Calc. Var. 16, 283–298 (2003) DOI (Digital Object Identifier) 10.1007/s005260100151
Calculus of Variations
Annalisa Baldi · Bruno Franchi
A Γ -convergence result for doubling metric measures and associated perimeters Received: 27 August 2001 / Accepted: 29 November 2001 / c Springer-Verlag 2002 Published online: 10 June 2002 – Abstract. In this paper we study the notion of perimeter associated with doubling metric measures or strongly A∞ weights. We prove that the metric perimeter in the sense of L. Ambrosio and M. Miranda jr. coincides with the metric Minkowski content and can be obtained also as a Γ -limit of Modica-Mortola type degenerate integral functionals. Mathematics Subject Classification (2000): 49Q20, 28A12
1. Introduction In the Euclidean space Rn , denote by B Euc (x, r) = {y ∈ Rn ; |x − y| < r} the Euclidean ball centered at x of radius r > 0, and let µ be a positive doubling measure (i.e. we assume there exists a constant c > 0 such that for all balls B Euc = B Euc (x, r) in Rn we have µ(2B Euc ) ≤ c µ(B Euc ), where 2B Euc = B Euc (x, 2r)). Following Semmes [20], we define a canonical quasi-metric D for x, y ∈ Rn as follows: D(x, y) := {µ(B Euc (x, |x − y|)) + µ(B Euc (y, |x − y|))}1/n ,
(1)
and we say that µ is a doubling metric measure if there exists a constant C > 0 and a metric d(x, y) on Rn such that C −1 d(x, y) ≤ D(x, y) ≤ Cd(x, y).
(2)
In [20], Semmes proves that a positive doubling measure µ satisfies (2) if and only if µ = ωdx, where ω is a strong-A∞ weight in the sense of David & Semmes [9] and dx stands for Lebesgue measure. In fact, the notion of strong-A∞ weight A. Baldi: Dipartimento di Matematica, Universit`a di Bologna, Piazza Porta S. Donato, 5, 40126 Bologna, Italy (e-mail:
[email protected]) B. Franchi: Dipartimento di Matematica, Universit`a di Bologna, Piazza Porta S. Donato, 5, 40126 Bologna, Italy (e-mail:
[email protected]) Investigation supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy. The authors are very grateful to Luigi Ambrosio and Francesco Serra Cassano for making their preprints available to them, for listening with patience and for many unvaluable suggestions.
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requires few preliminaries, and we want to state it carefully, since it will be used throughout all the present paper. Recall that a weight function ω (i.e. a function ω ∈ L1loc (Rn ), ω ≥ 0) is said to be an A∞ weight if for each ε > 0 there exists a δ > 0 so that, for any ball B Euc ⊆ Rn , if E ⊆ B Euc satisfies |E| ≤ δ|B Euc | then ω(E) ≤ εω(B Euc ). Here, our notations are standard: for any Lebesgue measurable set E, we denote by |E| its Lebesgue measure and we set ω(E) := E ω(x)dx. It is well known that the measure µ = ωdx is doubling. Then we can state now the definition of strong-A∞ weight, as it is given in [9]. For sake of simplicity and since we shall deal only with this class of weights, let us suppose that the weight ω is a bounded continuous function. If ω is a A∞ weight, its measure distance is defined by 1/n δ(x, y) :=
w(u) du
,
(3)
Euc Bx,y
Euc denotes the ball with diameter |x − y| containing x and y (notice the where Bx,y definition is equivalent to that of (1), by doubling). On the other hand, given a rectifiable arc γ(t), t ∈ [0, 1], we can define its 1 ω-length by the formula lω (γ) = 0 ω(γ(t))1/n |dγ(t)|, where |dγ(t)| denotes arclength measure on γ, whereas we shall denote the Euclidean length of γ by l(γ). The geodesic distance dω (x, y) is the infimum of the ω-lengths of all curves joining x to y.
Definition 1. A weight function ω ∈ A∞ is said to be a strong-A∞ weight (briefly s-A∞ ) if there exists C > 0 such that 1 (4) dω (x, y) ≤ δ(x, y) ≤ C dω (x, y) for all x, y ∈ Rn . C Notice, the left inequality in (4) always holds for any A∞ - weight. If x ∈ Rn and r > 0 are given, then a metric ball B(x, r) with respect to the metric dω is the set B(x, r) := {y ∈ Rn : dω (x, y) ≤ r}. Constant functions provide trivial examples of s-A∞ weights. Less trivial examples are given by ω(x) = |x|a , a ≥ 0; more generally, Jacobian determinants of quasiconformal maps are s-A∞ . In addition, any weight in the Muckenhoupt class A1 is a s-A∞ weight. We point out that the s-A∞ condition prevents ω from vanishing on rectifiable curves; however it can vanish on Cantor sets of large Hausdorff dimension (see Proposition 4.4 in [19]). Thus, typically a s-A∞ weight can be obtained by taking positive powers of the distance from a set of points that are sufficiently scattered, like the uniformly disconneted sets studied by Semmes, [19] Definition 4.2. Since A∞ weights yield doubling measures, it is easy to prove the following fact. Proposition 1. If K is a compact set, then there exists ε > 0 and constants cK , CK > 0 such that for x, y ∈ K cK |x − y|1/ε ≤ dω (x, y) ≤ CK |x − y|.
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It is proved in [20], (B.4.16) and (B.4.13), that the measure ω dx is a doubling measure with respect to the balls associated with dω , and even more that it is Ahlfors-regular of dimension n, more precisely it is said that there exist a, A > 0 depending only on n and on the constant of (4) such that for any x ∈ Rn , r > 0 we have (5) arn ≤ ω(B(x, r)) ≤ Arn . Moreover, there exists c > 0 such that for any r > 0 there exists R = R(r) such that B Euc (x, cR) ⊆ B(x, r) ⊆ B Euc (x, R) (6) for any x ∈ Rn . We are interested here to the notion of hypersurfaces measure associated with a s-A∞ weight (for a while, we have to be vague). Indeed, the key property of s-A∞ weights established in [9] consists of an isoperimetric inequality associated with a s-A∞ weight that reads as follows:
ω(x) dx ≤ C Ω
n n−1 1 ω 1− n (y)dHn−1 (y) ,
(7)
∂Ω
where Ω is a regular (say smooth for sake of simplicity) bounded open set and Hn−1 denotes the (n − 1)−dimensional Hausdorff measure concentrated on ∂Ω. Notice we can not expect any reasonable isoperimetric inequality like (7) for general A∞ weights, since these weights can vanish on hypersurfaces. The aim of the present paper is to compare different notions of codimension one measure associated with s-A∞ weights and that agree with the measure in (7) for regular hypersurfaces. In fact, following [15] and [2], we can give a notion of perimeter in the sense of De Giorgi for very general metric spaces endowed with doubling measures. To this end, let us start by defining the weighted BV space associated with a s-A∞ weight ω as it is given in [15], i.e. the space of bounded variation functions with respect 1 to the measure ω 1− n (x) dx (which is doubling since ω ∈ A∞ ), by means of a relaxation argument. Definition 2. Let Ω be an open subset of Rn and let ω be a s-A∞ weight function. 1 We shall say that a function u ∈ L1 (Ω; ω 1− n ) (i.e. the space of functions which are 1 1 integrable with respect to the measure ω 1− n (x) dx) belongs to u ∈ BV (Ω; ω 1− n ) 1 if there exists a sequence (uh )h∈N ⊂ Liploc (Ω) ∩ L1 (Ω; ω 1− n ) converging to u 1 in L1 (Ω; ω 1− n ) and satisfying 1 |Duh | ω 1− n (x) dx < ∞. lim sup h→+∞
Ω
Moreover, we put Du
1 ω 1− n
(Ω) := inf
1 lim inf h→+∞ Ω |Duh | ω 1− n (x) dx : (uh ) ⊂ Liploc (Ω) ∩ L1 (Ω; ω
1 1− n
1
), uh
L1loc (Ω;ω 1− n )
−→
u ,
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and u
1
BV (Ω;ω 1− n )
= u
1
L1 (Ω;ω 1− n )
+ Du
1
ω 1− n
1
(Ω).
1
We shall say that u ∈ BVloc (Rn ; ω 1− n ) if u ∈ BV (Ω; ω 1− n ) for any bounded open set Ω in Rn . We denote by Du the variational measure of a function u in the classical BV (Ω) space (see eg. [11]). Definition 3. A measurable set E is said to have ω-finite perimeter in Ω if χE ∈ 1 BV (Ω; ω 1− n ) and we write DχE 1− n1 (Ω) = ∂E 1− n1 (Ω). ω
ω
The theory of BV functions in metric spaces endowed with doubling measures developed in [2], [15] relies on the Poincar´e inequality for these spaces. On the other hand, the following Poincar´e inequality holds for dω -balls. This result is basically contained in [9]. Theorem 1. There exists CP > 0 such that, if x ∈ Rn and r > 0, then, for u ∈ Liploc (Rn ), n−1 n n 1 |u − uB(x,r) | n−1 ω(ξ)dξ ≤ CP r |∇u| ω 1− n (ξ)dξ, (8) B(x,r)
where uB(x,r) =
B(x,r)
B(x,r)
u(y)dy.
We recall now an important result contained in [2] (Theorem 3.3) and in [15] (Theorem 3.4): 1
Theorem 2. For any u ∈ BV (Rn ; ω 1− n ), the set function A → Du 1− n1 (A) ω is the restriction to the open subsets of Rn of a positive finite measure. We shall show in this note that it is possible to approximate the perimeter ∂E 1− n1 of a measurable set E ⊂ Rn via degenerate elliptic functionals Fε : ω
1
L1 (Ω; ω 1− n ) −→ [0, +∞] of the form 2 W (u) 2 1− n ω dx ε |Du| ω + ε Fε = Ω +∞
2
if u ∈ S 1,2 (Ω; ω 1− n )
(9)
otherwise,
where W (u) = u2 (1 − u)2 and 2 2 2 1,1 S 1,2 (Ω; ω 1− n ) := u ∈ L2 (Ω; ω 1− n ) ∩ Wloc (Ω) : |Du| ∈ L2 (Ω; ω 1− n ) . This result is a version, adapted to the new geometry of our space, of the well known theorem proved by Modica and Mortola ([17], when ω ≡ 1) which proposes a way to approximate a functional in BV via more manageable elliptic functionals. The notion of convergence here is the so-called Γ -convergence (see, e.g., [10], [1], [7], [8]). The definition and many properties of the Γ -convergence can be found in [1], [7] and [8]. Let us state now our main Theorem.
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Theorem 3 (Modica-Mortola type Theorem). Let Ω be a bounded open set in Rn with Lipschitz boundary. For any ε > 0 let Fε be defined as in (9) and set 1 2σ∂E 1− n1 (Ω) if u = χE ∈ BV (Ω; ω 1− n ) ω F = (10) +∞ otherwise, 1 with σ = 0 W (t)dt. Then the functionals Fε Γ −converge to F in 1 L1 (Ω; ω 1− n ) as ε → 0. The paper is organized as follows. In Section 2 we recall and prove some proper1 ties concerning the BV (Ω; ω 1− n ) space. In Section 3, Theorem 3 will be obtained after proving an equivalence of different notions of perimeter (see Proposition 2, where we define the Minkowski content of regular sets) and two approximations lemmas (Lemma 2, Lemma 3) . Let us conclude this Introduction by stressing some motivations of our work, and of the interest on the geometry associated with strong-A∞ weights. First of all, there is a general project intended to develop a Geometric Measure Theory in metric spaces. A general account of the current research in this field can be found for instance in [3], [4], [13], [14]. The notion of perimeter, as well as the equivalence of different definitionss for sufficiently suitable regular subsets of codimension one, is a keystone in this program. Basically, so far the study of such a notion has been carried on for two separate situations: on one side, for objects that are bilipschitz deformations of the Euclidean space, on the other side for codimension one regular hypersurfaces in the so-called Carnot-Carath´eodory spaces associated with the structure of Carnot groups. In particular, for this last class of metric spaces, a Γ limit characterization corresponding to our Theorem 3 has been recently proved by R. Monti & F. Serra Cassano in [18] (our presentation here is largely inspired to their paper). The notion of perimeter associated with a s-A∞ weight provides a third independent class of metric spaces where analogous arguments can be carried on (and a little bit more, as we point out below). In addition, there is another deep reason that attracted our attention toward the study of geometric properties of the distance associated with s-A∞ weights: indeed these metric spaces endowed with the measure ωdx are in some sense a general model for metric spaces of homogeneous type (i.e. metric spaces endowed with a doubling measure), because of a beautiful result proved by S. Semmes, [20] (Proposition B.20.2), asserting that if (M, ρ(x, y)) is any metric space endowed with a doubling measure, then we can find a positive integer n, and a metric doubling measure µ on Rn , such that (M, ρ(x, y)) is bilipschitz equivalent to a subset of (Rn , D(x, y)), where D(x, y) is associated with µ as in (1). In particular, the weight can be chosen of the form ω(x) = distEuc (x, E)s , for suitable E ⊂ Rn and s > 0, and hence for a continuous weight. This last fact gives in a certain sense a justification of our somehow restrictive assumption.
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2. Preliminary results The first property that we state is an Anzellotti-Giaquinta’s density - type theorem ([6]) for our space. Indeed the theorem follows pretty immediately from Definition 2. 1
Theorem 4. Let u ∈ BV (Ω; ω 1− n ), then there exists a sequence (uh )h∈N ∈ Liploc (Ω) such that 1
uh
L1 (Ω;ω 1− n )
−→
u
,
Duh
1
ω 1− n
(Ω)−→Du
1
ω 1− n
(Ω).
We need now a few technical facts we shall use later in order to prove Theorem 3. First of all, in the Euclidean setting it is well konwn that the distance function from compact sets is almost everywhere differentiable; moreover its gradient has modulus one almost everywhere. We need to show a corresponding result in our case. Theorem 5. If E is a compact set, then the function x → dω (x, E) is differentiable a.e. in Rn and |∇dω ( · , E)(x)| = ω 1/n (x) (11) for a.e. x ∈ Rn \E. Proof. Since, by a doubling argument, |dω (x, E) − dω (y, E)| ≤ dω (x, y) ≤ 1/n ω(ξ) dξ , then C δ(x, y) ≤ C |x − y| B(x,|x−y|) lim sup y→x
|dω (x, E) − dω (y, E)| < ∞, |x − y|
(12)
for any x ∈ Rn ; thus, the a.e. differentiability follows from Stepanoff’s Theorem. We are left with the proof of (11). It is easy to check that |∇dω ( · , E)(x)| ≤ ω 1/n (x) a.e. in Rn . Set now p = |∇dω ( . , E)(x)| and suppose by contradiction p < ω 1/n (x); write ω 1/n (x) = p + 4δ, for δ > 0. By continuity, min|ξ−x|
(13)
Take now r ≤ r0 and y ∈ E such that dω (x, y) = dω (x, E); let γr be a path joining x and y, where lω (γr ) < dω (x, y) + δ r. Let us denote by yr the first point where γr meet ∂B(x, r). Since x = y, such a point exists provided r is small enough. We have dω (x, yr ) + dω (yr , y) ≤ lω (x, yr ) + lω (y r , y) = lω (γr ), where ξ, η denotes an arc-piece of γr , with ξ, η ∈ γr . Therefore dω (x, yr ) + dω (yr , E) ≤ lω (γr ), hence dω (x, yr ) ≤ lω (γr ) − dω (yr , E) < dω (x, y) −
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dω (yr , E) + δ r = dω (x, E) − dω (yr , E) + δ r. By the differentiability of d it follows from above dω (x, yr ) ≤ < ∇dω (. , E)(x), x − yr > +o(|x − yr |) + δ r ≤ p r + o(r) + δ r = (p + δ + o(1)) r < (p + 2δ) r, if r < r(δ). In conclusion (14)
dω (x, yr ) < (p + 2δ) r, if r < r(δ) On the other hand, we show that
dω (x, yr ) ≥ inf {lω (σ), σ connects x with a point on ∂B(x, r), σ ⊂ B(x, r)} Indeed, any curve, starting from x and ending in yr , must go out of B(x, r) (in case, just at the last time) and then its ω−length is greater than the length of this arc contained in B(x, r). Therefore dω (x, yr ) ≥ p+3 δ r; by (13) this is a contradiction with (14) and we are done.
The following co-area formula is proved by [15] (Proposition 4.2) and it is an essential ingredient in proving Theorem 3. 1
Theorem 6. If u ∈ BV (Rn , ω 1− n ) and we set Et = {x; u(x) > t}, we have +∞ Du 1− n1 (A) = ∂Et 1− n1 (A) dt, (15) ω
ω
−∞
for every open set A. We give now the following representation Theorem that involves both spaces 1 BV (Ω) and BV (Ω; ω 1− n ). 1
Theorem 7. If Ω is a bounded open set, then BV (Ω) ⊆ BV (Ω; ω 1− n ). Moreover, if A ⊆ Ω is an open set and u ∈ BV (Ω) then 1 ω 1− n d Du. (16) Du 1− n1 (A) = ω
A
In particular, if E ⊆ Ω is a Caccioppoli set, i.e. χE ∈ BV (Ω), then 1 ω 1− n d∂E. ∂E 1− n1 (A) = ω
(17)
A
Proof. Let u ∈ BV (Ω) and let (uh )h ∈ BV (Ω) ∩ C ∞ (Ω), uh −→ u in L1 (Ω) and Duh (Ω) → Du (Ω) as h → +∞. By definition of Du 1− n1 and by the lower semicontinuity, we get that Du
ω
1
ω 1− n
1 1− n
1
(Ω) ≤ maxΩ¯ ω 1− n Du(Ω),
). and then u ∈ BV (Ω; ω Moreover, if we repeat the argument above for any open set A ⊆ Ω we get Du
1
1
ω 1− n
(A) ≤ max ω 1− n Du (A).
(18)
A
By Theorem 3.4 in [15] (see Theorem 2), there exists a positive finite measure µ in Ω so that Du 1− n1 (A) = µ(A) for every open set A ⊆ Ω and by (18) it follows ω
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that µ is absolutely continuous with respect to Du (that is a Radon measure). Now, by Radon-Nikodym’s theorem, there exists gu ∈ L1 (Ω, d Du) such that d µ = gu d Du.
(19)
1
We want to show that Du−a.e. gu = ω 1− n in Ω. To this end, let B Euc = 1 B (x, r) be given. If uh ∈ Lip (B Euc ) and uh → u in L1 (B Euc , ω 1− n ), then uh − uL1 (B Euc ) → 0 as h → ∞. Hence, again by definition of Du 1− n1 and by the lower semicontinuity of ω Du, we get Euc
1
inf ω 1− n Du(B(x, r))
B(x,r)
(20)
≤
gu dDu ≤ sup ω
1 1− n
Du(B(x, r)).
B(x,r)
B(x,r)
Notice that the set S = {x ∈ Ω; Du(B(x, r)) = 0 f or 0 < r ≤ rx } has zero Du-measure. Thus, if x ∈ Ω\S, we can divide (20) by Du(B(x, r)) and take the limit as r → 0+. By Lebesgue-Besicovitch’ differentiation theorem and 1 keeping in mind that ω is continuous, we obtain gu (x) = ω 1− n (x) for Du- a.e. x ∈ Ω which, together with (19), proves assertion (16) and we are done.
1
In particular, if u is Lipschitz continuous, then u ∈ BV (Ω; ω 1− n ) and 1 Du 1− n1 (Ω) = |Du| ω 1− n dx. ω
(21)
Ω
Corollary 1. If E has ω-finite perimeter and in addition ∂E is a C 2 -manifold, then for any open set A ⊆ Ω 1 ∂E 1− n1 (A) = ω 1− n dHn−1 . (22) ω
∂E∩A
1
Corollary 2. Let Ω be a bounded open set. If u ∈ Liploc (Ω) ∩ BV (Ω; ω 1− n ), then 1 Du 1− n1 (Ω) = |Du|ω 1− n dx. (23) ω
Ω
Proof. By Theorem 2, and as it is shown in the proof of Theorem 3.4 in [15], Du 1− n1 (Ω) = sup Du 1− n1 (B) : B open, B ⊂⊂ Ω . Hence, there exω ω ists a sequence (Ωk )k , Ωk ⊂⊂ Ω such that Du
1
ω 1− n
(Ωk ) ≤ Du
1
ω 1− n
(Ω) ≤ Du
1
ω 1− n
(Ωk ) +
1 . k
Without loss of generality, we may suppose Ωk ⊂ Ωk+1 for any k. Now, since u ∈ Lip(Ωk ) ⊆ BV(Ωk ), by (21) we have 1 Du 1− n1 (Ωk ) = |Du| ω 1− n dx ω
and the equality (23) follows.
Ωk
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3. Main convergence results Arguing as in [5] and in [18], the proof of Theorem 3 will be obtained once we have shown the following facts: i) The dω -Minkowski content of a smooth hypersurface ∂E equals its ω-perimeter in any open set Ω, provided ∂E is transversal in the measure-theoretic sense to ∂Ω (see Proposition 2). ii) If Ω is a Lipschitz domain, then any set of finite ω-perimeter in Ω can be approximated via smooth open sets with transveral boundaries as in i) (see Lemma 3). Once the above two steps are achieved, then the proof of Γ -convergence can be carried out repeating almost verbatim the arguments of [18] and relying on a general Reduction Lemma for metric spaces (see [1] and [17]) that reads as follows. Lemma 1. Let (M, d) be a metric space, let Fh , F : M −→ [−∞, +∞] with h ∈ N; consider D ⊂ M and x ∈ M . Let us suppose that 1) for every y ∈ D there exists a sequence (yh )h∈N ⊂ M such that yh → y in M and lim sup Fh (yh ) ≤ F (y); h→∞
2) there exists a sequence (xh )h∈N ⊂ D such that xh → x and lim sup F (xh ) ≤ h→∞
F (x); then there exists a sequence (xh )h∈N ⊂ M such that lim sup Fh (xh ) ≤ F (x). h→∞
1 1− n
In our case, we shall take M = L1 (Ω; ω ), and D is the set of all functions χE , where E is a bounded open set with smooth boundary, such that Hn−1 (∂Ω∩∂E) = 0. Let us start by proving point i) holds. Proposition 2. Let E ⊆ Rn be a compact set such that ∂E is a smooth manifold. If r > 0 define a tubular neighborhood Ir = Ir (∂E) = {x ∈ Rn ; dω (x, ∂E) < r} . Moreover, if Ω ⊆ Rn is an open set, put M + (∂E)(Ω) = lim sup r→0+
ω(Ir ∩ Ω) 2r
,
M − (∂E)(Ω) = lim inf r→0+
ω(Ir ∩ Ω) . 2r
Then, if Hn−1 (∂E ∩ ∂Ω) = 0 , M + (∂E)(Ω) = M − (∂E)(Ω) = ∂E where ∂E
1
ω 1− n
1
ω 1− n
(Ω)
(Ω) is represented as in formula (22).
Proof. Denote by dω (·) = dω (·, ∂E) the signed distance from ∂E, i.e. put x ∈ E, dω (x, ∂E) dω (x) = −dω (x, ∂E) x ∈ Rn \E,
(24)
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and define ϕρ (x) =
1 d (x) 2ρ ω
|dω (x)| < ρ,
dω (x) ≥ ρ, dω (x) ≤ −ρ.
1 0
1 1− n
Since ϕρ → χE in L1 (Ω; ω ), by the lower semicontinuity of the variation ([15], Proposition 3.6), by Theorem 7 and by (11) we get ∂E
1
ω 1− n
(Ω) ≤ M − (∂E)(Ω).
Let us prove now that M + (∂E)(Ω) ≤ ∂E
1
ω 1− n
(Ω).
(25)
The proof is divided in several steps. Step 1 Fix η > 0 and put ωη/2 = max{ω, η/2}. Then for ρ < ρ(η) and for any x ∈ Iρ ∩ {ω > η} we have dω (x, ∂E) = dωη/2 (x, ∂E) Proof. We can always reduce ourselves to Iρ ⊆ I1 = K, K being a compact set. Since ωη/2 ≥ ω, clearly dωη/2 ≥ dω . Call δ := distEuc ({ω ≥ η} ∩ K, {ω ≤ η/2} ∩ K) > 0. Let γ be a curve connecting x ∈ Iρ ∩{ω > η} with a point y ∈ ∂E such that dω (x, y) = dω (x, ∂E). If ε > 0, we can choose γ such that lω (γ) ≤ (1 + ε)dω (x, y) ≤ 2dω (x, y) ≤ 2ρ. Let us show that γ ⊆ {ω > η/2} ∩ K. Clearly γ ⊆ K. Suppose by contradiction γ is not contained in {ω > η/2}. Thus, let t1 be the first point such that γ(t1 ) ∈ ∂{ω > η/2} and denote by t0 the last point t0 < t1 such that γ(t0 ) ∈ ∂{ω > η}. t1 t η On the other hand, lω (γ) ≥ t01 ω 1/n |γ˙ (t)| dt ≥ ( )1/n |γ˙ (t)| dt ≥ 2 t0 δ η η ( )1/n δ, which is a contradiction if ρ < ( )1/n ; hence γ ⊆ {ω > η/2} ∩ K. 2 2 2 We conclude by observing that ωη/2 ≡ ω on {ω > η/2}, hence we find that dωη/2 ≤ lωη/2 (γ) = lω (γ) ≤ (1 + ε)dω (x, y) and the assertion follows.
Step 2 There exists a family of functions (σε,η )ε>0 ∈ C ∞ such that η ≤ σε,η ≤ ωη/2 ≤ σε,η + 2ε (ε < ε(η)); i) 4 ii) σε,η → ωη/2 uniformly as ε → 0 on compact sets. Proof. Take σε,η = ωη/2 ∗ Jθ(ε) − ε, where J is a standard regularization kernel, and use the local uniform continuity of ω.
Step 3 M + (∂E)({ω > η} ∩ Ω) ≤
∂E
1
ω 1− n d Hn−1
(26)
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Proof. Combining Steps 1 and Step 2, if x ∈ {ω > η} ∩ Iρ , then dω (x, ∂E) ≥ dσε,η (x, ∂E) if ε < ε(η), ρ < ρ(η) and hence Iρ ∩ {ω > η} ∩ Ω ⊆ {dσε,η (x, ∂E) < ρ} ∩ {ω > η} ∩ Ω for ε < ε(η) fixed and ρ < ρ(η). Then ω {dσε,η (., ∂E) < ρ} ∩ {ω > η} ∩ Ω M + (∂E) {ω > η} ∩ Ω ≤ lim sup . 2ρ ρ→0+ (27) Let now K be the compact set introduced in step 1 and let ψ1 , ψ2 be smooth functions, 0 ≤ ψi ≤ 1, with ψ1 ≡ 1 on {ω > η} and suppψ1 ∩ K ⊆ {ω > η/2}, whereas ψ2 ≡ 1 on Ω and suppψ2 is compact. Put ψ = ψ1 ψ2 ; by definition we have χ{ω>η}∩Ω ≤ ψ. Denote now by dσε,η the geodesic distance associated with σε,η . Since the σε,η ’s are smooth functions, which are locally bounded and bounded away from zero, dσε,η is a Riemannian distance. We shall denote by dσε,η the corresponding signed distance from ∂E, that is smooth in a neighborhood of ∂E (depending on η). Continuing from (27), if ρ is small enough and keeping in mind (11) with σε,η instead of ω, we have 1 ω {dσε,η (., ∂E) < ρ} ∩ {ω > η} ∩ Ω 2ρ 1 = ωη/2 {dσε,η (., ∂E) < ρ} ∩ {ω > η} ∩ Ω 2ρ 1 ≤ (σε,η + 2ε) {dσε,η (., ∂E) < ρ} ∩ {ω > η} ∩ Ω 2ρ 1 −1/n ≤ (σε,η + 2ε)σε,η |Ddσε,η | ψ dx 2ρ |dσε,η (·, ∂E)|<ρ ρ 1 −1/n = dt (σε,η + 2ε)σε,η ψ dHn−1 , 2ρ −ρ dσε,η (·, ∂E)=t −2/n
by usual coarea formula. For sake of simplicity write ψ(σε,η + 2ε) σε,η =: g, which is a smooth function. The set dσε,η = t := dσε,η (·, ∂E) = t is a 1/n
smooth manifold for any t, since |Ddσε,η (·, ∂E)| = σε,η > 0. Hence we have (t > 0) { dσε,η
=
{dσε,η =t}
div (g Dσε,η ) dx
g < Dσε,η ,
Dσε,η > dHn−1 − |Dσε,η |
g < Dσε,η , ∂E
Dσε,η > dHn−1 . |Dσε,η | (28)
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The left hand side of (28) tends to zero as t → 0+ so that, from previous expression we get lim
t→0+
g < Dσε,η ,
{dσε,η =t}
Dσε,η >dHn−1 |Dσε,η |
g |Dσε,η | dHn−1 =
= ∂E
∂E
−1/n ψ (σε,η + 2ε) σε,η dHn−1 .
Analogously, when t → 0−, lim
t→0−
{dσε,η =t}
= ∂E
g < Dσε,η ,
Dσε,η > dHn−1 |Dσε,η |
−1/n ψ (σε,η + 2ε) σε,η dHn−1 .
−1/n Finally, M + (∂E) {ω > η} ∩ Ω ≤ ∂E ψ (σε,η + 2ε) σε,η dHn−1 , so that, since suppψ1 ⊆ {ω > η/2} and ψ1 ≤ 1, we get M + (∂E) {ω > η} ∩ Ω ≤ ∂E∩{ω>η/2}
−1/n ψ2 (σε,η + 2ε) σε,η dHn−1 .
Letting ε → 0 from above (remember σε,η > η/4) +
1− 1
M (∂E) {ω > η} ∩ Ω ≤ ωη/2n ψ2 dHn−1 ∂E∩{ω>η/2} 1 1 = ω 1− n ψ2 dHn−1 ≤ ω 1− n ψ2 dHn−1 . ∂E∩{ω>η/2}
∂E
n We can take now ψ2 −→ χΩ pointwise in R \∂Ω. Since Hn−1 ∂E ∩ ∂Ω = 0, ψ2 −→ χΩ Hn−1 - a.e. on ∂E and hence, by 1 1 1− n Dominated Convergence theorem, ω ψ2 dHn−1 −→ ω 1− n dHn−1 , ∂E ∂E 1 1− n + so that M (∂E)({ω > η} ∩ Ω) ≤ ω d Hn−1 = ∂E 1− n1 (Ω) and
(26) is proved.
∂E
ω
Step 4 If ρ < ρ(η), then for any x ∈ Iρ ∩ {ω < η} there exists a Euclidean ball BxEuc = B Euc (ξx , rx ) such that i) x ∈ BxEuc and |BxEuc ∩ E| ≈ |BxEuc \E| ≈ |BxEuc |; ii) ω(BxEuc ) ≤ c0 ρn where c0 is a geometric constant independent on ρ, η and x; in addition, ω < 2 η in BxEuc .
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Proof. By compactness, there exists ξx ∈ ∂E such that dω (x, ξx ) = dω (x, ∂E). Take now rx = 2 |x − ξx |. Since ∂E is uniformly Lipschitz, then i) holds. Property ii) can be proved as follows. By doubling Euc ω(Bx ) = ω dy ≤ c ω dy B Euc (ξx , 2 |x−ξx |)
x, B Euc ( x+ξ 2
|x−ξx | ) 2
= c δ n (x, ξx ) ≤ c c1 dnω (x, ξx ) = c0 dnω (x, ∂E) < c0 ρn , and we are done. Finally, denoting by σ the modulus of continuity of ω, if z ∈ BxEuc , |ω(z) − ω(x)| ≤ σ(4|ξx − x|) ≤ σ(c2 dεω (x, ξx )) ≤ σ(c2 ρε ) < η if ρ < ρ(η), which concludes the proof of Step 4.
Step 5 There exists a geometric constant c3 such that ω Iρ ∩ {ω < η} ≤ 1
c3 ρ η 1− n Proof. Let {BjEuc } = {BxEuc } be a Vitali family extracted from the family j Euc Bx ; x ∈ Iρ ∩ {ω < η} defined in Step 4. We get ω Iρ ∩ {ω < η} ≤ j ω(5BjEuc ) ≤ c j ω(BjEuc ) 1 1 1 1 = c j ω(BjEuc ) n ω(BjEuc )1− n < c ρ η 1− n j |BjEuc |1− n . From Step 4-i), |BjEuc | ≈ min{|BjEuc ∩ E|, |BjEuc \E|} and hence, by a standard n−1 isoperimetric inequality, |BjEuc | n ≤ c Hn−1 BjEuc ∩ ∂E ; recall that BjEuc ∩ BkEuc = ∅ when j = k, thus 1 1 Hn−1 BjEuc ∩ ∂E ≤ c ρ η 1− n Hn−1 ∂E ; ω Iρ ∩ {ω < η} ≤ c ρ η 1− n j
the proof of step 5 is complete.
We can now conclude the proof of proposition by showing that (25) holds. From previous steps ω Iρ ∩ {ω < η} M + (∂E)(Ω) ≤ M + (∂E)({ω > η/2}) + lim sup 2ρ ρ→0 1 ≤ ∂E 1− n1 (Ω) + c η 1− n . ω
Since η > 0 is arbitrary we get (25) hence we obtain (24) and we are done.
The following approximation results make possible to apply in our context the Reduction Lemma for Γ -convergence (see [1] and [17]), and hence to restrict ourselves to smooth domains for which the previous Proposition applies.
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Lemma 2. Let Ω be a bounded Lipschitz domain in Rn . Let u ∈ Liploc (Ω) ∩ 1 BV (Ω; ω 1− n ), then for any ε > 0 there exists a function w ∈ Lip(Ω) such that w − u 1 < ε , Dw 1− 1 1− 1 (Ω) − Du 1− 1 (Ω) < ε. n n n L (Ω;ω
)
ω
ω
Proof. To make the proof more readable we divide it in several steps. Step 1 We can assume without loss of generality that u is bounded. Step 2 Define L := {x ∈ ∂Ω; ω(x) = 0} and, for δ > 0, set Lδ = {x ∈ Rn ; dEuc (x, L) < δ}. Then for any ε > 0 there exists δ = δ(ε) > 0 and 1 w = w(ε) ∈ Liploc (Ω) ∩ BV (Ω; ω 1− n ) such that i) w − u 1 < ε , Dw 1− 1 1− 1 (Ω) − Du 1− 1 (Ω) < ε n n n L (Ω;ω
ii) w ≡ 0 in Lδ ∩ Ω.
)
ω
ω
To show this fact, let us take a smooth function ψδ such that 0 ≤ ψδ ≤ 1, ψδ ≡ 0 in (−∞, δ], ψδ ≡ 1 in [2δ, +∞) and |ψδ | < 2δ , and put wδ (x) = ψδ (dEuc (x, L))u(x). 1 Clearly, wδ ∈ Liploc (Ω) ∩ L1 (Ω; ω 1− n ) and wδ ≡ 0 in Lδ ∩ Ω for any δ > 0, so that we just need to show that i) above is satisfied provided δ is small enough. Since ψδ (dEuc (x, L)) → 1 as δ → 0 for any x ∈ Ω, the first statement of i) follows trivially by dominate convergence theorem. Now, by Corollary 2, Dwδ ω1− n1 (Ω) − Duω1− n1 (Ω) 1 ≤ Ω |Du| − |ψδ (dEuc (., L))Du + uψδ (dEuc (., L))∇dEuc (., L)|ω 1− n dx 1 1 ≤ Ω (1 − ψδ )|Du|ω 1− n dx + 2δ {δ≤dEuc (x,L)≤2δ}∩Ω |u(x)|ω 1− n dx, and again by dominate convergence theorem the first term goes to zero as δ → 0, so that we have only to show that the second term vanishes when δ → 0. Therefore, keeping in mind that u is bounded, it is enough to prove that 1 2 ω 1− n (x) dx → 0 (29) δ Lδ as δ → 0. Let x ∈ Lδ , then there exists ξx ∈ L so that |x − ξx | = dEuc (x, L) < δ, therefore, if we denote by σ the modulus of continuity of ω, for any x ∈ Lδ we get ω(x) = ω(x) − ω(ξx ) ≤ σ(|x − ξx |) ≤ σ(δ) and hence 1 1 1 2 2 2 ω 1− n (x) dx ≤ σ 1− n (δ) |Lδ | ≤ σ 1− n (δ) | dEuc (x, ∂Ω) < δ |, δ Lδ ∩Ω δ δ Euc since so that dEuc (x, L ⊆n ∂Ω, ∂Ω) ≤ d (x, L). Thus (29) follows, since 2 Euc | x ∈ R , d (x, ∂Ω) < δ | goes to the Euclidean perimeter of Ω as δ → 0 δ ([12], Theorem 3.2.39). Hence, i) is fully proved and hence, from now on, we can assume without loss of generality that u vanishes in Lδ for some δ > 0.
With the notations of Step 2, let δ be fixed (depending on ε and u). By definition, for any x ∈ ∂Ω there exists an open neighborhood Ox of x of the form Ox = Tx ({(y , yn ), |y | < r, |yn | < ρ})
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297
where r = rx , ρ = ρx , Tx = Ux + bx is the sum of a rotation Ux and a vector x = Tx ({(y , yn ), |y | < 2r, |yn | < 2ρ}), we can bx ∈ Rn , such that, if we put O write x = Tx ({(y , yn ), |y | < 2r, |yn | < 2ρ yn > fx (y )}) , Ω∩O with fx a Lipschitz continuous functions in {|y | < r}. Take now x ∈ Γδ := {x ∈ ∂Ω; dEuc (x, L) ≥ δ}. If x ∈ Γδ then ω(x) > 0, since Γδ ⊆ ∂Ω and hence Γδ ∩ {x; ω(x) = 0} ⊆ Γδ ∩ Lδ = ∅. Thus, we can assume also inf {ω(y), y ∈ Ox } > 0. Step 3 By means of a localization argument and Step 2, we can reduce ourselves to prove that, if O is an open set such that ∂Ω ∩ O = ∅ and such that ω > 0 in 1 O, and if ψ ∈ D(O), then ψu can be approximated in BV (Ω; ω 1− n ) by Lipschitz continuous functions. Take a small t > 0, and define for x ∈ Ω ∩ O wt (x) = (ψu)(x + tT (en )), where T = Tx and en is the n-th vector of the canonical orthonormal basis of Rn . The and hence u(x+tT en ) is well definition above is well posed, since x+tT en ∈ Ω∩O defined. In addition, if t is sufficiently small, the map x → ψ(x+tT en ) is supported in O, and hence wt can be defined on all Ω by putting wt (x) = 0 if x ∈ Ω\O. It is u(x+tT en ) easy to see now that wt is Lipschitz continuous in Ω, since the map x → is Lipschitz continuous in Ω∩O, because ξ = x + tT en , x ∈ Ω ∩ O ∩∂Ω = ∅. 1 To achieve the proof we want to show that wj,t → ψj u in L1 (Ω; ω 1− n ) as t → 0+ and 1 1 1− n |Dwj,t |ω dx → |D(ψj u)|ω 1− n dx (30) Ω
Ω
as t → 0+. The first statement follows in a straightforward way by the dominate convergence theorem, since ψj u ∈ L∞ (Ω) ∩ C(Ω). To prove the second assertion, we notice first that, by definition, inf O ω > 0 and hence we can reduce ourselves in (30) to unweighted L1 -norms, so that we can apply standard L1 -mean convergence. This completes the proof of the Lemma. Lemma 3. Let Ω be a bounded open set in Rn with Lipschitz boundary ∂Ω. Let E ⊂⊂ Ω be a measurable set such that 0 < ∂E 1− n1 (Ω) < ∞. Then there exists ω a sequence of smooth bounded open sets (Es )s∈N in Rn with Hn−1 (∂Es ∩∂Ω) = 0, 1 χEs → χE in L1 (Ω; ω 1− n ) and ∂Es 1− n1 (Ω) → ∂E 1− n1 (Ω). ω
ω
Proof. Thanks to the previous Lemma, the proof is reduced to the corresponding unweighted analogous ([16], Lemma 1), since u ∈ Lip(Ω) can be extended on all
Rn . As we said at the beginning of this Section, thanks to the above results, the proof of Theorem 3 has became only a minor variant of the one given in [18] and therefore we will omit it.
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