subharmonic, so U I - h is :::; 0 in n and its normal derivative at points of an is ~ o. Thus, balayage outwards preserves positivity of charge. However, the balayage inwards discussed earlier, does not in general do so. We shall see examples of this later. Combining Herglotz's principle with Prop. 2.1 we see that, when UI is harmonically extendable to Rn \K (K compact) the balayage inwards can be done onto any smooth hypersurface surrounding K. REMARKS:

2.3 Quadrature domains. The most extreme case of balayage inwards is, when U I is harmonically extendable to Rn\{xO}, the complement of a single point. This happens when f is the characteristic function of a ball. Let's take as our ball B = B(O; 1) (of center 0, radius 1) in Rn. Its potential is, apart from a constant factor, the (unique) solution to .6.u = XB (where XE shall always denote the characteristic function of a set E) satisfying the conditions (2.2)

(2.3)

u(x)

--+ 0

as

u(x) = log Ixl + 0(1)

Ixl --+ 00 as

if n ~ 3

Ixl--+ 00,

if n = 2.

Let's calculate u when n = 3. On B, u equals Ix1 2 /6 plus a harmonic function and for Ixl > 1 it is a radially symmetric harmonic function, and it is in CI(R3); moreover it's easy to see that these conditions uniquely characterize u. Thus, we shall have u(x) = {

(lxI 2 /6) + a, b/lxl,

14

2. LOGARITHMIC POTENTIAL

if we can adjust the constants a, b so that the two pieces join in a Cl fashion at Ixl = 1. Clearly this requires a = -1/2, b = -1/3, and this choice gives u. In particular, u(x) = -(3Ixl)-1 for Ixl > 1, and this extends harmonically to R3 \ {o}. Another way to write the last formula IS

(2.4)

IBI- 1 l l x - yl-l dx

= Iyl-l,

Iyl > 1

where IBI denotes the volume of B( = 471"/3). Thus, the uniform mass distribution on B produces the same external gravitational field as a point mass at O. Yet another interpretation of (2.4) is (2.5)

IBI- 1 l

v(x)dx

=

v(O)

whenever v is a finite linear combination ofthe functions x 1---+ Ix-yl-l with Iyl > 1. It is fairly easy to show that these are dense in Ll(B), the integrable harmonic functions in B, so (2.5) holds for v E Ll(B). This is, of course, just the Gauss mean value theorem for harmonic functions; the point is that modulo an approximation theorem (whose proof we won't give here) it is equivalent to the formula (2.4) for the potential of XB outside B. Now, (2.5) is the simplest case of what can be called a "quadrature identity" i.e. one which reduces an integration (in the present instance, of a harmonic function) to a finite number of point evaluations (classically, such formulae, like that of "Gaussian quadrature" were one-dimensional, and of the form

for polynomals f of a certain degree). In 1907 C. Neumann (whom, I conjecture, was the author of the question studied in [Herglotz, 1914]) discovered a remarkable case of "balayage inwards": the uniform mass distribution on a certain oval-shaped lamina in R2 has the same logarithmic potential outside the lamina as that generated by two point masses inside the lamina. Although he (and Herglotz who shortly thereafter greatly generalized the result) did not interpret this result as a quadrature formula, one may do so, obtaining for Neumann's oval

(2.6)

110 udxdy = (Inl/2) [u(-a,o)+u(a,o)]

2.3 QUADRATURE DOMAINS

for all harmonicintegrable u on 0 (which contains the points (±a,

15

0».

It is easy to see that (2.6) is equivalent to

(2.7)

flo

fdxdy =

(101/2) (f(-a)+f(a»)

for all f E L~(O), the analytic integrable functions on O. This can in turn be formulated in terms of "balayage inwards" for what could be called the Cauchy potential of the uniform lamina 0, namely C«() := ffo(x + iy - ()-l dxdy which satisfies ac/a, = -1!"Xo. We see that (especially in 2 dimensions) there are many equivalent concepts to describe the same phenomenon (of "balayage inwards"). Although, for historical reasons, we have thus far mostly used potentialtheoretic formulations, the reformulation in terms of "quadrature identities" (which is of much later date, arising in [Davis, 1965] and, independently [Aharonov and Shapiro, 1974, 1976]) will usually be preferred by us in the remainder of these lectures. These identities are a simple and natural point of departure for function-theoretic studies, and moreover by varying the setting (e.g. varying the classes of "test functions" allowed in the quadrature identity, allowing more general functionals than point evaluations, etc.) one arrives at new problems. The fruitfulness of this point of view has shown itself in recent works of Gustafsson and Sakai, even when the object of study is classical potential theory and balayage. For an overview see [Gustafsson, 1989]. Remark. The idea to study analytic continuation of potentials into the region occupied by the charges in terms of the solution of a Cauchy problem for the Laplace operator, as in § 2.1, goes back at least as far as the paper "Bemerkung zur Potentialtheorie" by Erhard Schmidt in Math. Annalen 68 (1910) pp. 107-118, where also references to earlier works of Bruns and Stahl are given.

Chapter 3 Examples of "Quadrature Identities" 3.1 About the terminology. It was known already at the time of Newton that "quadrature", i.e. the evaluation of integrals, could sometimes be effected, approximately or exactly, by replacing f(t)dt with a sum of the type 2:;:1 cif(tj) where a ~ t1 < t2 < ... < tm ~ b and the Cj are real constants. For example

J:

(3.1)

3

ill

f(t) dt = f( -1) + 4f(0)

+ f(l)

is valid when f is any cubic polynomial as well as for many other functions (e.g. all odd continuous functions on [-1,1]). Moreover, for smooth f the difference between the left and right hand sides can be bounded by a constant times the supremum of If(4)(t)1 on [-1,1]. This formula can be scaled to an arbitrary interval; if we do so for a number of consecutive equal "small" intervals and add the resulting equations we get "Simpson's rule". The "quadrature identities" we speak of in these lectures have formal similarities with (3.1), but, also these differences: the domain of integration will be an open set in Rn for some n ~ 2; and the family of f for which the formula is to be exact will be some class of harmonic, or analytic functions. Moreover, the point of view is different: we are not here concerned with quadrature identities (q.i.) as a method ofnumerically integrating harmonic (or other) functions over the domains involved. In the next chapter we'll give a formal definition of q.i. (It will turn out that there are several definitions, according as to which subclass of the harmonic functions is to satisfy the q.i. and also whether we care to allow functionals more general than "point evaluations" into the "right hand side" of the formula. It will, in any case, turn out that only rather special domains admit any q.i. at all, and those will be called quadrature domains in the wide sense, or q.d.w.s.). In the present chapter we want only to motivate these definitions, and the theory in the next chapter,

17

3.2 EXAMPLES

by giving some examples. As already discussed in Chapter 2, these examples could also be presented in terms of "balayage inwards" . 3.2 Examples. "The grand-daddy of them all" is the mean value formula, which we have already spoken of: (3.1) where B = B( XO ; R) is a ball centered at the point XO E Rn, and IBI is its volume. The formula holds for all u harmonic and integrable over B(u E Ll(B». It is less well known that for any distinct points xl, ... , xm in Rn there is a bounded connected open set 0 containing all the x j and positive numbers Cj such that

(3.2)

i

(}

u dx =

t

Cj

u(x j ),

Vu E

Ll (0).

j=l

Without the requirement that 0 be connected this would be trivial, one could take 0 = UB( x j ; Rj) where the Rj are so small that these balls are mutually disjoint. The existence of a connected 0 satisfying (3.2) is, however, nontrivial. It follows from results in [Gustafsson, 1981] and [Sakai, 1983], based on variational inequalities (those authors worked in R2, but their methods apply as well in Rn). When n ~ 3 one does not know any nontrivial such domain (Le. with m ~ 2) explicitly. For n = 2 however, one can give explicit constructions (starting with the oval discovered by C. Neumann, mentioned above, and to be considered in a moment). Moreover one knows that all such domains are bounded by algebraic curves which have further remarkable properties (see below). For n ~ 3 one does not even have much qualitative information (e.g. concerning topological structure, or regularity ofthe boundary) concerning 0 that satisfy (3.2).

In the remainder of this chapter we'll work in R2. One construction of q.d. is based on the following two propositions, due to Philip Davis ([Davis, 1974, Ch. 14]; see also [Aharonov and Shapiro, 1976, Thm.1]).

18

3. EXAMPLES OF "QUADRATURE IDENTITIES"

PROPOSITION 3.1. Let n be a bounded plane domain with smooth boundary. Suppose there exists a function S analytic and single-valued in n except for simple poles at Zl, ... , Zm with residues at, ... , am respectively, and continuously extendable to each point ( of an where

it satisfies S(O

= C.

f

(3.3) for every

Then

10 IE L!(n)

Ido-=7r'fajl(zj) j=l

(analytic integrable functions on n).

REMARKS: By "smooth boundary" we mean here so regular that (a) Stokes' theorem can be applied and (b) functions analytic on a neighborhood of n are dense in L!(n). Since this chapter is mainly heuristic and we don't aim at maximum generality, we won't here enter into a discussion of conditions for (b); the reader can think of I in (3.3) as being analytic on a neighborhood of n. In case n is a Jordan domain and the aj are real, (3.3) gives

(3.4)

i

udo-

= 7r

t..

aj u(Zj)

for harmonic functions u which are the real parts of IE L!(n). Since these are dense in Ll(n), (3.4) then holds for u E Ll (n); in this manner we can get q.d. for harmonic functions also from Prop. 3.1, so in the remainder of the chapter we'll only deal explicitly with q.d. for analytic functions. PROPOSITION 3.2. Let 'P be a rational function without poles in 0 (where D denotes the open unit disk) and injective in D. Then, there is a function S analytic on n := 'P(D) except for poles, continuously extendable to each point ( E an and satisfying S( 0 = C there. Moreover, if WI, ••• ,Wm are the distinct poles of 'P, the poles of S in n are at the points Zj := 'P(wj) where wj = l/wj, and the pole of S at Zj has the same order as that of 'P at Wj. (Note that Wj = 00 is allowed). In particular, if the poles of 'P are simple, n satisfies the hypotheses of Prop. 3.1, and the q.i. (3.3) holds. PROOF OF PROP. 3.1: By a form of Stokes' theorem, for on a neighborhood of n

10 I do-

= (2i)-t

t I(O(

d(

I

analytic

3.2

EXAMPLES

19

where r is the positively oriented boundary of O. The last integral is (by residues)

which is (3.3). REMARK: We could of course have allowed S to have multiple poles, and then there would be terms on the right side of (3.3) involving derivatives of f at the Z j. PROOF OF PROP. 3.2: We can define a function S from 0 to the extended complex numbers by

(3.5)

S(IP(w») = 1P*(l/w),

wE 0

where 1,0* denotes the (rational) function defined by 1,0*( w) = IP( w).

S is analytic in 0 except at points Z = lP(t) where t is a pole of 1P*(l/w). Since the poles of 1,0* are {Wj}J!=ll the poles of S are at {1P(Wi)}J!=1 and it's clear that the pole of S at lP(wi) has the same order as that of 1,0 at Wj. Finally, for' E ao we have, = lP(w) with Iwl = 1 so, from (3.5), SeC) = S(IP(w» = 1P*(l/w) = 1P*(w) = lP(w) = (, completing the proof. Combining Prop. 3.1 and Prop. 3.2 gives COROLLARY 3.3. Let 1,0 be as in Prop. 3.2, and suppose all its poles are simple. Then, 0 = 1,0(0) is a q.d. in the sense that (3.3) holds for all f E L!(O). In (3.3), Zj = lP(wi) where {wt, ... ,wm } are the poles of 1,0, wi := Wj -1 and the aj are complex numbers depending

only on 0 (not on I). Let us illustrate this by constructing a non-trivial q.d. We start from the familiar fact that .,p(w) = (1/2)[w + (l/w)] conformally maps {iwi > 1} onto the domain C\[-l,l]. The image of the circle {iwi = R} where R> 1 is the ellipse ER whose parametric equations are

ER : x = (1/2)(R + R- 1 ) cos t, y = (1/2)(R - R- 1) sin t. Hence .,p(Rw) maps {lwl > 1} conformally onto the exterior of this ellipse. If N R is the curve obtained by inversion of E R with respect to o (i.e. by the map Z 1-+ l/z), .,p(Rw)-l conformally maps {Iwl > 1}

20

3. EXAMPLES OF "QUADRATURE IDENTITIES"

onto the interior OR of NR and so, finally .,p(R/w)-1 maps 0 onto OR, i.e.

maps 0 conformallyonto OR (which is "C. Neumann's oval"). Observe that cp has its poles at WI = iR, W2 = -iR. Thus the Schwarz function of NR has its poles at Zj = cp(l/wj),j = 1,2 i.e. at the points ±2(R2 - R-2)-li. The residues at these poles can easily be calculated using the parametric representation

of the Schwarz function S of N R. For, in general, if we have a parametric representation Z = A( w), S = B( w) and B has a simple pole at Wo with residue p, S (as a function of z) has a simple pole at Zo := A( wo) with residue pA'(wo), when A is analytic near Wo. In the present case Wo = i( or i), p = R-I and A'(wo) = 2R(R2 + R-2)(R2 - R-2)-2. If we introduce the abbreviations (3.7) for the semi-axes of the ellipse ER we thus obtain: the Schwarz function of N R is analytic in the interior of N R except for simple poles at the points ±(2ab)-I.i, at each of which it has the residue (a- 2 + b- 2)/4. Applying (3.3) with 0 the domain bounded by NR and f = 1 we see the area enclosed by NR is (1l"/2)(a- 2+b- 2). (Note that a 2 _b2 = 1 since the foci of each ellipse ER are at the points Z = ±1). We have thus obtained (in a somewhat different way) the result of C. Neumann: inversion of an ellipse about its center gives a curve (denoted NR above) which bounds a domain admitting a two-point quadrature identity, or in other words, the uniform lamina bounded by N R induces the same logarithmic potential outside N R as two suitably placed point masses (each with half the mass of the lamina). It is easy to check that N R is an algebraic curve of degree 4 whose equation in Cartesian coordinates is

(3.8)

3.3 EXAMPLES, CONTINUED

21

This is one of a class of what Herglotz calls "bicircular curves", and investigates in great detail in the cited work. 3.3 Examples, continued. Implicit in the preceding paragraph is the Schwarz function of the ellipse ER. The most direct way to get it is to substitute into the equation of E R

(3.7) where (a, b are given by (3.7)) x = (z + z)/2 and y = (z - z)/2i and solve the resulting equation for z. Alternatively, we can represent the Schwarz function S(z) of ER parametrically by

Since the expression for z = z(w) in (3.8) maps {Iwl > I} conformally to the exterior of ER we see that S is analytically continuable throughout the exterior of ER except for a pole at 00 where it behaves like S(z) = R-2 z + Co + CIZ-1 + .... Moreover, z(w) is locally univalent except near w = ±R-l where z = ±1. Thus, S(z) is analytically continuable along every path in C\{ -1, I}. It is easy to check that S has algebraic branch points of order 2 at z = -1 and z = 1 (the foci of the ellipse ER)' By direct calculation from (3.7) we see, in fact

(3.9)

S(z) = (2a 2 - l)z - 2ab(z2 _ 1)1/2

where we take the branch of (.)1/2 such that S(z) '" R-2 z at 00. From these formulae we can deduce interesting quadrature formulae for the ellipse. The first (known, in equivalent forms, to MacLaurin and perhaps earlier) is obtained as follows. Letting n denote the interior of the ellipse (3.7), if J is analytic on n,

{ jZ dz = (2i)-1 { J(z)S(z)dz. Jn{ Jda- = (2i)-1 Jan Jan We can shrink the contour of integration, until it "just barely" surrounds the segment [-1,1], and obtain in the limit

(3.10)

22

Since

3. EXAMPLES OF "QUADRATURE IDENTITIES"

n

has area

Inl =

nab we can write this

(3.11) Since the right side is the same for all ellipses n with foci at -1,1 we can state: The mean value of an analytic (or harmonic) function over each elliplle of a confocal family ill the .'!ame. This proposition (a generalization of "Gauss' mean value formula") can be extended to Rn. Another interesting formula, perhaps first noted in [Sakai, 1981] is: If f ill analytic and integrable on D := C\n where n ill an elliplle, then fD fda = o. To prove this, let D(t) be the domain bounded by ER and r t := {Izl = t} where t is large. We shall assume f extends smoothly to aD. Also, the hypotheses imply fez) = O(lzl- 3 ) for large Izi. Hence, { fda

= (

JD

fda

+ o( 1)

JD(t)

and {

= (2i)-1

fda

JD(t)

= (2i)-1

(

f(z)zdz

JaD(t)

(

JER

f(z)S(z)dz

+ O(t-l)

where S is the Schwarz function of ER. By Cauchy's theorem, we can move the contour in the last integral outwards (since S has no finite singularity outside E R ), replacing it by r t and then we see this integral is O(t-l). We conclude, letting t -+ 00, that fD fda = o. (The result also holds for f integrable and harmonic, and in this form is valid also in Rn (see [Friedman and Sakai, 1986] and references there). ) 3.4 Example of a q.d. with singular boundary point. Whenever n is a bounded domain whose boundary consists of finitely many nonsingular analytic Jordan curves, we can obtain a kind of q.i. for fn f dO' (with f analytic in n) by writing this as (2i)-1 fan fz dz, replacing z on each component of an by the corresponding Schwarz function, and then using Cauchy's theorem to move the integration contours inwards to n, thus replacing the integral of f over n by line integrals over some curves in the interior of n.

3.5 AN EXAMPLE OF "BALAYAGE INWARDS"

23

It may happen that this procedure even works when an is analytic but has isolated singular points. In this case the Schwarz function will not be regular in a full neighborhood of an but, may be analytic and bounded in a one-sided neighborhood of an, near each singular point. Here is an example. The function z = ep(w) := w + (w 2 /2) is analytic and univalent in {Iwl $ I}, it maps this disk conformally on a Jordan domain n (so-called cardioitl). The Schwarz function S of an is given parametrically by Z

= w + (w 2 /2),

S = w- 1

+ (w- 2 /2).

Clearly this uniquely defines a branch of S(z) that is holomorphic in except for a double pole at z = o. Since epl( -1) = 0, the map w 1-+ z is not injective on a full neighborhood of w = -1, hence S has a branch point (of order 2) at z = -1/2, the singular boundary point of n. We'll see in the next chapter that this behaviour is typical: if n is a bounded simply connected domain and there is a function analytic on n n {z : dist (z, an) < e} for some e > 0 and continuously extendible to an where it coincides with z, then n is the union of non-singular analytic arcs and at most a finite number of singular points, all of which are inward-pointing cusps, as in the example of the cardioid.

n

3.5 An example of "balayage inwards". We have seen how, starting from rational conformal maps of the unit disk, we can construct domains n satisfying q.i. of type (3.4). Now, in (3.4) some aj can well be negative (we leave it to the reader to construct an explicit example). Hence, if we do "balayage inwards" of the uniform mass (or better, charge) distribution on n we can sweep the charges ultimately to the finite set {Zl, ... , zm}. However, this cannot be done keeping the charges positive: they will be positive at the beginning, but beyond a certain point negative charges will be required. This supplies an example that was promised in Chapter 2.

Chapter 4 Quadrature Domains: Basic Properties, I 4.1 Notations,etc. Let us review some notations that we repeatedly use. By 0 we shall always denote an open set in Rn, and LP(O) denotes the usual Lebesgue space with respect to Lebesgue measure dx. By Lt(O) and L~ (0) we denote respectively the subspaces of LP(O) consisting ofharmonic and analytic functions (the latter only in the case n = 2). We shall occasionally also invoke the Sobolev space Wm,P(O) of distributions 'II. in 0 such that {)a u E LP(O) for all multi-indices a with lal :5 m, and its subspace W;a'p(O) which is the closure in Wm,P(O) of COC(O) (the infinitely differentiable functions on Rn whose support is a compact subset of 0). We use the notations of [Adams, 1975] to which we refer for more details of these spaces. Although we consistently use the same multi-index notation in Rn as in the modem theory of distributions and p.d.e. we sometimes depart from this in R2 and use traditional "complex variable" notations: z = x + iy rather than x = (Xl,X2) for a point of R2 ~ C. We'll also write du for area measure in C. Finally, XA always denotes the characteristic function of a set A, and E (or En) shall denote the usual "fundamental solution" for the Laplace operator in Rn. By a quadrature domain in the wide sense (q.d.w.s) we mean an open connected set 0 C Rn such that there exists a distribution JI. with compact support in 0 for which the quadrature identity (q.i.). (4.1)

In

udx = (JI.,u)

holds for all 'II. E Ll (0). We'll say 0 is a quadrature domain (q.d.) and (4.1) a quadrature identity (q.i.) if (4.1) holds for some distribution whose support is a finite set. Lots of variation is possible in these definitions, since we may e.g. wish to require (4.1) to hold only for some subclass of Ll(O) (such as L~ (0), which is a subclass when 0 is bounded). It can be shown for bounded domains 0 subjected to certain regularity hypotheses that

4.2 QUADRATURE DOMAINS AND THE SCHWARZ POTENTIAL

25

L'r (n) is dense in L1 (n), so the choice of Lt (n) as the class of "test functions" u in (4.1), for any 1 ~ P ~ 00, leads to the same class of domains. Also, when n = 2 we will frequently want to require that (4.1) hold for u in L! (n) rather than Ll(n). Usually, /J will be a real measure, and n fairly regular, and then it is easy to see that this distinction is irrelevant since sums of functions in L! (n) and their complex conjugates are dense in L1 (n). By and large the questions that interest us in these lectures are those in which this kind of distinction does not make any difference. Both Gustafsson and Sakai usually work with a definition of q.d. which requires /J to be a bounded measure on n but not necessarily of compact support (so that every domain is trivially a q.d.). As an example of a nontrivial q.d. in this broader sense we have a triangle in the plane. It is not hard to show e.g. if K is the subset of a triangle n consisting of an interior point, and lines joining it to the three vertices, that there is a bounded measure /J supported on K such that (4.1) holds. Again, if we allow unbounded domains n the requirement of compact support is rather restrictive since e.g. a solid circular cylinder in R3 admits an identity (4.1) where /J is a measure supported on the axis of the cylinder. However, for brevity, we won't deal much with those situations in these notes. If in (4.1) (with n = 3, say) n is bounded and we take u(x) = Ix - yl-l where y is a point of R3\n, we see that the Newtonian potential of the uniform mass distribution on n extends harmonically "inwards across an to R3\ supp /J. Thus, as we have observed several times, the concept of q.i. is very closely related to that of "balayage inwards" . 4.2 Quadrature domains and the Schwarz potential. THEOREM 4.1. H n is a bounded q.d.w.s. (so that (4.1) holds) there is a distribution V on Rn satisfying

(4.2)

~V

= Xu - /J

(4.3) 2

I

where U E Wo ,p (n) (Vp' < (0) and supp wEn. REMARKS: The proof we'll give is not the most elementary, but has the advantage that the method employed can be adapted to unbounded

26

4. QUADRATURE DOMAINS: BASIC PROPERTIES, I

domains, although we won't go into that here. Note that from (4.2) ~V = Xo on a neighborhood of an. By well-known results on local regularity of solutions of elliptic p.d.e. this implies that V E Wj!{ (D) for some neighborhood D of an, for every p' with 1 < p' < 00, and the first-order partial derivatives of 11 are in Wj~{ (D). By the Sobolev embedding theorem these first derivatives are therefore Holdercontinuous with Holder exponent A, for each A < 1. In view of (4.3) we then have: V and grad V vanish on an. Before proving the theorem, we will outline a proof of the weaker result where, instead of (4.3), V is only shown to vanish on RN\n, since that is much more elementary. Practically speaking, when n is the interior of its closure, this weaker result is as useful as the stronger one. To this end, let E denote the standard "fundamental solution" for the Laplace operator. For y ¢ n, we may take '11.( x) = E( x - y) in (4.1), which then reads: E * (Xo - J.t) = 0 on Rn\n. Denoting by V the convolution on the left side of this equation, we have (4.2) and the vanishing of V off n, as claimed. (To get sharper information on V we could choose u(x) = E(x - y) in (4.1), with yEan; this is the method that mainly has been employed so far, and is perfectly adequate for bounded n, but as we said, we'll give an alternative proof of Theorem 4.1.) The proof of Theorem 4.1 is based on several lemmas. LEMMA

1

4.2. [Havin, 1968]. Let n be a bounded open set in Rn and = p/(p - 1). Suppose FE Lpl (n) satisfies

< p < 00, p'

(4.4)

J

uFdx = 0,

V'll. E

Lt (n).

Then, there is a distribution U E W;,pl (n) satisfying ~U = F. PROOF: To say'll. E Lt(n) is equivalent (in view of "Weyl's lemma") to saying'll. E LP (n) and J u~

4.2 QUADRATURE DOMAINS AND THE SCHWARZ POTENTIAL

27

we have (aa~j)J\(e) = (eO /leI2)(~~j)J\(e) where J\ denotes, as usual, Fourier transformation. In the language of singular integral operators, this says that aO~j arises from ~~j by a singular integral operator (of the Calderon - Zygmund class) which is known to map Lr(Rn) continuously into Lr(Rn), for every 1 < r < 00 (for details see e.g. [Garcfa-Cuerva and Rubio de Francia, 1985, Ch. II]). Hence each of the sequences {aO~j} with lal = 2 is Cauchy in LP'(Rn). By virtue of "Poincare's inequality" [Adams, 1975, p. 159] the sequence {~j} is therefore Cauchy in the norm of the space W 2 ,p' (Rn) and hence converges to an element U of this space, which is in wg,P' (0) and satisfies ~U = F.

an

= 2,

REMARKS: Although we won't require this, let us note that the converse of Lemma 4.2 is also true: if F = ~U for some U E wg,P' (0), then (4.4) holds. This is rather trivial, since then F is the limit, in LP' norm,ofasequence {~~j} with ~jECgo(O), and each I(~~j)udx vanishes.

a

A result analogous to Lemma 4.2 holds for the operator (and will be used later), cf Lemma 2.2 (p.338) of [Shapiro, 1984]. . LEMMA 4.3. H I' is a distribution with compact support K in Rn and k is a radial function (i.e. k(x) depends only on Ixl) in ego with support in B(Ojc) and k(x)dx = 1, then,

IRn

(4.5)

holds for every hharmonic on a neighborhood of Ke := {x: dist (x, K) ~ c}. (Note that h is harmonic on a neighborhood of supp (p. * k) so the right side of (4.5) is meaningful). PROOF: We may assume h E ego (Rn). Now, {p. * k, h} = {p., k * h} and it is easy to check that k * h = h on a neighborhood of K, which proves the lemma. LEMMA 4.4. If v is a distribution of compact support which annihilates every function harmonic on a neighborhood of supp v then ~w = v has a solution with supp w C supp v. PROOF: If E is the fundamental solution for ~, and y fj. supp v, the hypotheses imply that v annihilates the function x 1-+ E(y - x). That

28

4. QUADRATURE DOMAINS: BASIC PROPERTIES, I

means w := E follows.

*v

vanishes off supp v, and since .6ow = v, the lemma

PROOF OF THEOREM 4.1: Assume 4.1 holds. Then, for sufficiently small e > 0, and all u E L1 (n), u dx = (u, JI. * k) where k is as in Lemma 4.3. Fix p > 1. Then F := Xo - JI. * k satisfies (4.4), so by Lemma 4.2 there is U E wg'p' satisfying .6oU = Xo - JI. * k. Moreover, combining Lemma 4.3 and Lemma 4.4 (with v = JI. - JI. * k) we obtain a solution w of .6ow = JI. - JI. * k with supp w c Kc (C n if e is chosen small enough). Hence V:= U - w satisfies (4.2), and Theorem 4.1 is proven.

10

When n is a bounded q.d.w.s. the distribution V whose existence is furnished by Theorem 4.1 is called the modified Schwarz potential (and -n V + (lxI 2 /2) the Schwarz potential) of n. This terminology is consistent with that introduced earlier, since if n = 2 and part of the boundary of the q.d.w.s. n is a non-singular analytic arc r, the modified Schwarz potential of r qua analytic arc coincides with the (just defined) m.S.p. of n qua q.d.w.s. There is a corresponding tworather than D.. dimensional version of Theorem 4.1 for the operator Observe, however, that in Chapter 1 a slightly different normalization was used, e.g. (see (1.11)) the Laplacian of the m.S.p. was there taken as 2, rather than 1. The normalization in the present chapter will henceforth always be intended.

a

n

THEOREM 4.5. If is a bounded open connected set in R2 ~ C and JI. a distribution of compact support in n such that

(4.6) holds for all

f

E L! (n), there is a distribution v on R2 satisfying

av

(4.7)

Oz =Xo-JI.

(4.8)

v=U+w

where U E W~,pl (n) (Vp'

< 00) and supp wen.

We won't give the proof, since it is completely analogous to that of Theorem 4.1, involving analogs of the lemmas 4.2, 4.3, 4.4 and the

4.3 SOME APPLICATIONS

29

details may be left to the reader. It follows from (4.8) that v is Holdercontinuous on a neighborhood of a~, and vanishes on a~. Therefore S{z) := z-v is (because of (4.7» holomorphic in 0 outside of supp J1., satisfies ~ = J1. in 0, is continuously extendable to ao and satisfies S{z) = z on a~. For any q.d.w.s. this associated function is called the Schwarz function of O. (Clearly it is unique.) Again, if part of ao is an analytic arc, this definition is compatible with that in Chapter 1. It follows from Lemma 4.3 that when we have a q.d.w.s. the distribution J1. appearing in (4.1) (or (4.6» may always be replaced by a "smoothed-out" version J1. * k in err' (O), but it is nevertheless very convenient to allow J1. to be a general distribution, in the definition of q.d.w.s. 4.3 Some applications. Let us illustrate the use of the general theorems 4.1 and 4.5. We'll begin by proving (otherwise than in the original paper): THEOREM 4.6. [Epstein, 1962]. H 0 is a bounded open set in R2 and (4.9)

/nfdu=IOI·f{zo)

holds for all f E L! (O), where Zo is a fixed point of 0 independent of f, then 0 is a circular disk centered at Zo. REMARK: Actually Epstein only assumed (4.9) for f E L! (O), and 0 of finite area (not bounded), but his method required the assumption that 0 be simply connected (this assumption was removed in [Epstein & Schiffer, 1965] where the theorem also was extended to Rn). We won't discuss these nuances here. PROOF OF THEOREM 4.6: Clearly we may assume Zo = o. Apply Theorem (4.5): by assumption (4.6) holds with J1. = 101· 6 (where 6 is the Dirac measure), so as/8z = 101· 6 has a solution in 0 which vanishes on a~. Thus, since a/8z(1/{7rz» = 6, we have a/8z(S101{7rz)-1) = 0 in 0, so there is a function h holomorphic in 0 and continuously extendable to 0 satisfying

+ h(z) 101 + 7rzh(z),

S{z) = 101· (7rz)-l

in O. For z E ao this gives 7rZZ = hence zh, having real boundary values, is constant and therefore vanishes identically.

30

4. QUADRATURE DOMAINS: BASIC PROPERTIES, I

This shows that Izl = R := (Inl/,1I")1/2 on an, i.e. an is a subset of this circle and since n is bounded it must be the disk {Izl < R}, which completes the proof. It is considerably less simple to prove the following generalization to

R": THEOREM 4.7. [Epstein and Schiffer, 1965]. If n is a bounded open set in R" and

(4.10)

holds for all u E L1 (n), where x O is a fixed point of n independent of u, then n is a ball centered at xO. PROOF: We'll give a proof which is not the shortest, but based on a very fruitful idea of Sakai which can be used in many similar problems. Our presentation is based on [Gustafsson, 1989]. Again, we may assume x O = o. Thus, (4.1) holds with I-' = Inl· 6 and by Theorem 4.1, n has a modified Schwarz potential V which satisfies

(4.11) (4.12)

V and grad V vanish on

an.

Now, since B, the open ball centered at 0 of volume same conditions as n, it has a m.S.p. Va satisfying

Inl,

satisfies the

(4.13) (4.14)

Vo

and grad Vo vanish on

aB.

Now, we claim, moreover (4.15)

Va(x) > 0,

Vx E B\{O}.

This requires a calculation, which we'll do later. From (4.11) and (4.13),

(4.16)

6.(V - Vo)

= Xn

- XB.

4.3 SOME APPLICATIONS

31

Now, suppose 0 i= B. We'll derive a contradiction. Consider first the case where Xu = XB a.e. Then from (4.16) and Weyl's lemma V - Vo is harmonic on RR and, vanishing for large lxi, it vanishes identically. From (4.12) and (4.15), Vex) - Vo(x) < 0 at any boundary point x of 0 inside B, hence there are no such boundary points and B C 0 whence, since IBI = 101, B = O. This is a contradiction, therefore Xu - XB is non-zero on a set of positive measure, which implies there are points of 0, and hence of ao not contained in B. Let y be such a point. On some neighborhood D of y, ~ V = Xu so V is nonconstant and subharmonic with V(y) = 0, hence sup VID > o. Therefore, max(V(x)- Vo(x» for x ERR, is positive; let it be attained at xl. We must have V(xl) > 0 because of (4.15). Hence xl E 0 (since V vanishes outside 0). But then, for x on some neighborhood N of xl

so V - Va is subharmonic on N. This is impossible unless XB(X) = 1 a.e. on a neighborhood of xl, i.e. xl E B n 0 and V - Vo is constant near xl, and hence on the closure of the component of RR\(aB u aO) containing xl. Thus, there is a point x 2 E aB u ao where V - Vo attains its maximum. By the reasoning just conducted for the point xl, we deduce that x 2 E B n 0, and have arrived at a contradiction, which finishes the proof. There remains, however, the detail of verifying (4.15). We shall give several proofs. First proof of (4.15). By a simple scaling argument we may assume B = B(O; 1). Vo is uniquely determined on B\{O} as the solution of the Cauchy problem: ~ Vo = 1, with Vo and grad Va vanishing on aBo Let's just do the cases n ~ 3. Since we are looking for Vo with radial symmetry, it is fairly easy to guess the solution,

(4.17)

2nVo(x) =

Ixl 2 + (2/(n - 2»lxI 2 - n

-

n/(n -

2).

This clearly solves the Cauchy problem, and hence is the unique solution. We have thus to verify that F(r) := (n - 2)r2 + 2r 2- n - n is positive for r in (0,1), which is evident since F(l) = 0 and F'(r) < 0 on (0,1).

32

4. QUADRATURE DOMAINS: BASIC PROPERTIES, I

Second proof of (4.15). If En is the standard fundamental solution for .6. in Rn,

Vo

= (XB -IBI· 0) * En = XB * En -IBI· En

so we have to show

(4.18) for 0 < Ixl < 1. Now, the function y ~ En(Y - x) is subharmonic on Rn, so (4.18) follows, indeed for all x, by the sub-mean value property.

A third proof of (4.15) follows from a general proposition, essentially contained in [Sakai, 1982, §14], and which (as we'll see) has other applications: PROPOSITION 4.8. (M. Sakai) Let n be a bounded connected open set in Rn symmetric with respect to a hyperplane H. Suppose J.L is a positive measure whose support is a compact subset of n n H and V E GI(n\ supp J.L) satisfies

(4.19) (4.20)

.6.V = Xo - J.L

V

in

n

and grad V vanish on

an.

Then

(4.21)

Vex) > 0

for x E

n\ sUPPJ.L.

PROOF: Without loss of generality we may take H to be {x n = O}. We employ the notation x = (x'; x n ) where x' := (xt, ... , Xn-l). Let n+ := {x En: Xn > O}. We'll show first

anY ~ 0

(4.22)

on

n+.

Since .6. V = 1 on n+, an V is harmonic in n+, so (4.22) will follow if we show

(4.23)

lim sup (an V)(x) ~ 0, z-y

zEO+

Vy E

an+.

4.3 SOME APPLICATIONS

33

We consider three cases: (i) yEan, Yn ~ 0 (ii) Yn = 0, y' E n\ supp p (iii) Yn = 0, Y' E supp p. In case (i), an V(x) -+ 0 as x -+ y because of (4.20). Next, observe that V# (x) := V( x' j -x n) satisfies the same hypotheses as V, so V#V is harmonic in n and vanishes on an, hence identically in n, so V(x'j -x n ) = V(x'j x n ) and (an V)(x'j 0) = 0 for x' ¢ supp p. Thus

....

(4.23) is verified in case (ii). To handle case (iii), observe first that if V is the extension of V to Rn obtained by defining it as 0 off n, then '"

....

V = (~V) * En where En is a fundamental solution as above. Hence, for x E n+ we have V(x) =

(an V)(x) =

In En(x - z)dz - JEn(x - z)dJ1.(z).

In (an En)(x - z)dz - J(an En)(x - z)dp(z).

The first integral is continuous with respect to x and when Xn = 0 it vanishes because of the odd symmetry of z 1-+ (an En) (x - z) = (an En) (x' - Z'j -zn) with respect to Zn. As to the second integral, a simple computation shows that an En(x - z) > 0 for Xn > 0, Zn = 0 so this integral is ~ 0 for Xn > 0 and (4.23) is verified also in case (iii). Hence (4.22) holds. Moreover, anY < 0 on n+, otherwise, by the (strong) maximum principle an V = 0 on n+ whence, in view of (4.20), V = 0 on n+ contradicting (4.19). Since V = 0 on an and an V < 0 on n+ it is clear that V > 0 on n+. Since V is symmetric about {xn = O}, (4.21) follows, and the proof is finished.

The modified Schwarz potential of the ball B = B( X O j R) is positive on B\{xo}, i.e. (4.15) holds. COROLLARY.

If we examine carefully the proof of theorem 4.7, we see that the only properties of the ball B that were used are (i) B is the interior of its closure, and B has connected complement (ii) B is a q.d. whose m.S.p. is positive (i.e. (4.15)). We required (i) in two ways: (a) to assure that if Rn\B contains points of n, it also contains a boundary point of nj and this conclusion would fail if B had a "hole" in it, (b) to infer from n ::> B and Inl = IBI that there are points of n outside

34

4. QUADRATURE DOMAINS: BASIC PROPERTIES, I

Bj and this would be incorrect if B could be 11\K where K is some closed set of measure zero. Therefore, we can formulate the following more general uniqueness theorem. THEOREM 4.9. [Sakai, 1982] Let 11 be a bounded open set in Rn with connected complement and which is the interior of its closure. H it satisfies the q.i. (4.1) and its modified Schwarz potential (i.e. the function V furnished by Theorem 4.1) is positive on 11\ supp 1', then 11 is the unique bounded open set satisfying (4.1). As an interesting application of this, recall the q.i. for an ellipse, proved in Chapter 3:

f

(4.24)

10

holds for u E

-1

L1 (11)

where 11 is the elliptic lamina

11 = {(x, y) E R2 : (x/a?

(4.25) and a 2

u du = 2ab 11 (1 - x 2)1/2 u( x, O)dx

-

+ (y/b)2 < I}

b2 = 1 (so the ellipse has its foci at {-I, I}).

COROLLARY 4.10. [Sakai, 1982]. H 11 is a bounded open set ::) {-I, I} such that (4.24) holds, for all u E Ll(11) then 11 is given by (4.25). PROOF: By Theorem 4.9 we need only verify that the m.S.p. of the ellipse (4.25) is positive in the ellipse, outside the segment [-1,1], and that is guaranteed by Proposition 4.8. REMARK: This proof extends to ellipsoids in Rn. A q.i. analogous to ( 4.24) holds where n

(4.26)

11 = {x E R

n :

"[)xj/aj?

< I}

j=1

and al (4.27)

~

a2

~

... > an>

In

o.

Namely,

udx = JUdI',

Vu E LU11)

4.3 SOME APPLICATIONS

35

where I' is a certain positive measure supported in the set n-l

K:= {x ERn:

Xn

= 0

and

L (a~ - a~)-l x~ ~ 1}

j=l

the so-called "focal ellipsoid" of O. (For more details, see [Khavinson and Shapiro, 1989a].) Observe that K lies in the symmetry plane {xn = O} of 0, so Prop.4.8 is applicable and shows that the ellipsoid (4.26) is uniquely characterized among bounded open sets by the q.i. (4.27). There is another proof of Cor. -4.10 that is more elementary, based on reasoning from [Gustafsson, 1989]. Namely, suppose 0 is the elliptic lamina (4.25) and D any other bounded open connected set satisfying the same q.i. (4.24) as O. The first part of the argument is general and works in Rn. Let VD, Vo denote the corresponding m.S.p., and reason as in the proof of Theorem 4.7. We find that VD - Vo attains a positive maximum M at some point of Rn. Now, however, we won't assume that Vo > O. Instead, observe that (since ~(VD- Vo) = XD-XO), VD- Vo is subharmonic on D. If it took the value M in D, by the maximum principle it would be = M on all of D, and this is a contradiction since the set (aD)\O is non-empty and at points of this set VD - Vo is zero. Hence, the maximum is taken at a point y in O\D, and 0 = grad (VD - Vo)(y) = - grad Vo(y). Thus, we shall have arrived at a contradiction if we show grad Vo(x)

(4.28)

i= 0, 't/x

E

0\ supp 1'.

We'll only complete the proof in R2. We have to show that the m.S.p. of the ellipse (4.25) has no critical points in 0\ [-1, 1]. In view of the relation between the m.S.p. and the Schwarz function discussed in Chapt.1, we have to show: The equation S(z) = z, where S is the Schwarz function of 00 (0 being given by (4.25)) has no solution in 0\ [-1, 1]). This, however, is immediate: S satisfies the identity

so if S(z)

= z,

then z

= x + iy

satisfies

(x2/a 2) + (y2/b2) = 1

36

4. QUADRATURE DOMAINS: BASIC PROPERTIES, I

=

i.e. only points z on the ellip"e can satisfy S(z) z (indeed, this is true even for the multiple-valued function S after analytic continuation along any path in C\ {-I, I}).

4.4 Subharmonic quadrature domains. If 11 is a q.d.w.s. (i.e. satisfies (4.1» it has a m.S.p. V which may or may not be non-negative on 11\ supp J.I.. Sakai was the first to realize the significance of this distinction, and discovered the important consequences for uniqueness questions when V ~ 0, as we have illustrated in the last Section. He showed that, when J.I. is a measure

(a)

Vex)

~

0

on

11\ supp J.I.

is equivalent to

(b) for all "ubharmonic /unction" u integrable on 11 and continuous on "upp J.I.. That (b) '* ( a) is rather easy, choosing as test function in (b), u(x) = En(x - y) with y E 11\ supp J.I.. In the other direction, one requires a non-trivial approximation theorem which we won't enter into here, and refer to Sakai's book. In view of the implication (a) '* (b), and the results of the last Section, we get e.g.

(2/1r)

11

-1

(1 - x 2 ) u(x, O)dx

110

~ 1111- f

u

dO'

for u subharmonic inside the ellipse 11 of (4.25), i.e. a "ub-mean value property for ellipse", and the analogous inequality in Rn is also valid. A q.d.w.s. for which (a) (or what is the same, (b) )holds shall be called a subharmonic quadrature domain. It is easy to give examples of q.d. that are not subharmonic q.d. For example, as shown in Chap. 3, there is a q.d. whose associated J.I. is a finite sum of point masses. If one of these is negative (as can well happen, even if there are just 2 point masses) the domain can't be a subharmonic q.d. Suppose, e.g. that the q.i. is

4.4 SUBHARMONIC QUADRATURE DOMAINS

for harmonic u, and that el < we should have

o.

37

Then, if 0 were a subharmonic q.d.

(4.29) for continuous subharmonic functions u. But, taking for for

u(z) = { log Iz - zll loge

Iz - zll ~ e Iz - zll ~ e

in (4.29), we get a contradiction for sufficiently small positive e. However, even if 0 is a q.d.w.s. for a positive measure I-' it can fail to be a subharmonic q.d. In this case Theorem 4.9 is not applicable, and indeed 0 may then not be the only q.d. corresponding to 1-'. We give the following simple, instructive example, also due to Sakai. Let 0 be the open unit disk in the complex plane, and 0 the annulus {(5/12) < Izl < (13/12)}. Note that 101 = 101 = 7r. There is a uniquely determined number e such that

(4.30) holds for u E

1

L1 (0),

13/12

5/12

r dr

Indeed, the integral on the left equals

121r

u(re it ) dt =

1

0

13 / 12

(A log r

+ B)r dr

5/12

for some constants A, B (depending on u) that are independent of r, so (4.30) holds if

1

13/12

(4.31)

loge = 2

r log r dr.

5/12

It is easy to compute that (3/4) < e < 1. Let now I-' denote the normalized arc-length measure on {Izl = e} so that (4.32)

38

4. QUADRATURE DOMAINS: BASIC PROPERTIES, I

holds for u E L1 (0). Thus, 0 is a q.d.w.s. with respect to J.I., and so is n. It follows that the m.S.p. of 0 takes negative values, or what is equivalent, the formula

does not hold for all subharmonic integrable functions on OJ these assertions can be checked by simple computations, which we leave to the reader.

Chapter 5 Quadrature Domains: Basic Properties, II

5.1 Regularity of the boundary. The simplest regularity result is the following two-dimensional one. THEOREM 5.1. Let n be a simply connected plane domain, Zo E an and suppose for some e > 0 there is a function S holomorphic on N := n n De, where De = {z : Iz - zol < e} continuously extendable to N and satisfying S(O = (, V( E annD e • Let cp map the unit disk o of the w-plane confonnally on n. Assume cp extends continuously to points of aD sufIiciently close to w = 1, and cp(l) = zoo Then cp extends analytically to a neighborhood of 1. PROOF: S 0 cp is holomorphic at points of D near 1, and satisfies S(cp(w)) = cp(w) for Iwl = 1, w near 1. Let cp* be defined by cp*(w) = cp(w); it is holomorphic in D and extends to points of aD close to 1, and S(cp(w)) = cp*(l/w) holds on an arc of aD containing 1, so by the theorem of Painleve (cf. Proposition 1.4) the left and right hand functions are analytic continuations of one another across this arc. In particular, cp*(l/w) is analytically continuable inward across this arc, so cp*(w) (and hence cp(w)) are continuable outwards, which proves the theorem. In particular, any q.d.w.s. n which is simply connected has this sort of boundary, in view of Theorem 4.5 and the remarks following it. The same applies (by decomposition) if n is finitely connected. Recently in [Sakai, 1989] it was shown that every q.d.w.s. in the plane is finitely connected, a problem that had been open for some time. In the more particular case where n satisfies (4.6) with JL of finite support, a good deal more can be said. It was shown [Aharonov and Shapiro, 1976, Theorem 3] that an is algebraic, more precisely: there is a non-constant polynomial P(X, Y) with real coefficients, irreducible over the complex field, such that P(x, y) = 0 for x + iy E an. Thus, for example, an annulus, or a triangle, cannot be of this type since their boundaries (while algebraic) are given by polynomials that are reducible. The latter, moreover, is not a q.d.w.s. because of Theorem 5.1.

40

5. QUADRATURE DOMAINS: BASIC PROPERTIES, II

[Gustafsson, 1983] showed moreover that P above must have as leading terms a constant multiple of (X2 +y2)m for some mj moreover an must consist of all the real points (x, y) satisfying P( x, y) = 0 except possibly for finitely many. No relults of thu nature are known in n ~ 3 dimensionl. Valency of the Schwarz function. Let us again consider a bounded q.d. n in R2, for which the associated distribution has finite support, or what comes to the same, there is a function S analytic in n except for m poles (counting multiplicity) and continously extendable to all points of an, where S(C) = (. This function has a whole series of remarkable properties, the deepest of which are given in [Gustafsson, 1983, 1988] and [Sakai, 1988a]. In view of the introductory nature of these notes, I won't go into all those here, but just prove one simpler result from [A vei, 1977]. 5.2

THEOREM 5.2.

in

(a) (b)

With

n

and S as above, the equation S(z) = w has

n m

zeroes, if

m- 1

w E

zeroes, if

C\n

wEn.

PROOF: We give a proof based on the concept of Brouwer degree, following [Shapiro, 1987]. (For the basic topology see e.g. [Lloyd, 1978].)

Let C denote the compactified complex plane (Riemann sphere), and consider the map F: C -+ C defined by

F(z) = {S(Z) z

, ,

Note that F is continuous from C to itself. Since C has no boundary the Brouwer degree deg(F,C,w) is constant for w E C. Now, for w = 00 the value of this degree is -m + 1. Indeed, w = 00 is taken on by F exactly m times (counting multiplicities) in n and (because F is conjugate-meromorphic on n) each of these oo-points contributes -1 to the calculation of deg(F,C,oo). Moreover the only solution of F(z) = 00 with z E C \n is z = 00, and since F is the identity map

5.3 VARIATIONAL PROPERTIES OF Q.D.

on a neighborhood of 00, the contribution of z = +1. We conclude that

(5.1)

deg(F, C, w) = -m + 1,

\/w E

00

41

to the degree is

C.

Now, (a) follows because for wEe \n, F(z) = w has exactly one solution z E C \n (the map being the identity near z, hence orientationpreserving and contributing +1 to the degree) so it must have m solutions in n (each contributing -1). Likewise (b) follows, because for wEn all solutions to F( z) = w must lie in n: each contributes -1, so there are m -1 of them. The proof is complete. With a little more work we can prove a more complete result. Let's call a boundary point Zo of n regular if the conformal map cp in theorem 5.1 satisfies cp'(w o ) i- 0 at the point Wo of aD that maps to zoo (This is equivalent to saying that an is, near Zo, a non-singular analytic Jordan arc as defined in Chapter 1.) The alternative is that cpl(WO) = 0 in which case

cp(w) = Zo

+ c(w -

wo?

+ O(lw -

woll)

near Wo for some nonzero constant c (because cp" ( wo) = 0 would be incompatible with univalence in D) and an has a cusp at Zo, which points into n. We can now state: with the notations of Theorem 5.2, S(z) = w has in n: m - 1 zeroes if w is a regular point of an, and m - 2 zeroes if w is the vertex of a cusp on an. The proofs, which follow from (5.1) in a similar way as ( a), (b) above, are left to the reader. &.3 Variational properties of q.d.

Quadrature domains (of the special type considered in §5.2) have an interesting variational property, as the following result (generalizing one from [Shapiro and Ullemar, 1981]) shows.

n and S be as in §5.2, and assume moreover tbe distinct poles ZI, ••• , Zm of S in n are all simple. Let cp be any confonnal map of a domain D in tbe w-plane onto n, witb cp(Wj) = Zj (j = 1,2, ... ,m). H f is bolomorpbic in D and THEOREM 5.3. Let

(5.2)

f'(wj)

= cp'(Wj)

(j

= 1,2, ... m)

42

5. QUADRATURE DOMAINS: BASIC PROPERTIES, II

then

(5.3) Thus, the area of feD) (counting multiplicity if f is not univalent), f satisfying (5.2), is not less than Inl.

for any

PROOF: Let K>. denote the (Bergman) reproducing element at oX i.e. K>. E L~ (n) and

E

n,

(5.4) for all g E L~ (n). If the q.i. for n is

(5.5)

L ~ t, hdu

cjh(zj),

Vh E L! (ll)

then (specializing to h E L~ (n), and using (5.4) and (5.5)) we get (5.6) Thus, the quadrature property of n is equivalent to an identity involving Bergman kernels, an observation due to [Epstein, 1962]. Let us also denote the kernel K>.(z) by K(z, oX). (Observe that our convention, in defining K(z, oX) is that it is conjugate analytic in oX.) If k( w, t) denotes the Bergman kernel of D, then [Bergman, 1970]:

(5.7) Taking here t

= Wj

and using (5.6), we get m

(5.8)

cp'(w) =

L

Cj

cp'(Wj) -1 . k(w, Wj)

j=1

i.e. the derivative of the mapping function i.'J a linear combination of the kwj(j = 1, ... m). This important relation is due to [Avci, 1977, Theorem 8].

5.3 VARIATIONAL PROPERTIES OF Q.D.

43

Our theorem follows immediately. For, assuming as we may that the right side of (5.3) is finite, in the Hilbert space L~ (D), J' -cp' vanishes at WI, ••• , Wm so it is orthogonal to k W1 ' ••• ,kwm and hence (by (5.8)) to cp' so 1If'1I = IIcp' + (J' - cp')11 ~ IIcp'lI, and the proof is finished. Remarks 1. Originally, Aharonovand I elaborated the theory of quadrature domains in the hope that this would help us to solve certain extremal problems in conformal mapping. Thus far we have not been able to complete this program because of technical difficulties, see [Aharonovand Shapiro, 1973, 1974, 1978]. 2) Avci's result embodied in (5.8) gives another proof that a simply connected q.d. (of the special type considered in this section) is the conformal image of the open disk 0 under a rational map, since the r.k. of 0 is k(w,t) = (1/1I'"){1 - tw)-2 and then cp, as determined from (5.8), is rational. To Avci is also due a converse result: if D is any (not necessarily simply connected) domain with reproducing kernel k(w, t) and some single-valued univalent function cp has a derivative that is a finite linear combination of r.k., m

cp'(w) =

(5.9)

L

aj k(w, Wj)

j=l

for distinct points {Wj} C D, then cp(D) =: h E L~ (n) we have

In hdu Iv h(cp(w))lcp'(wWdu =

n

is a q.d. Indeed, if

= (H,cp')

where H(w) := h(cp(w))cp'(w) is in L~ (D) and (,) denotes the inner product in L~ (D). In view of (5.9) we thus have

(5.10)

1 =f hdu

n

~l

o'j H(Wj)

=

f

Aj h(Zj)

~l

where Aj := o'j cp'(Wj) and Zj := cp(Wj), so n is a q.d. (There is an obvious generalization of these results to q.i. where, in the right side of (5.10), we allow functionals like h'(zj), h"(zj), . .. ). It should be stressed that the essential idea here, to relate r.k. and quadrature identities, is due to Davis (see Chapter 14 of [Davis, 1974]).

44

5. QUADRATURE DOMAINS: BASIC PROPERTIES, II

5.4 Other varieties of quadrature domains. In this section we consider briefly, just to illustrate some possibilities, three other notions of q.d. that have been considered. 5.4.1. In this paragraph, we consider plane domains n with rectifiable boundary to which there is associated a distribution J.L with supp J.l C n such that

f f ds = (J.l,J)

(5.11)

Jan

holds for all analytic functions f in n of some suitable class, where ds denotes arc length measure on an. Again, when J.l is a point mass and n a circular disk we have a classical mean value formula. Domains satisfying (5.11) for J.l of finite support seem first to have been considered in [Aharonov and Shapiro, 1978, Appendix 2]. The simply connected case was studied in [Shapiro and Ullemar, 1981] and the general case in [Gustafsson, 1987]. Two natural choices for the class of test functions f in (5.11) are Hl(n) and El(n) (see [Duren, 1970, Chapter 10] for the definitions). For domains with regular boundaries, these classes coincide. The theory of q.d. of type (5.11) is greatly complicated by a remarkable counter-example due (in a slightly different form) to Keldys and Lavrentieff (see [Privalov, 1966]). Namely, there is a Jordan domain n with rectifiable boundary, and a point Zo E n, such that

(5.12)

f

Jan

f ds =

L(an)· f(zo)

holds for every polynomial f, L(·) denoting length (or Hausdorff 1measure), and n is not a disk, in fact there are such n with L( an) = 27r and arbitrarily small diameter. There is, namely, a univalent function <.p in the unit disk 0 such that <.p' is a singular inner function, /<.p/( e it )/ = 1 a.e. (for definitions, as well as references to later constructions simpler than that of K & L, see [Duren, 1979]). We then have for any polynomial f, assuming as we may that <.p(0) = 0 E n,

and since <.p' is non-constant n is not a disk. Thus, if we hope to get inverse theorems of the type of Epstein's theorem 4.6 above, we must

5.4 OTHER VARIETIES OF Q.D.

45

either use a very large class of test functions, like El(n), or else impose stronger hypotheses on n and then require (5.11) only for "nice functions". At bottom these are the same thing, since there are theorems guaranteeing that suitable regularity of an implies "nice" functions are dense in El(n). Without getting into these rather complicated matters, let us only state one rather special result from [Shapiro and Ullemar, 1981]. For simplicity, we'll just deal with the q.i.

(5.13)

f J8n

f ds =

f

Cj

f(zj)

j=1

where {Zj} are m distinct points of

n,

and all Cj=f.O.

THEOREM 5.4. If n is a Jordan domain with a rectifiable boundary, and n satisfies the Smirnov condition (see below) and (5.13) holds for all polynomials f, the conformal map cp of 0 -+ n is a rational function, more precisely

(5.14)

cp'(w) =

m

Pr(w)2 Wjw)2

TI j =1 (1 -

where r ~ m -1, {Wj} cO and P r is a polynomial of degree r with all its zeroes outside o. Conversely, if cp is any function univalent in o such that cp' has the form (5.14), n = cp(O) satisfies a q.i. of the form (5.13) for polynomials f. REMARKS: a) That n satisfies the Smirnov condition means that some (and hence, it can be shown, every) conformal map of 0 on n has a derivative which is outer, in the sense of Beurling. The meaning of this condition geometricl&lly is still not completely understood, but it is sufficient if e.g. an is composed of finitely many non-singular Cl arcs (see [Tumarkin, 1962], [Shapiro, 1966], [Makarov, 1989]). b) It is easy to see that cp' being of the form (5.14) is invariant with respect to the choice of map from 0 -+ n. c) For a more general result see Corollary 3.3 of [Gustafsson, 1987]. d) It is not hard to see that (5.14) is equivalent to: there is a meromorphic function in n (with poles at {cp(l/wj)}) whose value at each point of an is the complez conjugate of a directed unit tangent vector at that point, cf. [Shapiro and Ullemar, 1981]. It is immediate to see that if

(5.15)

46

5. QUADRATURE DOMAINS: BASIC PROPERTIES, II

where T«() is a positively directed unit tangent vector, and G is the boundary value of a function holomorphic in n, then n satisfies a q.i. since for I analytic in n and smooth up to the boundary,

using (5.15) and residue calculus. e) It's easy to see that when r.p satisfies (5.14), r.p has a branch that is analytic on a neighborhood of D. This easily implies that n is a quadrature domain in the wide sense as defined in the preceding chapter, for a certain distribution Jt supported in n (d. [Shapiro, 1987, Theorem 2.5]). Whether Jt has finite support depends on whether r.p is rational or not - both cases can occur. We won't prove Theorem 5.4, however we'll prove the following result which is a fairly simple corollary of it, and whose proof involves the same ideas that are needed to get Theorem 5.4. It is a kind of arc-length measure analog of Epstein's Theorem 4.6 above. THEOREM 5.5. If n is a domain satisfying the hypotheses of Theorem 5.4, and (5.12) holds for all polynomials I, n is a disk centered at zoo PROOF: We may assume Zo = 0 E n and L(an) = 211". Let r.p denote the conformal map of 0 on n with r.p(0) = 0 and r.p'(0) > O. In (5.12) we may take in place of I any function continuous on nand analytic in n, since on a Jordan domain such a function is uniformaly approximable by polynomials (Walsh's Theorem, see [Gaier, 1980].) Now, r.p is a homeomorphism of 0 on n so its inverse function 'l/J is a homeomorphism of n on D. Taking I = 'l/J", with k a positive integer, in (5.12) we get

for k = 1,2, ... hence 1r.pI(eit)1 = 1 a.e. Being an outer function, with r.p1(0) > 0, r.p' is the constant 1 so r.p( w) = w and n is the unit disk. One can prove versions of this theorem also for non- Jordan domains. In view of Theorem 5.5 it is natural to ask: Is there a domain n in R3 other than a ball, homeomorphic to a ball and with a "nice" bo'Undary s'Uch that Jan 'U dS = c'U(O) holds for all harmonic functions

47

5.4 OTHER VARIETIES OF Q.D.

u continuous in n P (Here dS is surface measure on an and C is the area of an.) The answer is: no. This was shown very recently in [Shahgolian, 1990] (and analogous result in Rn). 5.4.2 Quadrature domains for L! ". Another class of domains n which leads to interesting function-th~retic problems is that for which

1 f

(5.16)

fdq =

()

Cjf(Zj)

j=1

holds whenever f E L!,8 (n), the subclass of L!(n) consisting of functions with single-valued integrals, i.e. f which are derivatives of single-valued functions in n. If n is simply connected, then of course L!,,, (n) = L! (n) and we get nothing new, but if it is not L!,8 (n) is a proper closed subspace of L! (n). It is known [Aharonov and Shapiro, 1976, Lemma 2.4] that (5.16) for all f E L! ,,(n) implies: There exists a function H analytic in n except fo~ simple poles at {Zj} and continuously extendable to an such that H«()-( is constant on each connected component of an. Using this, we'll prove the following result, which in a more general form was proved in [Sakai, 1972]. The proof has been rediscovered independently by M. Schiffer and by D. Aharonov. THEOREM 5.6. If n is a bounded plane domain containing is a finite union of continua, and (5.17)

then

n

i

f dq =

Zo

and

an

Inl· f(zo),

is a disk centered at zoo

For those n to which the theorem applies this is, of course, an essential strengthening of Epstein's theorem. PROOF OF THEOREM 5.6: We may assume Zo = O. By the remarks immediately preceding the theorem, if r := an consists of components r 1, r 2, ... , r r there is a function H analytic in n except for a simple pole at 0, and complex constants at, ... , a r such that H extends continuously to I' and

H«()=(+aj,

(EI'j

(j=1,2, ... ,r).

48

5. QUADRATURE DOMAINS: BASIC PROPERTIES, II

Fix now a complex number ..\,1..\1 = 1, and let G.\(z) := 1H(z) + ..\z. Then G.\ is analytic in n except for a simple pole at 0 and for ( E rj, G.\(() lies on the line 1aj +R. Thus, K.\:= G.\(r) is the union of r horizontal segments so C\K.\ is connected (where C is the Riemann sphere) and the Brouwer degree of the map G.\ : n -+ C is constant on C\K.\. The value of this constant is 1 since G.\ takes the value 00 exactly once and so G.\ is a conformal map of n on C\K.\ (this result can of course also be obtained from the argument principle). Consequently, G~(z) is never 0 for zEn, i.e. 1H'(z) + ..\ =/: O. Since ..\ is an arbitrary number of modulus 1 we conclude: IH'(z)1 =/: 1 for zEn. It follows by continuity (since H has a pole at 0) that IH'(z)1 > 1 for zEn. Also, the regularity theorem 5.1 applies to the present situation to show that each r j is a non-singular analytic curve, except for, possibly, a finite exceptional set (of cusps pointing into n) and by a calculation done previously H' (() = T( () 2 at each regular boundary point (. Putting all this together we see that 1/ H' is analytic in n, with a double zero at 0 and no other zeroes. Moreover, it is of absolute value less than 1, and its boundary values are of modulus 1. Hence this function has a single-valued analytic square root F. We have F(n) c 0 (open unit disk), moreover the cluster set of F at each point of an is on aD. Again, by degree theory, the Brouwer degree of the map F : n -+ 0 is constant on 0, and this constant is 1 since F has precisely one zero in n. Consequently F maps n conformally on 0, from which it follows that n is simply connected. Theorem 4.6 now shows n is a disk centered at 0, and the theorem is proved. REMARK: The method employed does not seem to yield anything analogous for (5.16) with m > 1. However, there are some further results for quadrature domains with resped to L!,6 in [Gustafsson, 1983].

an.

5.4.3 Support of J.I. meets As remarked earlier, Sakai and Gustafsson usually work with a definition of q.d. in which J.I. is a measure whose support is allowed to meet Let's give just one example of such a domain, to see what kind of questions can arise. Let n be a triangle, whose boundary consists of oriented line segments The Schwarz function Sj of rj is of the form Sj(z) = ajz + bj for certain complex constants aj, bj. If f is analytic on n, we have

an.

rt, r 2, r3'

(5.18)

L

fda- = (2i)-1

3

~

i

3

j

fidz = (2i)-1

~

i

j

J(z)Sj(z)dz.

5.4 OTHER VARIETIES OF Q.D.

49

Now, since Sj is analytic everywhere, we can by Cauchy's theorem replace the path of integration r j by any other smooth arc with the same endpoints, and lying in O. For example, denoting the vertices of o by Zl, Z2, Z3 (so that r 3 is the segment from Zl to Z2, etc.) and letting Zo be any point of 0, let Cj be a smooth arc in 0 joining Zo to Zj such that C l , C 2 , C 3 are mutually disjoint except for their common endpoint Zoo Then r3 can be replaced in (5.18) by the path C l uC2 (with correct orientation), and similary for r 2, rl. Then (5.18) takes the form (5.19) where J.L is a certain complex measure living on the closure of U~=l Cj (in fact it is of the form (Ajz + Bj)dz along Cj, for suitable complex constants Aj, B j). So we have a great multiplicity of quadrature formula (5.19). However, one of these is distinguished: if Zo is the point where the angle-bisectors of the triangle 0 meet, and Cj is the line segment joining Zo to zj, then the measure J.L appearing in (5.19) turns out to be real. Gustafsson has shown (unpublished) that, subject to certain topological restrictions (so that supp J.L isn't "too massive"), this is the only case of a real measure satisfying (5.19). He has also generalized this to polyhedra in Rn (the role of angle bisectors then being played by hyperplanes bisecting the dihedral angles formed by pairs of adjacent faces). As another interesting example of an identity (5.19) where J.L measure whose support meets ao we have (5.20)

f f du =

10

c

IS

a

f f ds lao

where ds is arc-length measure, c = 101· L(aO)-l, and (5.20) is to hold for, say, all analytic functions f in 0 continuously extendable to O. The higher-dimensional analog is (5.21)

f

10

udx = c

f

lao

udS

for harmonic functions u, where dS is surface measure and c is the volume of 0 divided by the (n - 1 )-dimensional measure of a~. One

50

5. QUADRATURE DOMAINS: BASIC PROPERTIES, II

solution for n is a ball. If n is assumed homeomorphic to a ball, with sufficiently regular boundary, in (5.21) then it is known that n must be a ball. This follows from an argument of [Kosmodem'yanskii, 1981] applied to a theorem of [Serrin, 1971]. For further discussion see [Khavinson, 1987]. 5.5 Existence of q.d. Let us give a few remarks concerning methods to construct, or at least prove existence of, quadrature domains. In the plane, we can get simply connected q.d. by mapping the unit disk conformally by a function

10 f dO'

= (J.l,J)

for some distribution J.l with compact support in n. If moreover

2 variables, the only known methods that give general existence proofs are those by Sakai and Gustafsson (see especially [Sakai, 1983] and [Gustafsson, 1985]). These methods are based on variational inequalities and are close in spirit to those used to solve free boundary problems in hydrodynalnics. Despite the great interest and importance of this method, we won't be able to go into it in these lectures. Basic references are [Sakai, 1983] and [Gustafsson, 1981, 1985, 1989]. 5.6 Conclusion. In concluding this chapter, let us point out the close relation of q.d. with the partial differential equation {)2 u/&z2 = o. This has been

5.6 CONCLUSION

51

developed by Jacqueline Detraz [Detraz, 1988] although she does not use the terminology of q.d. or the Schwarz function. She showed that if is a simply connected domain in C with analytic boundary, there is a solution of this p.d.e. in 0, continuously extendible to a~, vanishing on ao and not identically zero, if and only if the conformal map of 0 on 0 is rational. (Her argument can be adapted to show that a corresponding result holds for multiply connected domains: the failure of uniqueness is then equivalent to 0 being a quadrature domain.) She also gives an example of a Jordan domain 0 whose boundary is a nonsingular Coo curve, analytic except at one point (0, and whose Schwarz function is meromorphic in 0 with poles clustering at (0. This corresponds to a quadrature identity like (3.4) with infinitely many points

n

{Zj}.

Chapter 6 Schwarz ian Reflection, Revisited

6.1 Reformulation in terms of harmonic functions. To better understand Schwarzian reflection, and the prospect of extending it to Rn, let us first give a formulation in R2 that does not explicitly involve conformal mapping, nor even holomorphic functions. The following is a generalization of Prop. 1.3. PROPOSITION 6.1. Let D be a plane domain part of whose boundary is a non-singular analytic arc 0', and suppose that the real-valued hannonic function u, continuous on the closure of D, vanishes on 0'. Then, there is an open set N C R2 containing 0' (except possibly for its endpoints) such that (i) u extends hannonically into DuN and (ii) for points zED sufficiently near 0', the extended function (still denoted by u) satisfies u(Ra(z)) = -u(z), if Ra(z) E N. (Here Ra is the anti-conformal reflection associated to 0', Section 1.3). Moreover, N depends only on n (not on u). It is easy to deduce Prop. 1.3 from this. Indeed, in proving Prop. 1.3 we may assume D, E are simply connected and map each of these conformally on the upper half-plane. This leads easily to a reduction of Prop. 1.9 to its special case where E = J(D) is the upper halfplane. But then u = 1m J vanishes on 0' so we may apply Prop. 6.1. We deduce, first that 1m J is harmonically extendible to DUN, and therefore J is holomorphically extendible to DuN, and the extended function (still denoted by 1) satisfies 1m J(Ra(z)) = - 1m J(z) for points zED near 0'. But -1m J(z) = 1m J(z), so J(Ra(z))-J(z) has vanishing imaginary part and, being conjugate analytic, is constant. The constant is 0 as we see by taking z E 0', so J(Ra(z)) = J(z) = Rr(J(z))j here Rr(w) = W is the anti-conformal reflection associated to 8E, the real axis of the w-plane. Thus, Prop. 6.1 implies Prop. 1.3.

6.1 REFORMULATION IN TERMS OF HARMONIC FUNCTIONS

53

Prop. 6.1 can be proven in various ways. Note first that it consists of two, essentially independent, assertions. First, that 1.£ extends harmonically into the larger domain D U Nj and secondly that this extension is related by a functional equation to the given 1.£ in D. The first assertion is a special case of a principle valid very generally, e.g. for any solution 1.£ of a linear elliptic equation with real-analytic coefficients, in a domain D C Rn, that is continuously extendible to a portion (7 of 8D which is a non-singular real-analytic hypersurface, and vanishes on (7. This follows from "regularity up to the boundary" estimates for elliptic equations. More specifically, under the stated hypotheses, if is a point in the relative interior of (7, and {xi} c D, xi -+ there are constants A, B depending only on D and the coefficients of the equation such that

e,

e

(6.1) holds for all j and all multi-indices a. If j is large enough this shows 1.£ extends holomorphically to a ball in en centered at xi, whose intersection with Rn extends beyond D, and provides the asserted extension of 1.£ (which then necessarily satisfies the same elliptic p.d.e. in the extended domain). We won't give a proof of (6.1), which is fairly cumbersome. References are given e.g. in [Courant-Hilbert, 1962, p. 347]. As to the second assertion, that is much more specialj it is intuitively not plausible that there should be something corresponding to "anticonformal reflection" with respect to general analytic hypersurfaces in Rn for n > 2, since as is well known, there are relatively few conformal maps in these spaces. In [Khavinson and Shapiro, 1989b] this is discussed in more detail. We shall return to this shortly. First, however, let's prove Prop. 6.1. PROOF OF PROP. 6.1: To simplify the proof we'll accept (from the p.d.e. literature) that our hypotheses imply that all 8 0 1.£, lal ~ 2, extend continuously to a neighborhood N of (7, that is independent of u. Now, for ZED, Z near (7, R/7(z) lies in N\D (by Prop. 1.2) and -(uoR/7) is harmonic in N\D and has the same boundary values on (7 as u(z) (namely, 0) so it is a good candidate to be a harmonic extension of u. The crucial thing is to verify (A) The first partial derivatives of 1.£ and of -(1.£ oR/7) match on (7. Indeed, assuming this, and letting 1.£# denote the harmonic extension of 1.£ into DuN, 1.£# and -(1.£ 0 R/7) are harmonic on N\D and have

54

6. SCHWARZIAN REFLECTION, REVISITED

the same "Cauchy data" on the boundary arc tT, i.e. the functions and their gradients coincide on tT and so, by Holmgren's uniqueness theorem, are identical. So, we have only to check (A). It is convenient notationally to write

R tT : (x,y)

f-+

(P(x,y),Q(x,y»

where P, Q are real and P + iQ = RtT(x + iy), and to let '1.£2 := ~;. We have to show that (with P z := 8P/8x, etc.)

hold on

tT,

0= 8z (u(x,y)

+ u(P,Q»)

=

'1.£1

+ '1.£1 Pz + '1.£2 Qz

0= 8y (u(X,y)

+ u(P, Q»)

=

'1.£2

+ '1.£1 Py + '1.£2 Qy

'1.£1

:= ~:,

i.e. that

(6.2) (6.3)

Py '1.£1

+ (1 + Qy)U2 =

0

hold on tT. Now, on tT the functions '1.£, P - x, Q - y all vanish, hence their gradients are proportional and we have (6.4) (6.5) on 0'. But since R is anti-conformal we have identically P z = -Qy, Py = Qz' Using these relations in (6.4), (6.5) we obtain (6.2), (6.3) and this proves (A), and hence Prop. 6.l. REMARK: We could reduce the input of "elliptic regularity theory" in the above proof to: '1.£ and its first partial derivatives extend continuously to 0'. In this case, the above calculations show that the function obtained by piecing together '1.£ and -('1.£ 0 R tT ) is in C 1 (N) for some neighborhood N of 0', and harmonic on N\O'. The proof can then be concluded by a "removable singularities" lemma analogous to Prop. 1.4. This is, again, a general result so we'll state it in Rn.

6.2 STUDY'S INTERPRETATION OF SCHWARZIAN REFLECTION

55

CI(B), where B is an open ball in RR, and harmonic on B\r, where r is a non-singular CI hypersurface. PROPOSITION 6.2. Let u bein

Then u is harmonic in B. This proposition can be proved in the same way as Prop. 1,4, so we'll just sketch the argument. The crucial thing is that the analog of Lemma 1.5 holds in the setting of Prop. 6.2. This is because a function in CI(n) which, together with its first partial derivatives, extends continuously up to a portion r of an which is a nonsingular CI hypersurface (indeed, even the "Lipschitz graph" regularity of r suffices) can be Cl_ extended across r, [Smith, 1971, Ch. 17]. Thus, by the technique we employed in proving Lemma 1.5 (since each :~u. is in C(B)nCl(B\r» 1 the distributional derivatives aiajU can be computed "piecewise" to be the locally integrable functions 8!2;~i on B\r. Since ~u = 0 on B\r, the distributional Laplacian of u in B vanishes, and then Weyl's lemma gives the harmonicity of u in B. Although Prop. 6.2 is very well known, and often useful, it is not easy to find it stated in the literature. One finds in [Carleson, 1967] results of similar character, but apparently not this one. 6.2 Study's interpretation of Schwarzian reflection. An elegant geometric interpretation of anti-conformal reflection was sketched in [Study, 1907], and we give an account of this, with more details, in the present section (see also [Khavinson and Shapiro, 1989b]). The key idea is that anti-conformal reflection, and Schwarzian reflection of harmonic functions vanishing on an analytic curve (as given by Prop. 6.1) have simple geometric interpretations when we look at the picture in C2 rather than in C ~ R2. This is precisely analogous to the situation in classical geometry, where geometrical properties of "real" figures become more simple and unified when we extend those figures from R2 to C2. To give a familiar illustration: consider two circles in the plane, neither of which encloses the other. Then there are points from which the tangents to the two circles have equal lengths. The locus of such points is immediately seen to be the line passing through the points of intersection of the circle, if they intersect. But what if they do not? The answer is, if we consider the coordinates in the Cartesian equations of the circles to be complex rather than real numbers, there are still two (finite) intersections in C2, and the (complex) line joining them still meets R2 in a line, which is still the desired locus ("radical axis" of the circle pair).

56

6. SCHWARZIAN REFLECTION, REVISITED

Since this section is included mainly for its aesthetic appeal, we introduce some simplifying assumptions that alter nothing essential in the discussion. Thus, suppose we have a real algebraic curve r in the plane, described by the equation cp(x, y) = where cp is a polynomial with real coefficients, and irreducible over C. To ensure r is nontrivial we'll assume cp(O, 0) = and grad cp(O,O) does not vanish. Let now u be a real-valued harmonic function in some "sufficiently large" neighborhood of (0,0), say in DR : {x 2 + y2 < R2}. Then it is well-known that u extends analytically to a function in the ball 1 2 B of radius 2- / . R about in C2, where it is holomorphic and satisfies the partial differential equation

°

°

u

°

(6.6) where X, Y denote complex variables: X = x + ix', Y = y + iy' with x,y,x',y' real. There is a holomorphic function fez) in DR whose real part is u/2, so u(x, y) = f(x

+ iy) + f(x + iy) =

where g, defined by g(z) =

(6.7)

fez)

u(X, Y) = f(X

f(x

+ iy) + g(x -

iy)

is holomorphic in DR. We thus have

+ iY) + g(X -

iY)

for (X, Y) E B, where f,g are holomorphic in DR. For the next part of the discussion we require a basic geometric figure in C2, well known in classical projective geometry. Let (Xl, Yt) be a point of C2 and consider the pair of (complex!) lines through (Xl, Y 1 ) defined by (6.8)

Rl := {(X, Y) E C 2

:

(6.9)

B1 := {(X, Y) E C 2

:

X

+ iY =

Xl

+ iYd

X - iY = Xl - iYd.

We have thus through the given point a "red" line Rl with direction numbers (1, i) and a "blue" Bl with direction numbers (1, -i). In classical geometry these lines often occur (usually drawn from points of R2) where they are variously called "minimal", "isotropic", "the null

6.2 STUDY'S INTERPRETATION OF SCHWARZIAN REFLECTION

57

circle (X - X l )2 + (Y - Yi)2 = 0", or "the lines joining (Xl, Yd with the circular points at infinity" (cf. [Struik, 1953]). We may also remark that these lines have a completely different role as the complex bicharacteristics of the differential operator appearing in (6.6) ("complexified Laplacian"). Now, choose a second point (X3, Y 3 ) distinct from (Xt,Yd and consider the analogous lines R 3,B3 through it. Then Rl meets B3 in a (finite) point (X2' Y 2 ) and Bl meets R3 in a (finite) point (X4' 1'4). These four points are then the vertices of a quadrilateral which, to have a convenient name, we'll call a Study quadrilateral. It is clear that the vertices numbered 2 and 4 are given by:

(6.10) (6.11) Assuming that the whole quadrilateral lies in B we then get, using (6.7)

+ U(X3' Y3) = I(Xl + iYd + g(Xl - iYl ) +I(X3 + iY3) + g(X3 - iY3) = I(X2 + iY2) + g(X4 - iY4) +I(X4 + iY4) + g(X2 - iY2) = U(X2' 1'2) + U(X4' Y4). U(Xl' Yl

)

Thus, the extension to C2 of a harmonic function satisfies the functional equation (6.12) where (Xj,Y;) (j = 1,2,3,4) are the successive vertices of a Study quadrilateral contained in the domain of holomorphy of Suppose now that u happens to vanish on r. Then (because

u.

u

r := {(X, Y) E C

r

2 :

r


Suppose now two opposite vertices (## 1 and 3) of a Study quadrilateral lie in R2, in DR and the other pair lies on Then the right side of (6.12) is 0, and u(Xj,Y;) = u(Xj,Yj) for j=1,3 so we have:

r.

58

6. SCHWARZIAN REFLECTION, REVISITED

PROPOSITION 6.3. H u is hannonic in a sufIiciently large neighborhood

of an arc of an irreducible algebraic curve r in R2 and vanishes on r, then at a pair of points of R2 which are opposite vertices of a Study quadrilateral, with the remaining vertices on u takes values that are negatives of one another. Moreover, if the initial pair is sufIiciently close to r, these points are one another's anti-conformal reiIections with respect to r.

r,

This is Study's version of the reflection principle. Let us hasten to add that it is not necessary that r be algebraic (and indeed, Study does not assume this), we have only made this assumption for simplicity (in discussing irreducibility). In a local version, it is easy to check that it suffices for r to be non-singular and analytic. It remains only to check that the initial pair are the anti-conformal images of one another, in the sense of Chapter 1. To see this, let (x}, YI) denote a point of R2 near r. The "red" line through it has the equation X + iY = Xl + iYI. Let it intersect in (X2, Y2) and let the "blue" line through (X2 ,Y2) meet R2 in (X3, Y3). Thus

t

(6.13) (6.14) Moreover, if S denotes the Schwarz function of r, the equation of can be written locally as X - iy = S( X + iy), and that of as

t

X - iY = S(X

(6.15)

r

+ iY).

Thus, (from (6.14»

+ iY2) S(XI + iyt}

= S(X2 =

(from (6.15), since (X2' Y2 ) E t) (from (6.13».

t

In like manner, if (X4, Y4) denotes the intersection with of the "blue" line through (Xl, YI) and (a, b) the intersection with R2 of the "red" line through (X4' Y4) we have X4 X4

-

iY4 =

Xl -

iYI

+ iY4 = a + ib

6.2 STUDY'S INTERPRETATION OF SCHWARZIAN REFLECTION

iYI

SO Xl -

= X. -

iY.

59

= S(X. + iY.) = S(a + ib),

or writing R = S, we have

+ iY3 = Xl + iYI = X3

R(XI + iYI) R(a + ib).

Assuming all points in the equation lie in a neighborhood of r, sufficiently small so that R is an involutive homeomorphism (cf. Prop. 1.2), the last equation implies R( Xl + iYI) = a + ib, hence a + ib = X3 + iY3 and we have shown that (Xl, YI) and (X3, Y3) are one another's anticonformal reflections in r. Moreover they are opposite vertices of a Study parallelogram, the other two vertices of which are on Because they are real points, we see that we can get from one to the other either by first travelling by the "red" line to t, then the "blue" back to R2j or the reverse procedure. Both lead from (Xl, YI) to the same point

t.

(X3, Y3). Of course, this argument is only correct "near r". Moreover, in general the "red" line through (Xl, yt) meets in many points that can serve as (X2 ,Y2) above, each of which yields a different branch of the anti-conformal reflection of (Xl, YI) in r (in general, Schwarz functions are multiple-valued). In any case, if the intersections of the "red" line through (Xl, YI) with are distinct (equal in number to the degree of cp) all branches of the Schwarz function near (Xl, YI) are holomorphic. It is not hard to see that if a "red" line through (Xl, YI ) is tangent to (which, because of the reality of the coefficients of cp implies a corresponding behaviour for the "blue" line) some branch of S has a singularity at (Xl, yd. We also get a singularity if the red line at infinity, since then there is no "blue" line through the point meets of contact and Study's construction breaks down. In terminating this Section, we make a few observations. First, note that Schwarz ian reflection extends naturally to holomorphic solutions of (82w/8X2) + (8 2w/8y2) = 0 that vanish on a non-singular "curve" i.e. holomorphic variety of complex dimension 1, in C 2, as Study's construction shows. Secondly, Study's argument is based essentially on the fact that + e~, the "symbol" of the two-dimensional Laplace operator, splits into linear factors over C. Since + ... + e! is irreducible over C for n ~ 3 it is clear that we cannot expect the above results to generalize in a straightforward wayj and indeed, Schwarzian reflection in Rn, n ~ 3 is very limited [Khavinson and Shapiro, 1989b]. One

t

t

t

t

a

a

60

6. SCHWARZIAN REFLECTION I REVISITED

aspect of this is presented in the next Section; the presentation is from an early version of the paper just cited.

6.3 Failure of Schwarz ian reflection in RS. If u is harmonic in a domain n of Rn that has a hyperplane H as a portion of its boundary, and u is continuously extendable to H and vanishes there, it extends harmonically to interior points of n U (H nan) u n', where n' is the reflection of n in H. Moreover the extended function satisfies the functional equation u(x) + u(x') = 0 where x, x' are "mirror images" of one another with respect to H. If in place of H we have a hypersphere 5, a similar result holds. For example if 5 is the unit hypersphere, the corresponding functional equation (discovered by Kelvin) is u(x) + Ixl 2- n u(x') = 0 where x' := Ixl- 2 x is the inversive image of x with respect to 5. These results are easily established using the technique of §6.1. This, plus what we know of reflection in analytic curves in R2, might lead one to suppose that analogous functional equations hold for other analytic hypersurfaces. However, this is not so. Here we shall only show the failure of Schwarzian reflection with respect to circular cylinders in RS.

u

Let r := {x = (Xt,X2,XS) E RS : x~ + x~ = I}. Tbere is a neigbborbood N of r sucb tbat for any two points a = (aI, a2, as), b = (bl , ~, bs ) in N\r tbere is a function u barmonic in N, vanisbing on N n r and satisfying u(a) = 0, u(b) = l. PROPOSITION 6.4.

Thus, no functional equation of the type Au(a) + Bu(b) = 0 can exist for the class of u in question, with A and B not both zero. PROOF OF PROP. 6.4: First, observe that if as f. bs the function uo = (xs - as) In( x~ + x~) is harmonic off the axis of r, vanishes on r and at a, and does not vanish at b. Taking a constant multiple of Uo, we obtain the desired function u. Thus, assume without loss of generality that as = bs = o. Let us switch to polar coordinates and write a = (rl,Ol), b = (r2,02). If 102 - 011 f. 0,1r, then taking u = (r - ~) sin(O - Od, u(a) = 0 and u(b) f. o. If 102 - 011 = 0 or 1r, but rlr2 f. 1 then we look for u(r,O) = cl(r _r- l ) cOS(O-Ol) +c2(r2r- 2 ) cos [2(0 - Odl, where Cl,C2 are determined from the system

(6.16)

{

u(rt, OI) =

u(r2' ( 2)

=

cl(rl - rIl) + c2(r~ - rI2) = 0 ±cl(r2 - ril) + c2(r~ - ri 2 ) = 1

6.3 FAILURE OF SCHWARZIAN REFLECTION IN

R3

61

choosing + or - according to whether 192 - 91 1 = 0 or 7r. Since in both cases the determinant D of (6.16) equals (rl - r t l )(r2 - ri l )[(r2 + ril) T (rl + r t l )] #= 0 because r1r2 #= 1, (6.16) has a unique solution. Moreover, if r2 = r t l but 192 -91 1= 7r,D #= 0 and hence, in that case (6.16) also has a unique solution. Finally, let us assume that rl = ri l , 91 = 92 = O. Look for U = ul(r, 9)X3(X3) where ut,X3 are regular functions, ul(1,9) = O. Then,

o=

~U =

X3 ~Ul +UIX~'

implies that

(6.17)

~Ul

=

X~'

=

const

= -A 2 < O.

Ul X3 Writing Ul(r,9) = R(r)T(9) we obtain from (6.17) (6.18)

(T( 9) must be periodic with period 27r.) The left hand side of (6.18) is the well known Bessel equation with parameter A. Hence, its regular solution is given by

(6.19)

In(Ar)(Acos n9 +Bsin n9)

(A,B E R).

Here, In(x) = E~o k!r~;~~+k) (x/2)2k+ n denotes the Bessel function of order n. As R(l) = 0, A must be equal to a root of In(x) = O. We may take for Ul any linear combination of terms (6.19); we shall take the swn of terms with n = 1, B = 0 and A = a, f3 respectively, getting

Ul (r, 9) = (Cl J l ( ar) + C2 J l (f3r)) cos 9, where a,f3 are distinct roots of Jl(x) = 0 and Ct,C2 are to be determined by

{ ul(rt, 0) = clJl(art) + c2 J l(f3rt) = 0 ul(r2,0) = ul(r t l , 0) = clJl(a/rt) + C2Jl(f3/rt) Asswning this system has a unique solution, we find (6.20)

U = Cl J l (ar) cos 9 eQZa + C2Jl (f3r) cos 9

= l.

e{Jx a ,

u(a) = 0, u(b) = l.

Now, the determinant of the system (6.20) is analytic in r1 and hence is nonzero for rl sufficiently near 1, unless it vanishes identically, so the proof of our theorem is reduced to the following simple assertion.

62

6. SCHWARZIAN REFLECTION, REVISITED

The function ~(z) = J1 (az)J1 (f3/z) - J1 ({3z)h(a/z) where a and {3 are distinct complex numbers does not vanish identically.

LEMMA 6.5.

PROOF OF LEMMA 6.5:

:=

Observe that the meromorphic function, F(z) 00. Now, if ~ == 0

J 1 (az)/J1 ({3z) has an essential singularity at

then

F(z)

J 1 (a/z)

= J1 ({3/z) = F(l/z)

and, therefore, F has an essential singularity at o. However, any meromorphic function F satisfying F( z) = F( 1/ z) must be a rational function, since for each w in the Riemann sphere, the number of w-points in {Izl > I} of F equals the number of w-points in {Izl < I}, hence is bounded by a number independent of w, so 00 is a regular point or pole of F. But F is not rational, having infinitely many zeroes. The proof is now complete.

Chapter 7 Projectors From L2(80) to H2(80) 7.1 Introduction. This and the next chapter are devoted to topics in one-variable complex analysis where the Schwarz function finds applications. 0 shall denote a bounded domain of finite connectivity in C whose boundary r is everywhere smooth (at least, to start with; a Holder-continuous unit tangent vector certainly suffices for our needs, but we won't discuss here the question of minimal smoothness assumptions). 7.2 The Hilbert operator of a plane domain. Let F denote a Holder-continuous complex-valued function defined on r, and consider the "integral of Cauchy type" (7.1)

{211"i)-1

J{~

-

z)-1 F{()d(.

This defines a pair of analytic functions, one (denoted li{Z» in 0 and one (denoted le{z» in Oe := (:\0. Note that Oe may have several components. It is well known (see, e.g. [Muskhelishvili, 1968]) that both Ii and Ie extend continuously to r. Note also that le{oo) = O. Moreover, for (0 E r the limits (7.2) and (7.3) exist and satisfy the "Plemelj-Sokhotski relations"

+ (211"i)-1 P.V.

i {( i«-

(7.4)

Fi{(O) = (1/2) F«(o)

(7.5)

Fe«o) = -(1/2)F«0) + (211"i)-1 P.V.

(0)-1 F(Od(

(0)-1 F(Od(

64

7. PROJECTORS FROM L'(80) TO H'(80)

where P. V. denotes "Cauchy principal value", i.e. lim e-O

f

J1'-'I>e

... d(.

Therefore

(7.6) We can thus state PROPOSITION 7.1. With the above-stated regularity assumptions concerning r and F, F can be expressed as Fi - Fe, where Fi, Fe are m C(r), and: (a) Fi is the boundary value of a function holomorphic in n. (b) Fe is the boundary value of a function holomorphic in ne and vanishing at 00. Moreover, Fi and Fe are the unique elements of C(r) satisfying (a), (b) and F = Fi - Fe.

We have only to check the uniqueness and, in view of linearity it suffices to consider F = O. But, if Fi - Fe = 0, then the functions in ni, ne whose boundary values are Fi, Fe respectively combine to form a single entire function which, vanishing at 00, is O. Hence Fi = Fe = O. From the theory of singular integrals it is known that the principal value integral in (7.4) has the same qualitative behaviour as the Hilbert transform, e.g. it maps LP(r; ds) -+ LP(r; ds) for 1 < p < 00 and preserves the class of functions Holder-continuous of order A > 0 (for references see [Garcia-Cuerva and Rubio de Francia, 1985, p.225]). In particular the maps taking F 1--+ Fi and F 1--+ Fe defined by (7.4), (7.5) initially for smooth F, extend continuously to L2(r) and we define in this way bounded linear maps from L2(r; ds) to the closure (in L2(r; ds» of the smooth functions which are boundary values of functions holomorphic in n, and analogously to the closure of the smooth functions which are boundary values of functions holomorphic in ne and vanishing at 00. We denote these closures by H2(r) and H;(r) respectively, so that H2(r) is just the (boundary values of) the Hardy space H2 associated to n, [Duren, 1970]. We have thus the direct sum decomposition (writing henceforth L2(r), or L2 for L2(r; ds» :

(7.7)

7.2 HILBERT OPERATOR OF A PLANE DOMAIN

65

and correspondingly, a contin'Uo'U3, linear idempotent map (projector) from L2(r) --+ H2(r) which we shall denote by H, and call the Hilbert operator (or projector). Note that it is, in general, an "oblique" projector, i.e. not self-adjoint or, in other words, the decomposition (7.7) is not orthogonal. The orthogonal (or Szego) projector from L2(r) --+ H2(r) shall be denoted by S. In [Kerzman and Stein, 1978] the relation between H and S was studied. Although their main program concerned several complex variables, they also studied the one-variable case, and specifically, how to express S in terms of H. The idea is that H can, in principle, be computed directly from its singular integral representation (because the "Cauchy kernel" is directly available), whereas S cannot; conceived as a singular integral operator, it is based on the Szego kernel which is not known a priori. This is an important theme in function theory: to compute, or estimate "deep" entities (like Szego or Bergman kernels, Riemann maps, harmonic measures, solutions of boundary value problems) in terms of "naive" ones that are easily computed such as Cauchy integrals, potentials, or geometric quantities like lengths, angles, areas. The classical Neumann-Poincare problem was of just this character, to compute the solution of Dirichlet's problem in terms of double-layer potentials (which one "just writes down"). PROPOSITION 7.2. [Kerzman and Stein, 1978]. The Szego projector can be expressed in terms of the Hilbert projector by

(7.8)

S = H [I + (H - H*)]-1

where I denotes the identity operator on L2(r). PROOF:

(7.9)

(7.10)

We have the immediate relations

HS=S,

SH* = S,

SH=H

H*S = H*.

Hence S(H - H*) = H - S, whence Sri + (H - H*)] = H. Since the bracketed operator has its spectrum on the line {Re A = I} it is invertible, and that gives (7.8).

66

7. PROJECTORS FROM L2(8n) TO H2(8n)

Now, H is expressible in tenns of a singular integral and so, therefore, is its adjoint H*. Kerzman and Stein made the important observation that in the integral kernel representing the operator H - H* the "singular" parts cancel and we are left with an integral operator having a smooth kernel (in particular, H - H* is compact), and that makes possible in principle effective modes of attack for computing numerically the inverse operator in (7.8). We refer for details of this to their paper as well as [Kerzman and Trummer, 1986], and shall instead follow up another of their results. THEOREM 7.3. [Kerzman and Stein, 1978]. H H = 5, i.e. if H self-adjoint, n is a circular disk.

1S

The converse is evident: when n is a disk H2(r) and H:(r) are mutually orthogonal. Actually, in the cited paper very strong regularity hypotheses are imposed: n is assumed simply connected, and r is COO. Our main purpose in this chapter is to prove the following generalizations. THEOREM 7.4. H, for some Zo E orthogonal to H2(r), n is a disk.

n,

the function z

1--+

(z - zo)-l is

This implies Theorem 7.3 because if H is self-adjoint, every function in H:(r) (in particular (z - zO)-l, restricted to r) is orthogonal to

H2(r). THEOREM 7.5. H H - H* is of finite rank,

n is a

disk.

The proofs will be given in Section 7.4. First, however, I want to discuss some related matters. 7.3 Relation to the Neumann-Poincare problem. The "Hilbert operator" of a domain, as defined in §7.2 is closely related to the Neumann-Poinca.re-Fredholm solution of the Dirichlet problem in tenns of a double-layer potential. If F is a (for the moment) real-valued smooth function on r, its double-layer potential is defined for z E C\r by

(7.11) where a/an denotes differentiation in the direction of the outer nonnal. This defines a pair of harmonic functions Ui, U e in n, ne respectively.

7.3 RELATION TO THE NEUMANN-POINCARE PROBLEM

67

It's easy to see that the above integral equals

Re (211'i)-1

i «-

Z)-1 F«)d(.

Thus, Ui(Z) has boundary values on r given by Re (HF). Consider a smooth complex-valued function F + iG on reF, G real). The boundary value (from n) of its double-layer potential is, denoting by C the operator of complex conjugation: DOW

Re (HF)

+ iRe (HG)

=(1/2)[HF + CHF + iHG + i(CHG)] =(1/2)(H + CHC) (F + iG). Hence, for any complex-valued smooth function on r the boundary values from n of its double-layer potential are obtained by applying to it the operator (1/2)(H + CHC). Thus, an L2(r) version of the Dirichlet problem is: given F on r, solve (7.12)

(H

+ CHC)G =

2F.

The double-layer potential of G is then harmonic in n and has boundary values F. The essence of Fredholm's solution of the Dirichlet problem (which we are here, for convenience, looking at in the context of L2(r) rather than the usual c(r), but that is not really essential) is then that (7.13)

H+CHC=I+K

where K is a compact operator (so that the Fredholm-Riesz theory applies to I + K). When n = 0, it is easy to check that H+CHC maps each FE c(r) to F + (211')-1 Jo27r F(eit)dt, so the K in (7.13) is just the rank one operator of orthogonal projection onto the constants. Conversely: THEOREM

7.6. If the rank of H

+ CHC -

I is finite,

n

is a disk.

The special case of rank one was also obtained by D. Khavinson (unpublished). Theorem 7.6 will be proved in the next Section. Before

68

7. PROJECTORS FROM L2(80) TO H2(80)

turning to it, we wish to remark that the solution to the Dirichlet problem in two dimensions, embodied in (7.12), doesn't require explicit reference to potentials, only to the operator H (or what is the same, to Cauchy's integral). This gives a rather neat way to "package" Fredholm's solution of the Dirichlet problem. Of course, one must first show (under suitable hypotheses on S1) that K in (7.13) is compact, and this is not trivial, and we'll assume this result (of Fredholm) here. A rather simple computation shows that, if r o, rt, ... r m denote the components of r (with ro outermost) the kernel of H + CHC consists of functions equal to 0 on ro and to some complex constant Cj on each rj(j = 1,2, ... ,m). In particular H + CHC (which is a Fredholm operator of index 0) is injective when r = ro (i.e., S1 is simply connected) and hence surjective. Thus, every FE L2(r) is the sum of a function in H2(r), and the complex conjugate of one, so it is the boundary value of a harmonic function in S1. H m ~ 1, Fredholm-Riesz theory tells us H + CHC has codimension m. If we choose a point Zj inside the contour rj (j = 1, ... , m) the functions {log Iz - zjlli=1.2 .... m are linearly independent modulo ker (H + CHC), so every function in L2(r) can be expressed as a function of the form Ei=l a j log 1Z - Z j 1 and one in the range of H + CHC, leading also in this case to the solution of Dirichlet's problem.

7.4 Proofs of the preceding theorems. We now turn to the proofs of the preceding theorems, in which the Schwarz function plays an essential role. Perhaps the simplest proof is that of Theorem 7.6, so we'll start there. PROOF OF THEOREM 7.6: To bring out the idea in its simplest form, suppose first that S1 is simply connected and K := H + CHC - I has rank one. (Note that since K does not annihilate constant functions its rank is always ~ 1.) Multiplying on the left by H in H +C H C = 1+ K gives HCHC = HK, hence HCHC has rank ~ 1, and so = 1. For any f E H2(r), HCHC] = Hf. Now, the function ] = a + hz, for suitable choice of a,b in C (not both 0), annihilates HCHC, so a+hz E ker H, which means there is a function 9 in H2(S1 e ) satisfying g(z) = a + hz for z E r. Hence b :I 0, otherwise 9 = a on r and hence in S1 e , so g( 00) = a :I 0, a contradiction. Now g(z) = O(lzl- 1) near 00, so g(z)(a+bz) =: h(z) is holomorphic in S1 e and bounded, and regular at 00. Its boundary values on r are la + bzl 2 , so 1m h is harmonic in S1e and zero on r = aOe whence it

7.4 PROOFS OF THE PRECEDING THEOREMS

69

°

is == and h is constant. Hence la + bzl 2 = C for z E r where C > 0, and so r must be a subset of the circle {Iz + ab-11 = c1/ 2 ·lbl- 1}. For topological reasons, r must be the whole circle, so 0 is a disk and the proof is finished in this case. H 0 were not simply connected, Oe would consist of several components and the above argument would only show that h is constant in each component, so h(z) = Cj for z E rj (j = 0,1, ... m) where r 0 is the outermost component of r and r 1, ... , r m (where we may suppose m ~ 1) are the remaining components. Thus ro is a whole circle, and each rj is an arc of a circle concentric with roo (Possibly that of least radius, say r m, is the whole circle; for topological reasons rj, with 1 ~ j ~ m - 1 are proper arcs.) Now, the presence of a boundary component that is a proper arc could be excluded by simply saying, we demanded more regularity than that (i.e. continuously turning tangent); but there is no need to resort to such low tactics, we can exclude this case as follows. Fix any fo E H2(r), then reasoning as before with a + bfo instead of a + bz, we conclude that a + bfo(z) is, for z E r, the boundary value of a function holomorphic in Oe. Let Zl, Z2 be distinct points on the same component of a~, satisfying IZll < 1, IZ21 < 1, and let fo be a branch of [(z - zJ)/(z - Z2)P/3. This is in H2(O). There is no loss of generality in assuming ro is {Izl = 1}. For zEro, fo(z) = [(1 - zlz)/(1 - Z2Z)]l/3 and this function is not holomorphic in Oe, so we have a contradiction to the assumption that ro is not all of a~. Finally, let's suppose rank K = k ~ 1 (and hence, rank HCHC ~ k). Consider n:::;=oCjzj : Co, ... ,Ck E C}. By linear algebra there is a choice of {Cj}, not all 0, such that the E lies in the kernel of H (reasoning as above), so with this choice of {Cj}, c.p := E;=o Cj zj satisfies: ~Ir is a boundary value of H;(r). Moreover Cj i= for at least one j ~ 1 (as before). Now we play the same game with linear combinations of the functions z, z~, ... zc.pk and conclude: there are complex constants ao, ... ,ak (not all zero) such that z(ao + al ~ + " .ak~k)lr is a boundary value of H;(r). Thus, in view of what we know about c.p, zlr is the boundary value of a function meromorphic in Oe. Call this function S(z). Now, S can have no poles in Oe. Because, if S has a pole at Zo E Oe, then E~=l CjSj must have a pole there. But this function coincides on r with
°

70

7. PROJECTORS FROM L2(80) TO H2(80)

This finishes the proof of Theorem 7.6. PROOF OF THEOREM 7.4: Let T«() denote an oriented unit tangent vector to 'Y at ( (encoded as a complex number), so that T( 0 = d( / ds where ( runs along r and s denotes arc length. The assumed orthogonality can be written

0= (7.14)

=

i

i

I«()(' - %o)-lds

I«()T«()«( - %o)-ld(

for all I E H2(r). This implies (see Remark 1 following): there is a function 9 E H2 (r) such that

(7.15) Now, suppose r is a non-singular analytic curve (or finite union of such) - we shall justify this assumption later - and let S denote its Schwarz function. (In case r has more than one component S nonetheless exists and is holomorphic on a disconnected neighborhood of r.) From (7.15),

g«()2 = T«()2 «( _ %0)-2 = S'«() (S(O

_ %0) -2,

(E

r.

Thus, (d/dz) (S(z) -%0)-1 coincides on r with the boundary values of the function g2 holomorphic in o. Let G denote a primitive function of g2 in 0, i.e. G' = g2. Then, the above shows that G is .~ingle-valued since Jc G' (z) dz vanishes around every closed curve sufficiently close to a~. Since (S(z) - %0)-1 = G+ const., we see that S has a singlevalued analytic continuation throughout 0 except for finitely many poles, and S( z) - %0 has no zeroes. By Theorem 5.2 S has then exactly one pole in 0, so by the reasoning used in proving Theorem 4.6, 0 is a disk, and the theorem is proved, modulo the following. REMARKS 1: The above proof used the following (well known) proposition: II F E L2(r) satisfies

(7.16)

7.4 PROOFS OF THE PRECEDING THEOREMS

71

then F E H2(r) (the converse is also true, and easily checked). To prove this proposition, let F = Fi - Fe be the decomposition of Proposition 7.1. Then from (7.16), (7.17)

«( -

for every f E H2(r). Choose f«() = z)-l where z E Oe. Since the first term on the right of (7.17) vanishes,

The last integral equals, by Cauchy's integral theorem, (-1) times the function holomorphic in Oe that has Fe as its boundary value. Hence Fe = 0, and F = Fi E H2(r). 2: Let us now show how the analyticity of the boundary can be deduced from the hypotheses of Theorem 7.4. It is clearly enough to show: if 9 is analytic in 0, continuously extendible to 0, and

g«() = T«() then

r

«( - zo)-t,

(E

r

is analytic.

PROOF: If 0 is multiply connected, let ro be the outermost component of r. We'll show it is analytic. ( A similar argument can be given for the other components; of course if 0 is simply connected, r = r o.) Let z = c.p( w) be a conformal map from 0 on the bounded domain 0 0 with boundary roo Now,

T«()

= d(/ds = (dc.p/dt) + (ds/dt)

where ( = c.p(e it ) is a point of ro(t E R) and s is the arc-length parameter on r o. Thus,

Without loss of generality, we may take Zo = 0, and then

72

7. PROJECTORS FROM L2(OO) TO H2(80)

It follows from our hypotheses that this function on the right is the boundary value of a function analytic in {p < Iwl < I} for some p < l. Our task therefore reduces to showing: If cp is analytic and univalent in D, smoothly extendible to D and

is the boundary value of a function analytic in {p is analytically continuable across all of aD.

< Iwl < I},

then cp

We're going to assume r is sufficiently smooth so that cp'(e it ) is continuous and non-vanishing. Then F( eit)cp( e it ) is continuous on aD and has modulus 1, and being the boundary value of an analytic function in {p < Iwl < I} it is analytically continuable across all of aD. Hence, there is some G holomorphic on a neighborhood Nt of aD satisfying PROOF:

(7.18) Also, F(eit)cp'(eit)/cp(eit) is real on aD, so F(w)cp'(w)/cp(w) is analytically continuable across aD and there is a function H holomorphic on a neighborhood N2 of aD satisfying (7.19) Combining (7.18), (7.19) gives that H/G which is meromorphic in Nt n N2 coincides on aD (and hence in Nt n N 2) with cp' /cp2. Since this function can't have poles on aD, we see that (d/ dw)( cp-t) is analytically continuable to a neighborhood of aD, and hence so is cp-t, and cp itself, which concludes the proof. As to Theorem 7.5, the proof is a hybrid of the methods used in proving Theorems 7.4 and 7.6. Since no really new ideas enter we won't give further details here.

7.5 A property of the Szego projector s. One can ask whether the analog of Theorem 7.6 holds, with H replaced by S. Here, however, we have

7.5 A PROPERTY OF THE SZEGO PROJECTOR S THEOREM 7.7. For every m ~ 1 there is a bounded domain

that S + csc - I has rank m.

73

n

such

We'll present the proof, and a fuller discussion, elsewhere. The domains in question, it turns out, are precisely "quadrature domains" of the type considered in §5.4.1 where integration is over the boundary, with respect to arc length. Just to indicate the ideas involve we'll sketch the proof in one direction. Suppose that rank (S + esc - I) = m. Then, the rank of S times this operator is ~ m : rank (SeSC) ~ m, or rank (SS#) ~ m where S# := cse is the orthogonal projector on CH2(r) (the conjugate analytic functions). Now, there are complex numbers Co, Cl, ••• Cm not all 0 such that (co + Cl Z + ... c m Z m) Ir is in ker (SS#) and hence in ker S, i.e. P(z) := Co + ... + Cm zm satisfies P . f ds = 0 for all f E H2. Clearly deg P ~ 1. Let ZI, ... , Zr (r ~ m) be the zeroes of P in n (multiple zeroes counted multiply). Then one can show (with some slight hypotheses of boundary regularity on r) that, given g E H2 (r) vanishing at {z j }, there is a sequence {Ii} c H2(r) such that liP --+ g in H2(r). Hence, g ds = 0 for every g vanishing at {ZI, ... , zr} (with multiplicity p if Zj appears p times) and this is equivalent to n admitting a q.i. of the type considered in §5.4.1.

Ir

Ir

Chapter 8 The Friedrichs Operator

8.1 Introduction. In this chapter 0 shall denote a connected open set in the complex plane (later to be subjected to further regularity conditions)j as earlier, du = dx dy denotes planar Lebesgue measure and LP(O) = LP(Oj du) the usual Lebesgue space, and L~(O) its closed subspace consisting of the holomorphic functions in LP(O). In [Friedrichs, 1937] that author studied the bilinear form

(S.l)

(j,g)

1-+

Re

l

J(z)g(z)du

on L! r (0) X L! r (0), where L! r (0) denotes L! (0) regarded as a vector'space ove; R. (Note that is not conjugated in (S.l)!) With suitable regularity hypotheses on ao he derived certain bounds for the form (S.l) as well as spectral properties. His motivation was that the form (S.l) is closely related to boundary value problems for the biharmonic equation, and hence to certain questions of planar elasticity. (This connection is developed very sketchily in the cited paper, and some further discussion may be found in [Norman, 19S7].) While Friedrichs develops spectral theory in terms of quadratic Jor'Tn3 a la Hilbert and Hellinger - Toeplitz, I prefer to formulate the question in terms of an operator which is essentially of the type nowadays called a Hankel operator. The reason for including this material in the lectures is that it bears an intimate relation to the Schwarz function and quadrature domains.

9

8.2 The Friedrichs operator. Let P = Po denote the orthogonal projector of L2(0) on L!(O), and (as earlier) C the map "complex conjugation" on L2(0). Then the Friedrichs operator associated to 0 is defined as

(S.2)

T=To =Poci

L~(O)

.

8.2 THE FRIEDRICHS OPERATOR

75

In other words, Tn f = Pnl for f E L!(n). Observe that Tn is continuous, and real-linear, but not complex-linear since

Tn(>' J) =

(8.3)

1. Tn f,

>. E C.

Henceforth we'll usually suppress the subscript n when there is no danger of confusion and write simply P or T. From (8.3) we see that T2 i3 complex-linear. The following are easily verified, where <, > denotes the inner product in L2(n), i.e. < f,g > = f'9 dO' :

In

J

(8.4)

< f, Tg > =

(8.5)

=

(8.6)

< CF,G >=< CG,F >

f gdO'

(f,g E L~

(n»)

(f'9EL~(O») (F,G E L 2 (n»)

< T2 f,g >=< PCPC f,g > = < CPC f,g >=< Cg,PCf > = < PCg,PCf >, so

Consequently,

(by (8.6»

Hence T2 is a linear, positive (i.e. non-negative; hence in particular self-adjoint) operator on L!(n) of norm ~ 1. The norm is certainly 1 if n is of finite area since then constant functions belong to L!(n). T2 has also a geometrical interpretation. Since Q := C PC is the orthogonal projector from L2(n) to CL!(n) (the conjugate-analytic functions), PQP is a self-adjoint operator on L2(n) having L!(n) as an invariant subspace, and its restriction to this invariant subspace is T2. Thus the spectral properties of T2 reflect the geometry of the pair ofsubspaces L!(!l) and CL!(n) within L2(n). Here we shall only give a few results. For the purposes of this chapter we'll say 00 is smooth at one of its points Zo if, for some e > 0, r(zo; e) := (00) n D(zo; e) is a Jordan arc with a Holder-continuous oriented unit tangent vector (where D(zoi e) denotes {z : Iz - zol < e}), and n n D(zo; e) lies to one side of r( Zo; e).

76

8. THE FRIEDRICHS OPERATOR

THEOREM 8.1. [Friedrichs, 1937]. H 0 is bounded and ao is smooth at each of its points, To is compact. That is, if {In} C L~(O) and In -+ 0 weakly, then liTo In II -+ o.

We'll deduce this later from some more general results, based on [Shapiro, 1981]. Since that paper is rather inaccessible, we'll give the details.

8.3 Weak· limits of sequences in L!(O). In this section we consider the isometric embeddings (8.8)

where 0 denotes the closure of 0, and M(O) is the Banach space of bounded complex measures on 0, conceived as the dual of the Banach space C(O) of continuous complex-valued functions on o. In this manner L! inherits from M(O) a weak* topology which, for sequences, takes the form (for notational reasons, we denote by I du the image of IE L!(O) in M(O) under the imbedding (8.8»: In du -+ I du

(8.9) {:=:}

(weak*)

10 In u du 10 lu du, for all u -+

E C(O).

When no confusion is to be feared, we express (8.9) simply by saying In -+ I (weak*). Let now 0 be any bounded open set, r = a~, and define r reg to be the set of points of r in whose neighborhood r is smooth in the sense defined in §8.2. Observe that r reg is relatively open, and so the set rirr := r\rreg, the set of "irregular boundary points of 0" is compact. THEOREM 8.2. Let Un} C L!(O),

J..I.

E M(O), where 0 is any

bounded open set, and suppose (8.10)

Then, there exists

In du -+ dJ..l.

IE L!(O) and v

(weak*). E

M(O) such that

(8.11)

dJ..l. = Idu +dv

(8.12)

suppv C

rirr.

8.3 WEAK· LIMITS OF SEQUENCES IN L~(O) COROLLARY

77

1. H rjrr is empty, the unit ball of L~(n) is weak* closed

in M(n). REMARK l: If T denotes the circle, the analogous proposition that the unit ball of the Hardy space HI (T) is weak* closed in M(T) is equivalent to the F. and M. Riesz theorem [Duren, 1970, p.4l]. REMARK 2: If an is smooth except for a "corner" at Zo of angle a 1= 7T' or 27T', then it is easy to show there exist Un} C L~(n) such that {In dO'} tends weak* to 6(zo)' Suppose, for simplicity, that Zo = 0 and n is contained in the wedge Wa := {z : 0 < (J < a} (where z = re i8 , and n n 0 = Wa n D. Let g(z) = (z - .\)-m where .\ E C\W a and m ~ 3 are chosen so that 9 E L~(Wa) and fw", gdO' 1= O. It's easy to see that this is possible if a is not 7T' or 27T' (note that if a = 7T', i.e. Wa is a half-plane, fw", 9 dO' vanishes for all 9 E L~(Wa». If now

cp E c(TI) we have, writing In(z) = n 2 g(nz), that In E L~(n) and

{ I/n(z)ldO' ::; {

in

iw",

I/n(z)ldO' = {

iw",

Ig(z)ldO'

and

which says that Un dO'} tends in the weak* topology of M(n) to a non-zero constant times a 6-mass at O. COROLLARY

2. H nrr is empty, the following are equivalent for Un}, I

in L~(n) : (i) In -+ I (weak*) (ii) supn II In II < 00 and In(z)

-+

I(z) lor each zEn.

PROOF: That (i) => (ii), for any n, is easy and left to the reader. Suppose now (ii) holds. It is enough to show that any subsequence of {In} has, in turn, a subsequence which converges (weak*) to f. But, any subsequence of {In} has a further subsequence which converges weak* to some p. E M(n), and by Corollary 1 this p. may be identified with an element of L!(n), dp. = gdO' for some 9 E L!(n). Now, this subsequence converges pointwise to g(z), and so 9 = I, which completes the proof.

78

8. THE FRIEDRICHS OPERATOR

More generally, for any bounded open for {In}, I in L!(n) when supn II In II

n

the following are equivalent

< 00 :

(a) In(z) -+ I(z), all zEn (b) 10 Incpdu -+ Io/cpdu for all cp E C(n) vanishing on rirr. The proof is nearly identical with that of Corollary 2, hence omitted. To prove Theorem 8.2, we require a lemma.

LEMMA 8.3. Let u E Cl(n) and u(C) = 0 for all , E an. Assume moreover that u vanishes on a neighborhood of rirr. Then ~ E A(n), where (8.13)

A(n) := {u E Loo(n) :

In ul du

= 0,

all f E L!(n)}

PROOF: The hypotheses imply that u satisfies a Lipschitz condition

on

n.

(8.14) Extend u to all of R2 by taking it to be 0 outside n. Then, it is easy to see that the extended function (still denoted by u) satisfies (8.14) for all z}, Z2 in R2. In particular, u belongs to the Sobolev space W 1 ,P(R2) for all p < 00. Since also supp u c n and u vanishes on a neighborhood of the irregular boundary points, it is easy to deduce that u E W~,p(n). Henceforth we shall take p = 3. Thus, there is a sequence {CPn} in cOX'(n) such that CPn -+ U in the norm of Wl,3(n). n -+ ~ in L3(n), and so for all 9 E L3/2(n) This implies

8li

(8.15)

lim n-+oo

f

10

g(z) a:,n du vZ

= f g(z): duo 10

vZ

Choosing, in particular, g(z) = (z-a)-l where a is any point in C\n, we have 9 E L:/2(n) and, since I g8!Jin du = 0 for each n, (8.15) tells us that I 9 ~ du = O. Hence I ~ f du = 0 for every rational function f with simple poles, all outside of n, and since [Bers, 1965] these functions are dense in L!(n), and ~ E Loo(n), this last relation

8.3 WEAK· LIMITS OF SEQUENCES IN Ll(O)

79

extends to all IE L!(n), that is to say, ~ E A(n) and the lemma is proven.

In

In

8.2: By hypothesis In r.p du --+ r.p dJ1. for all r.p E C(n). In particular, choose here r.p = ~ where .,p E C8"(n). Then, since I In~ du = 0 for all n, we get that I ~dJ1. = 0 for all such .,p. Let us now write J1. = J1.i + v where J1.i is the restriction of J1. to n, and supp v c an. Then I ~dJ1.i = o. Considering J1.i as a distribution in n, this says = o. By "Weyl's lemma" this implies that the distribution J1.i is identified, in the canonical way, with a holomorphic function I on n. Since J1.i is a bounded measure, we have I E L!(n). Hence we have In du --+ I du + dv (weak*). Thus, v E M(an) and we have 9n du --+ dv (weak*) where 9n := In - / E L!(n). To complete the proof, we have to show supp v c rjrr. Let (0 denote any point of rreg. It follows from the definition of regular point that there is a function v E Cl(~), where ~ is some neighborhood of (0 in R2, such that v(e) = 0 for all ( Ern ~ and grad vko =f:. o. We may take v real, so the last relation is equivalent to ~ ko =f:. o. Multiplying v by a "cut-off function" in coo(R2), with support in ~ and equal to one on a neighborhood of (0, we may assume that v is defined and of class CIon all of R2 and vanishes on all of r and on a neighborhood of rjrr. Finally, let P denote any polynomial in x, y with real coefficients, and set u = Pv. By the lemma, ~ E A(n). Hence PROOF OF THEOREM

W

Since supp vCr and v vanishes on r, the first integral on the right vanishes and we have I P ~ dv = o. This holding for all polynomials P, the measure ~ dv vanishes. Since ~ =f:. 0 on a certain neighborhood of (0, supp v cannot meet this neigborhood. Since (0 was any point of r reg, supp v n r reg = 0 and Theorem 8.2 is proven. Implicit in the above is an approximation theorem which may be of some interest (the case n = unit disc was proved by [Reich, 1976] using Fourier series): Let Q be any bounded open set, and let C#(r) denote the set 0/ functions in C(r) which vanish on rjrr. Then, the restrictions to r of functions in C(Q) n A(Q) are dense in C#(r).

COROLLARY 3.

80

8. THE FRIEDRICHS OPERATOR

is any measure on r which annihilates those restrictions then, as was shown in the proof of Theorem 8.2, v is supported on rjrr, and hence annihilates elements of C#(r). Hence, by a standard duality argument, the result follows. PROOF: If

V

We can now easily prove Theorem 8.1. Suppose, then, that fl satisfies the hypotheses there and Un} C L!(fl) is a sequence tending weakly to o. We must show

IIT/nll-+ o.

(8.16)

There is no loss of generality to assume T/n =1= o. Define IIT/nll-1 so that 9n E L!(fl), 119nll = 1 and (T/n,9n) = that, using (8.4)

9n = T/n . liT/nil, so

(8.17) Now, hn := by II· lit)

/n 9n

is in L!(n) and (denoting the norm in that space

for some constant C. Moreover, for each Zo

En

so {h n } tend to zero pointwise in n. By Corollary 2 to Theorem 8.2, {hndO'} -+ 0 (weak*) in M(n), hence

so, in view of (8.17), (8.16) holds and Theorem 8.1 is proven. REMARK: It's easy to see that Theorem 8.1 remains true if we allow cusp singularities on an which point into n (i.e. angles of 271").

8.4 The Friedrichs operator, geometry, and the Schwarz function. Under the hypotheses of Theorem 8.1, T2 has a sequence of eigenvalues 1 = AO > Al ;:::: A2 . .. decreasing to zero, and a corresponding

8.4 FRIEDRICHS OPERATOR, GEOMETRY, SCHWARZ FUNCTION

81

sequence of eigenfunctions {CPn(z)}n>O which form an orthonormal basis for L~(n) and are easily seen ~ satisfy the remarkable "double orthogonality" relations (8.18)

(8.19) Clearly ).0 = 1, corresponding to CPo = constant. There may be degeneration in the sense that 0 is an eigenvalue of T2, even of infinite order. In the case of the disk we have in fact rank (T2) = 1 and ).1 = ).2 = ... = o. In general, ).1 is a measure of the "angle" between L! and C L!. It is characterized by

(8.20) where sup

IS

over

As was already indicated, these eigenfunctions enable one to write down solutions of the basic boundary value problems for the biharmonic equation. Friedrichs also studied the essential spectrum of T2 in cases where T2 is not compact. [Norman, 1987] showed that this essential spectrum contains 0 if an is smooth near at least one point. Moreover, extending analysis of Friedrichs, he showed that if an consists of finitely many smooth arcs joined so as to for~ "corners" with interior angles Ql, ••• Q n

then each of the numbers 1sm Qj 12 is in the essential specQj

trum of T2. Thus, the spectral properties of T2 reflect the geometry of n in an interesting way. Norman also investigated the injectivity of T2, and obtained the following result.

n be a bounded domain whose boundary is smooth near two of its points (1, (2. Suppose that an coincides near these points with nonsingular analytic arcs 1'1,1'2 having

THEOREM 8.3. [Norman, 1987]. Let

82

8. THE FRIEDRICHS OPERATOR

Schwarz functions S10 52. H 51 and 52 are analytically continuable along paths in n into some neighborhood where they do not coincide, then T2 is injective. Thus, for example, if an contains two segments of different straight lines, T2 is injective. We won't give the proof here, because it involves technical difficulties concerning boundary behaviour, but shall outline a heuristic argument which indicates the main idea of the proof. So, suppose n satisfies the hypotheses of Theorem 8.3 and that I is a nontrivial function in ker T2. In view of (8.7), T I = 0 and that means, by (8.4) (8.21)

10 I gdu

= 0,

lor all 9 E L!(n).

a

By virtue of Lemma 4.2, or more accurately its analog for the operator, there is a function 11. E W~,2 (n) satisfying Bu/az = I in n. Thus (a/az)(u - zJ) vanishes in n, and so for some 9 holomorphic in n,

(8.22)

u(z) = zl(z) - g(z)

in

n.

If we can deduce that I = 0 we'll now have a contradiction, which would prove the theorem. So far, the analysis has been rigorous, but now we'll argue heuristically. Since 11. "vanishes on an " we get from (8.22) (il I and 9 had boundary values in a suitable sense this would be rigorous, but functions in L~(n) don't in general have boundary values, which is why we use the adjective "heuristic"): z = g(z)/I(z) on an. Thus, if r l is any analytic arc of an its Schwarz function has a meromorphic continuation throughout n given by 9 / I. This continuation is the same for r 2, and our hypotheses now imply the sought-for contradiction. Similar reasoning shows that if an is a single analytic arc, its Schwarz function extends meromorphically throughout n, and n is a quadrature domain. As we'll see in the next section, T2 has finite rank in a q.d., so we get the surprising result: if an is a non-singular analytic Jordan curve, and T2 is not injective, then T2 is of finite rank. Thus, if T2 maps any nontrivial function to 0, it maps almost everything to

O! 8.5 The Friedrichs operator and quadrature domains. In [Shapiro, 1984] the following theorem was proved:

8.5 FRIEDRICHS OPERATOR AND Q.D.

83

8.4. Let n be a planar open set whose boundary consists of IS finite number of continua. Then, the following assertions are equivalent: (i) n is a quadrature domain, in the sense that

THEOREM

for all h E L~(n). Here the non-negative integers numbers Cij and points Zj En do not depend on h.

rj,

complex

(ii) The "Friedrichs operator" To is of finite rank. It is of interest that in a quadrature domain (even one "in the wide sense", in which case it needn't be of finite rank) the Friedrichs operator has a property much stronger than compactness, as given by the following result from [Shapiro, 1987]. The open set n is a q.d.w.s. if and only if there exists a compact set Ken and constant C such that the Friedrichs operator T of n satisfies THEOREM 8.5.

We refer to the cited papers for the proofs of the last two theorems. The last inequality implies exponential decrease of the eigenvalues of

T2. It seems there are numerous problems of interest connected with relating spectral properties of T6 to the geometry of n. These problems can of course be generalized, in the spirit of "Hankel operators", by introducing a "symbol function" cp E LOO(n) into the definition of T, i.e. considering the operator f 1-+ Po MtpC f where Mtp denotes multiplication by cpo Similary, one can introduce Friedrichs-like operators in the context ofthe "boundary spaces" L2 (r; ds) and H2 (r; ds) that Were studied in the last chapter.

Chapter 9 Concluding Remarks 9.1 The Schwarz potential in en. We first encountered the Schwarz potential as attached to a nonsingular real-analytic arc in R2 (a concept which immediately permits generalization to a real-analytic "hypersurface" (i.e. of dimension n -1) in Rn). We have seen that if the arc happens to be a closed curve bounding a lamina of uniform density in R2, the singularities of the Schwarz potential are precisely those encountered by the logarithmic potential of this lamina, considered initially in the exterior of the lamina (where it is harmonic) when it is extended harmonically inwards across the boundary of the lamina; the nature of this extension was the theme of the "Preisaufgabe" that Herglotz studied in his paper which we have often cited. Once we re-interpret this problem (as we have done, and which Herglotz never did) in tenns of a Cauchy problem (for the Laplace equation) we are outside the usual territory of potential theory and in the domain of partial differential equations, especially the question of propagation of singularities for solutions to Cauchy problems. This subject has been extensively studied for (linear) hyperbolic equations in Rn, less so for elliptic equations. Since the solutions of an elliptic Cauchy problem, in case the partial differential equation is linear with real-analytic coefficients, and the initial manifold in Rn as well as the Cauchy data are real-analytic, extends always some distance into en, it is natural from the beginning to place the action in en rather than Rn in the elliptic case, i.e. to study the question in the framework of the holomorphic Cauchy problem, and that is what we shall do in this Section. It is a remarkable fact that the singularities (in Rn) of the Schwarz potential are much better understood when seen from the vantage point of en, as we'll learn shortly. Concerning the holomorphic Cauchy problem in general, there still does not seem to exist an adequate global theory. The Cauchy-Kovalevskaya theorem gives the local existence and uniqueness theory near non-characteristic points of the initial manifold. The local theory of singularities that "arise" at characteristic points of the initial manifold is the subject of the seminal paper [Leray, 1957]. Later, this

9.2 THE ELLIPSE, REVISITED

85

theory was reworked and extended in [Garding, Kotake and Leray, 1964] and other works. As to global results, much is known when the Cauchy data is given on a hyperplane. (e.g [Miyake, 1981], [Persson, 1971], [Johnsson, 1989]) but almost nothing when it is given on a more general (even an algebraic) manifold. For the Laplace operator, there are some rudimentary results in [Khavinson and Shapiro, 1989a] and a fairly complete treatment for data on quadric surfaces in [Johnsson, 1990], where ideas of Leray et al are adapted and extended into a global context. The following discussion is very sketchy, intended only to illustrate some of the ideas that come into play.

9.2 The ellipse, revisited. As an illustration of the "cn viewpoint" (here with n = 2) let us seek to "explain" why the Schwarz potential of the ellipse has singularities at the two foci, and nowhere else in R2. We'll take as our ellipse (9.1)

r:= {(x,y)

E R2: Ax2 +By2

= I}

where A = a- 2 ,B = b- 2 and a > b> o. The foci of r are at ±c, where c = (a 2 - b2 )1/2. Along with r we'll consider its "complexification"

(9.2)

r:= {(X, Y)

E C 2 : AX 2

+ By 2 = I}

where X = x + ix', Y = y + iy' and x, y, x', y' are real. From here on we'll take for granted familiarity with the Cauchy-Kovalevskaya theorem (C-K theorem) and related notions like "characteristic point", etc. (cf. [Garabedian, 1964]). The (modified) Schwarz potential of r is the (unique) solution to the Cauchy problem

r

(9.3)

.6.v = 1 near

(9.4)

v and grad v vanish on

r.

Since there are no characteristic points on r, v exists as a real-analytic function in a neighborhood of r by virtue of the C-K theorem. Hence it extends to a holomorphic function on some neighborhood N (in

v

86

9. CONCLUDING REMARKS

v vex, Y) solves the holomorphic (meaning

e2 )

of r.1t is clear that = here: in e 2 ) Cauchy problem (9.5)

(9.6)

v,

av

ax

and

av

ay

vanish on

t n N.

v

t

Now, is holomorphically extendible along every path on that does not go through a characteristic point. For a complex one-dimensional non-singular variety ("curve") in e 2 defined by cp{X, Y) = 0 where cp is holomorphic near (XO, yO) and vanishes at (XO, yO), the condition for this point to be characteristic for the operator in (9.5) is

(9.7) Hence, the characteristic points on tions

t

are gotten by solving the equa-

(9.8)

(9.9) This shows there are 4 characteristic points on independent choices of sign in

t,

corresponding to

(9.10)

v

t

Now, analytic continuation of along (so that the Cauchy data remains 0, as prescribed by (9.6» must lead to a singularity when we reach a characteristic point. This is a general phenomenon (also in en), and can be seen as follows. Suppose first we are in a neighborhood M of a point of cp(X, Y) = 0 where never simultaneously vanish. (In the present situation, cp = AX2 +By2 -1, but that special and its gradient vanish on it's choice plays no role.) Then since

U,

t:

v

-u.

r,

9.3 PROPAGATION OF SINGULARITIES, AN EXAMPLE

87

easy to see V =
(9.11)

v

where all terms in ( ... ) contain the factor <po If now were analytically continuable along a path in t that crosses a characteristic point, f would be continuable along this same path. But, from (9.5) and (9.11)

2[(!;)2+(;;?] ./=1

must hold along this path, and this gives a contradiction at a charac1vanishes there. teristic point, since [ The upshot of this is: it develops a singularity at each characteristic point 0/ t. The next insight is that these singularities propagate in some manner throughout C 2, in particular they can be expected to propagate to R2 where they will turn up as singularities of the original Cauchy problem (9.3), (9.4). But, how do the singularities propagate? In general this is one of the deepest questions in the study of P.D.E. In order to gain some insight into it, let's put the present discussion "on ice" (and return to it in §9.4) in order to look at a very simple and transparent example.

9.3 Propagation of singularities, an example. In this paragraph, let r denote a real-analytic non-singular Jordan arc in R2, defined by an equation
(9.12)

(9.13)

-- = 1

oxoy

v,ov/ox

and

nearr

ov/oy

vanish on r.

(This is, apart from a complex-linear change of variables, the same Cauchy problem as in the last Section, but easier to grasp geometrically since we can draw a picture in R2.) Characteristic points on rare those where (a

88

9. CONCLUDING REMARKS

vanishes. Suppose (0,0) is such a point, and acp/ax vanishes at (0,0) while acp/ay does not. Thus, in a neighborhood N of (O,O),r can be represented by the equation y = f(x) where f is real-analytic on an intenJal about x = 0 and f(O) = /,(0) = o. It's easy to see that the most general solution to (9.12) holomorphic on a neighborhood of a non-characteristic point (xo, Yo) of r n N is of the form

v=xy-A(x)-B(y)

(9.14)

where A, B are holomorphic on a neighborhood of (xo, yo). From (9.13) we get (9.15)

y - A'(x)

= 0,

x - B'(y)

=0

on

r n

N.

Now, (0,0) is the only characteristic point of r in N. Just as in the last Section, it's easy to see that v can't be analytically continued along r from (xo, Yo) across (0,0), so as to be real-analytic on a neighborhood of (0,0). For in that case, A and B would be continuable analytically to a neighborhood of o. From the second relation in (9.15)

x = B'(y) = B'(f(x» and, differentiating, 1 = B"(f(x» . /'(x). Clearly this cannot hold at x = 0, because /,(0) = o. Moreover, if we suppose, to fix ideas, that f(x) '" cx m near x = 0, with c i= 0 and m an integer ~ 2, the last equation shows that B"(t) '" clt-(m-l)/m for some constant Cl i= 0, as t -+ 0 through positive values. Thus B(t) has an algebraic branch point at t = o. On the other hand, (9.15) gives A'(x) = y = f(x) for x near 0, showing A has an analytic continuation to a neighborhood of x = o. Hence, looking at (9.14) we see that the singularity of v at (0,0) arises solely because of the B(y) term and propagates along the line {y = OJ, i.e. the singularity propagates along the tangent line to r at the characteristic point. It is a remarkable fact - and this is the essence of Leray's theorythat the behaviour noted in this simple example persists "generically" (meaning that there are exceptions, described by Leray, and depending on the local behaviour of the bicharacteristics) for all (linear) holomorphic Cauchy problems, with data on an analytic variety, in the neighborhood of characteristic points of the variety. Of course, in en with

9.4 THE ELLIPSE (CONCLUDED)

89

n > 2, the characteristic points are not in general isolated, but form an analytic subvariety of the variety r carrying the data, of dimension n - 2. In general the singularities do not propagate along straight lines as in our example above, but along bicharacteristics (associated to the p.d.e. in question) that are tangent to r at its characteristic points. In the case of the Laplace equation the bicharacteristics are however (complex) lines

{Z : where A E

ZI

= Al t, ...

cn\{o} satisfies

E~

,Zn = An t,

(t E C)}

Al = O.

In the case of the Laplace operator and n = 2 one can work out the details in an elementary manner, even globally, much as in the above simple example. This is because the "symbol" of the Laplace operator, e? + ... + e!, splits into linear factors for n = 2(n ~ 3 is an altogether more difficult affair.) In any case, we won't here aim at a complete analysis, but making reasonable assumptions complete the discussion of the ellipse from §9.2.

9.4 The ellipse (concluded). We have seen that the (complexified) Schwarz potential extends analytically along each path on that does not meet one of the four characteristic points (9.10), and develops singularities at these points. We have also motivated the idea that from the points (9.10) the singularities travel along bicharacteristic (complex) lines that are tangent to at the respective points. Calculating these lines, we are led to the following assertion.

v

r

r

THEOREM 9.1. The Schwarz potential of the ellipse (9.1) extends holomorphically throughout C 2 along every path that does not meet one of the four complex lines

{(X, Y) E C 2 : X ± iY =

±c}

(the two choices of sign being effected independently) each of which is bicharacteristic for the operator (9.5) and is tangent to (as given by (9.2)) at one of tbe four points defined by (9.10). Moreover, every point

r

90

9. CONCLUDING REMARKS

of each of these lines is a singularity for some branch of the (analytically continued) Schwarz potential. Note that the only points where these 4 lines meet R2 are (± c, 0), the (real) foci of the original ellipse. This is the promised "explanation" of those singUlarities. We won't insist on completing details of the proof here, as [Johnsson, 1990] contains much more general results in RR. Another way to describe the "four lines", which suggests a multidimensional generalization, is to consider the (two-dimensional) "isotropic cone"

K := { (X, Y) E C 2 : X 2

+ y2 =

°}

and its translates K" gotten by translating the apex of the cone from (0,0) to (= «(1,(2) in C2. Then, the four lines are the union of the cones K, and K" where (=(c,O) and ('=(-c,O) are the (real) foci of the ellipse. It is a known fact from classical geometry that these are the only points in R2 from which the corresponding isotropic cone is tangent to the (complexified) ellipse in two points, i.e. each of the two bicharacteristics through such a point exhibits this tangency. It is a remarkable fact that the corresponding description also characterizes the focal ellipse of an ellipsoid in 3 dimensions: the focal ellip3e i3 preci3ely the 3et of point3 in R3 from which the i30tropic cone (with apez at the point) i3 tangent to the (complezijied) ellip30id in (at lea3t) two point3. Moreover, just as the real foci of the ellipse should be seen as the real trace of the above four-line configuration, the focal ellipse of a 3dimensional ellipsoid is the real trace of a remarkable surface in C 3 (called by geometers the focal developpable of the ellipsoid) which can be interpreted as the union of all bicharacteri3tic3 that are tangent to the (complezijied) ellip30id at it3 characteri3tic point3. In C3 this surface admits an equivalent definition as the set of points from which the isotropic cone has at least one point of tangency with the ellipsoid. Gunnar Johnsson discovered that this figure is the precise set of singularities in C3 of the Schwarz potential of the ellipsoid. He also showed that the solution to each of a large class of Cauchy problems with entire data on the ellipsoid, corresponding to operators whose principal part is the (complexified) Laplace operator, has no singularities other than points of this same ''focal developpable" j he also extended this result

9.5 THE SPHERE, REVISITED

91

from ellipsoids to other quadrics, and arbitrary dimensions. (Leray's general theory gives these results but only locally, near the characteristic points of the (complexified) ellipsoid.) Here we will leave this subject for now, but not without venturing the opinion that interesting and remarkable things are waiting to be discovered here, especially concerning the singularities of the Schwarz potential when the underlying surface is algebraic of degree ~ 3. Because of the coincidence that the bicharacteristics of the (complex) Laplace operator are identical with the "minimal lines" of classical projective geometry, the description of singularities in en of solutions to Cauchy problems for this operator (which propagate along such lines) often involves configurations already discovered by classical geometers, most especially so-called focal figures, having great beauty and intricacy of structure. Further study of the indicated Cauchy problems is bound to deepen the contact with, and enrich, this branch of geometry. One is tempted to guess that, in some sense, the singularities of the Schwarz potential in en of any algebraic variety will be its "focal developpable" , although so far as I know there isn't even a suitable definition of this object for n ~ 3 in the literature, let alone a carefully worked out theory of its purely algebraico-geometrical properties. It is important that a correct definition be "projective" so as not to "miss" parts of the figure deriving from tangency at infinity, cf. the next paragraph.

9.5 The sphere, revisited. We recall that the Schwarz potential of a sphere in Rn has a singularity at its center. How is this to be explained by "looking into en"? Unlike the ellipsoid, the sphere (when "complexified") has no characteristic points. Let's illustrate with n = 3 and use now the notations Z = (Zt,Z2,Z3) for points of e 3 , Zj = xj+iYj with x = (Xt,X2,X3), Y = (y!, Y2, Y3) denoting points of R3. Writing

r

:= { x E R3 :

r := {

Z

E

x~ + x~ + x~ =

e 3 : z~ + zi + z~ =

r

1} 1}

we check immediately that lacks characteristic points. Thus, the solution to the Cauchy problem

.a. v =

1 near

r

v, grad v vanish on

r

92

9. CONCLUDING REMARKS

r

extends as a holomorphic function to a neighborhood of III C 3 , by the C-K theorem. In this example we see the limitations of Leray's theory, which only deals with propagation of singularities from (finite) characteristic points of However, singularities can turn up that are not from these sources; these seem to come in some sense "from infinity". Johnsson has provided an extension of Leray's theory which (within the context of that theory) explains the genesis of this second kind of singularities, but the details are too intricate to be entered into here. Already in pure geometry a similar dichotomy turns up. Thus, the center of a sphere is not an "ordinary" focus, since the isotropic cone {zi + zi + z§ = O} does not even meet much less have two points of tangency. However, it is an "asymptote" of r, touching it "at infinity" along a whole circle. The distinction between "ordinary" and "extraordinary" foci of algebraic curves is known in classical geometry, see for example [Salmon, 1879, pp.119 fr.]. But I don't know where this has been treated in dimension n ~ 3.

r.

r,

9.6 Further horizons. Hopefully the reader who has proceeded thus far and finds the subject attractive has noticed areas requiring further study. I want to summarize some of these, and add a few more questions. 9.6.1 Regularity problems. Suppose n is a bounded quadrature domain (in the wide sense) in Rn, so that u dx = (u, J.L) holds for all integrable harmonic functions on n, where J.L is a certain distribution in n with compact support. What regularity can we then a88ert of an? A priori one cannot exclude extreme pathologies, such as components of Rn\n that cluster at some (even all) boundary points. For n = 2, Sakai has shown recently in a fundamental paper [Sakai, 1989] that this cannot occur. Consequently one can then use conformal mapping techniques, as in Theorem 5.1, to get an adequate description of the boundary regularity and the nature of its (isolated) cusp singularities. However Sakai's methods do not seem susceptible of extension for n ~ 3. Moreover, even if we assume that n is topologically nice, conformal mapping is no longer available to get the fine structure of an (especially near singular points, which in general are no longer isolated). One doe8 know from general results concerning the regularity of free boundaries that an is real-analytic near any point where it is known to be a nonsingular Cl hypersurface.

In

9.6 FURTHER HORIZONS

93

9.6.2 Again, the Cauchy problem. With regard to the Cauchy problem

(*)

{

a'u

~+ U%l

11.,

au

a'u f ... !i'2= u%'"

au

aZ1 ' ••• , aZn

near

r

vanish on

r

where r := {z E en : cp(z) = O} and cp is (say) an irreducible polynomial, we have already singled out the solution (call it here 11.1) corresponding to f = 1 as the (modified) complex Schwarz potential of r. As we saw, it becomes singular at each characteristic point zO of r (this being true for any holomorphic f with f(zO) i= 0). Thus, it is tempting to conjecture that if f is very regular (say, to fix ideas, a polynomial), so it does not itself introduce any finite singularities into the solution of (*), then the corresponding 11. will not have singularities not already present in 11.1, or more precisely that 11. is analytically continuable, starting from an arbitrary point on r that is not characteristic, along every path in en along which 11.1 is analytically continuable. This conjecture, which is easily proven for n = 2, was verified in some cases for n ~ 3, in [Khavinson and Shapiro, 1989a]. Gunnar Johnsson in [Johnsson, 1990] proved a slightly weaker assertion for quadratic cp, and all n (even for entire I). A complete solution seems to require a fuller understanding of how singularities of holomorphic Cauchy problems propagate in en. The problem just stated has also a "real" interpretation in terms of potentials. If n is (say) a bounded connected domain in Rn, and n1 denotes the "body" consisting of a uniform mass distribution on n, while nf denotes the same spatial body with density given by f where f(x) is a polynomial (or, more generally an entire function on en, restricted to Rn), then : is the Newtonian potential of nf harmonically continuable inwards (from Rn\n) as far as that of n 1 (i.e. along every path in Rn along which the latter is continuable)? Another problem regarding (*) (when f = 1; we'll continue to call this solution 11.1) is: for which r can 11.1 extend as a holomorphic /unction to all of en ? Clearly each hyperplane has this property, and one may ask whether, conversely, such r must be a hyperplane. In case n = 2, if r n R2 contains a nontrivial arc the answer is yes in view of a theorem of P. Davis: namely, the Schwarz /unction of that arc (in R2 ~ C) is entire under our assumptions, and hence [Davis,

94

9. CONCLUDING REMARKS

1974, p.107] linear. Thus r n R2 is a line, etc. For general n, if

showing that the entire function (1/2)
9.7 CONCLUSION

95

H + CHC - I there have been several studies of its eigenvalues and eigenfunctions in the compact case, e.g. [Schiffer, 1957]. D. Khavinson has (in a letter) raised the very interesting question of the injectivity of the latter operator. Observe that if, for some I E H2(r) (notations as in Chapter 7) we have (H + CHC - 1)1 = 0, then H(CHC)I = o. Thus, g := HC I is in H2(r), and simultaneously its complex conjugate is in H~(r). Suppose for simplicity that r is an analytic Jordan curve. We then ask: Can there ezist a pair 01 junctions, one analytic inside r, the other outside r and vanishing at 00 and which have (say) continuous boundary values on r which are complez conjugates 01 one another? This indeed happens when r is a circle, and one asks: is the circle the only case? If so, we would have in all other cases injectivity of Fredholm's operator and hence a far-reaching extension of Theorem 7.6. In the same spirit, one could try to strengthen Theorem 7.5. Can one inler that 0 is a disk from the mere assumption that the ranges 01 Hand H* have a non-trivial intersection, i.e. that some I E H2(r), not the zero functions, is orthogonal to H~(r)? The hypothesis is equivalent to: for some IE H2(r) that is not the zero function, and g E H~(r), IT = g on r (where T is a unit tangent vector, as in Chapter 7). Also, for the Friedrichs operator (and its "boundary" analog, see below) spectral questions can be asked (all these are in the same spirit in which one asks what geometric information lurks in the spectrum of the Laplace operator, a very popular subject nowadays). The simples} result goes back to [Friedrichs, 1937], who showed that (in the notation of Chapter 8) the essential spectrum of TJ is in [O,p] for some p < 1 if 0 is a "polygon" made up of finitely many smooth arcs meeting at non-zero angles. (I pointed out [Shapiro, 1980, Theorem 5.1] that this remains true if 0 satisfies an interior cone condition.) It would be of interest to develop analogous results where the set-up of L2(0), L~(O), Po is replaced by L2(r), H2(r) and S, the Szego projector. Thus, the "boundary analog" of the Friedrichs operator on H2(r) is SCIH2(r). A natural setting here would be boundaries that are locally Lipschitz graphs, in view of Calderon's result, cf. [Murai, 1988]. 9.7 Conclusion. In conclusion, I apologize for having omitted certain very relevant topics. Especially, the connection with Hele-Shaw flows, free boundary

96

9. CONCLUDING REMARKS

problems, and variational inequalities is totally lacking, but fortunately excellent accounts of these are available. An important early paper of [Richardson, 1972], and several already cited papers of Gustafsson and Sakai, can be recommended, as well as [Friedman, 1982] for orientation in this area. For further connections with inverse problems of gravitation theory, see also [Zalcman, 1987].

Afterword. After the completion of this chapter, Boris Shapiro of Moscow kindly directed my attention to works by B. Yu. Sternin and V. E. Shatalov concerning the singularities of the solution to Cauchy's problem in en. In particular, the "limitation of Leray's theory" mentioned in § 9.5 is overcome by these authors, who work entirely in a projective framework and thus pick up all singularities that "come from infinity". The reader may consult "On an integral transform of complex analytic functions", an English translation of one of their papers, in Math. USSR Izvestiya vol. 29 (1987), No.2, pp. 407-427, for an account of their method.

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INDEX

relation to Schwarz function, 7-8,10,84 and Schwarz potential, 84-94 Circular arcs: Schwarzian reflection, 2 Circular cylinders, Schwarzian reflection, 60-62 Complex bicharacteristics, 57 Conformal mapping, 1-2,41,48, 50. See also Schwarzian reflection Continuability, of Schwarz potential, 9-11

A Anti-conformal reflection, 4-5, 55-60. See also Schwarzian reflection Arcs, Schwarzian reflection, 2-3 Avci's method, 42-43, 50

B

Balayage, 11-12. See also Quadrature domains; Schwarzian reflection inwards, 12, 13-15,23 outwards, 12-13 Balls: modified Schwarz potential of, 33-34 Schwarz potential, 91-92 Bergman kernels, 42-43 Bessel equation, 61 Bicharacteristics, 89 complex, 57 Bicircular curves, 21 Brouwer degree, 40-41, 48

D

Dirichlet problem, 65, 66 Distributional derivatives, 55 Distributional Laplacian, 55 Double-layer potential, 65, 66-67

E Ellipsoids: quadrature formula for, 20-22, 34-35,36 Schwarz function of, 21-22 Schwarz potential, 85-87, 89-91 Elliptic regularity, 54 Essential spectrum, 81

C

Cardioid, 23 Cauchy data, 54 Cauchy integral, 63-66, 68 Cauchy-Kovalevskaya theorem, 84 Cauchy principal value, 64 Cauchy problem: holomorphic, 84, 93

F Fredholm operator, 68 105

INDEX

L

Fredholm's solution, of Dirichlet problem, 67-68 Free boundary problems, 96 Friedrichs operator, 74-76, 95 and quadrature domains, 82-83 relation to Schwarz function, 80-82 Functional equations, 53

Laplace equation, 89 Laplacian operator, 57, 59, 85, 95 bicharacteristics, 89 distributional, 55 Leray's theory, 91, 92, 96 Lipschitz graphs, 55 Logarithmic potential, 9-11

G M

Gauss' mean value theorem, 22 Mean value formula, 17,44 Mean value theorem, Gauss', 22 Meromorphic functions, 62

H

Hankel operator, 74, 83 Harmonic functions, Schwarzian reflection, 52-61 Hele-Shaw flows, 95-96 Herglotz's principle, 11 Hilbert operator, 94-95 and plane domains, 63-66, 70-72 relation to Neumann-Poincare problem, 66-69 Hilbert transforms, 64 Holomorphic Cauchy problem, 84, 93 Holomorphic functions, 56, 64-65

N

Neumann-Poincare solution, of Dirichlet problem, 65, 66-69 Newtonian potentials, 11

o Orthogonality relations, 81

P

Inverse problems, 96 Inverse theorems, 44-45 Isometric embeddings, 76

Plane domains, 44-48 Hilbert operator of, 63-66, 70-72 Plemelj-Sokhotski relations, 63 Polynomials, 39-40, 44-46, 93 Projectors: Hilbert, see Hilbert operator Szego, 65-66,72-73,95

J

Q

Jordan arcs, 2-3

Quadrature domains, 13-15,44-47,

I

lOG

INDEX

48-50. See also Balayage; Schwarz function; Schwarzian reflection; Schwarz potential boundary regularity of, 39-40 ellipsoids, 20-22, 34-35, 36 examples of, 17-23 existence, 50 and Friedrichs operator, 82-83 functions with single-valued integrals, 47-48 regularity of differential, 92 and Schwarz potential, 25-29 with singular boundary point, 22-23 sub harmonic, 36-38 and SzegO projectors, 73 variational properties of, 41-43 "wide sense" of, 16,24-25 Quadrature identities, 16, 24 Quadric surfaces, 85, 93

and Hilbert operator, see Hilbert operator of triangle, 48-49 valency of, 40-41 Schwarzian reflection, see also Balayage; Quadrature domains circular cylinders, 60-62 generalized, 5-7 geometric interpretation, 55-60 harmonic function formulation, 52-55 in R3, failure of, 60-62 straight line boundaries in, 2 Schwarz potential, 7-8, 28 analytic continuability, 9-11 and Cauchy problem (en), 84-85,93-94 ellipses,85-87,89-91 singularity propagation, 84, 86-89,90,91-92 spheres, 91-92 modified, 8,28,33-34 and quadrature domains, 25-29 Simpson's rule, 16 Singular inner functions, 44 Singularities: propagation of, 84, 86-89, 90-92 removable, 54 Smirnov condition, 45 Spheres: modified Schwarz potential, 33-34 Schwarz potential, 91-92 Study quadrilaterals, 57 Subharmonic quadrature domains, 36-38 Sweeping, see Balayage SzegO kernel, 65 SzegO projector, 65-66, 72-73, 95

R R3, failure of Schwarzian

reflection in, 60-62 Reflection, anti-conformal, 4-5, 55-60. See also Schwarzian reflection Reflection, Schwarzian, see Schwarzian reflection Removable singularities, 54

s Schottky double, 50 Schwarz function, 2-4, 29, 57-58 and Cauchy's problem, 7-8, 10, 84

of ellipse, 21-22 and Friedrichs operator, 80-82. See also Friedrichs operator

107

INDEX

T

Variational inequalities, 96

Triangle, Schwarz function of, 48-49

W

Weak limits, of sequences,

v

76-80

lOS