Monotonicity formula and regularity for general free discontinuity problems

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1 Monotonicity formula and regularity for general free discontinuity problems Dorin Bucur Stephan Luckhaus Laboratoire de Mathématiques, CNRS UMR 57, Université de Savoie Campus Scientifique, Le-Bourget-Du-Lac, France, Universität Leipzig, Fakultät für Mathematik und Informatik, Mathematisches Institut Postfach 00 90, Leipzig, Germany, Abstract We give a general monotonicity formula for local minimizers of free discontinuity problems which have a critical deviation from minimality, of order d. This result allows to prove partial regularity results (i.e. closure and density estimates for the jump set) for a large class of free discontinuity problems involving general energies associated to the jump set, as for example free boundary problems with Robin conditions. In particular, we give a short proof to the De Giorgi-Carriero-Leaci result for the Mumford-Shah functional. Introduction We give a monotonicity formula which describes the behaviour of the energy associated to almostquasi local minimisers of general free discontinuity problems, near the points of the jump set. This formula is a key tool to prove both the closedenss of the jump set and of uniform density estimates of the associated Hausdorff measure. In this way we generalize the De Giorgi-Carriero-Leaci result on the Mumford-Shah functional [5] (see also [, 4]) to a large class of functionals possibly involving geometric quantities of the jump set and the traces of the SBV functions on the jump set. An important question is whether an SBV minimiser of a general free discontinuity problem is a classical one, i.e. its jump set is closed and so the function is locally smooth (in the complement of the jump set). A positive answer to this question was given by De Giorgi, Carriero and Leaci in [5] (see also [4] and []) for the Mumford-Shah functional. Precisely, they analyse properties of quasi minimisers of the Mumford-Shah functional, i.e. functions u SBV loc (R d ) satisfying for some α > 0, c α 0 and Λ = (x) x R d, ρ > 0, v SBV loc (R d ) such that {u v} (x) = u dx + H d (J u (x)) v dx + ΛH d (J v (x)) + c α ρ d +α. (x) The work of D.B. was supported by the programme Free Boundary Problems, Theory and Applications, MSRI Berkeley, January-May 0 and ANR-09-BLAN-0037 The work of S.L. was supported by the programme Free Boundary Problems, Theory and Applications, MSRI Berkeley, January-May 0

2 The main purpose of this paper is to study the behaviour of almost-quasi minimisers, i.e. to consider Λ. As the value of Λ is allowed to be greater than, the deviation from minimality becomes of the same order as the Hausdorff measure, d. The main consequence of this freedom is that one can deal with functionals involving quite general energies associated to the jump set, and still conclude that the solutions have a closed jump set with an associated Hausdorff measure satisfying upper and lower uniform density bounds. The first result of the paper is a monotonicity formula (Theorem.) which holds for almostquasi minimisers. Roughly speaking, a quantity of the form [ ( )] ρ d u dx + H d (J u ) c(d, Λ) + c(d, α)ρ α () is non decreasing in ρ, for small ρ. For ρ 0, around a regular point of the jump set, the limit of the quantity above is bounded from below by a positive constant depending only on the dimension of the space and Λ. In this way, one can obtain key information for proving its closure, and latter uniform density bounds. The constants appearing in the density bounds are obtained by refining the monotonicity formula and are given by explicit estimates. In the second part of the paper, we analyse a free boundary - free discontinuity problem with Robin conditions. A similar question for Dirichlet boundary conditions is treated by Alt and Caffarelli in []. In our example, the energy we minimize among all admissible functions u with values prescribed on a given compact subset of R d, is of the form u dx + β ( u + + u )dh d + {u > 0}. R d J u A variational framework based on SBV spaces which allows to deal with this type of problems was introduced in [3]. In order to prove that the solutions u of these free boundary problems are almost-quasi minimisers, and so they fit in the framework of our paper, the crucial difficulty is to establish uniform upper and lower positive bounds on their positivity set, i.e. to prove that they satisfy 0 < α < α, such that α u(x) α, a.e. x {u > 0}. If the jump set were closed and smooth, the lower bound would be a consequence of the Hopf lemma in relationship with the PDE satisfied by u. Nevertheless, these two properties are a priori unknown, being precisely the object of investigation. The monotonicity formula In the sequel d, (x) stands for the open ball of R d centered in x and of radius ρ (if ambiguity on the dimension occurs, the ball is denoted B d ρ(x)). If x = 0, we simple denote it. By d K (x) we denote the distance from the point x to set K in R d. For an open set Ω R d, by SBV (Ω) we denote the space of special functions with bounded variation and by J u the jump set of a function u SBV (Ω) (see [, Definition 3.67]). For every m N, by H m we denote the m-dimensional Hausdorff measure. The usual Lebesgue measure is denoted by. Definition. Let Λ, α > 0, c α 0. We say that a function u SBV (Ω) is a local (Λ, α, c α ) almost-quasi minimizer of a free discontinuity problem at the point x Ω, if 0 < ρ < d Ω (x), v SBV (Ω) such that {u v} (x) =

3 (x) u dx + H d (J u (x)) (x) v dx + ΛH d (J v (x)) + c α ρ d +α. We say that a function u SBV (Ω) is a (Λ, α, c α ) almost-quasi minimizer of a free discontinuity problem if it is a local almost quasi minimizer at every point of Ω. For simplicity, in the sequel we drop the mention to the triplet (Λ, α, c α ) and we simply refer to almost-quasi minimisers. Here is the main result of the paper. Theorem. (The monotonicity formula) Let u SBV (Ω) be a local almost-quasi minimizer of a free discontinuity problem at 0. Then the mapping ρ E(ρ) := [ ρ d ( is non decreasing on (0, d Ω (0)). )] u dx + H d (J u ) c dλ d d + (d )c α α ρα () We give two lemmas relating SBV functions on spheres with the harmonic extensions of their H -approximations. The boundary of the ball (0) is denoted and by SBV ( ) we denote the SBV space on the sphere. The tangential gradient is denoted τ. Lemma.3 For d, there exists a constant c d such that for every function u SBV ( ) which satisfies ε(u) := τ u dx + H d (J u ) c d Λ d ρ d (3) there exists w H ( ) such that τ w dx + Λ(d ) H d ({u w} ) ε(u). (4) ρ Lemma.4 Under the hypotheses of Lemma.3, there exists w H ( ) such that w = 0 in and w dx + ΛH d ({u w} ) ρ ε(u). (5) d The proof is done simultaneously for Theorem. and Lemmas.3,.4, in four steps, using a cyclic inductive argument over the dimension of the space. Step. Proof of Lemma.3 for d =. It is enough to take c <. Indeed, since H d (J u ) H 0 (J u ) and by hypothesis H 0 (J u ) <, then J u =. So we can take directly w := u. Step. Proof of the implication: Lemma.3 in R d = Lemma.4 in R d, for every d. Let d and assume Lemma.3 is true. Let u SBV ( ) such that ε(u) c d Λ d ρ d and w H ( ) be the function given by Lemma.3. We denote by w the harmonic extension of w in. Since d w dx τ w dh d, (6) ρ relation (5) follows directly. 3

4 Step 3. Proof of the implication: Lemma.4 in R d = Theorem. in R d, for every d. Assume Lemma.4 is true. In order to prove the monotonicity formula, we introduce the function E : (0, d Ω (0)) R, E(ρ) := u dx + H d (J u ). Then E is a non decreasing function, and its distributional derivative can be written as ρ E(ρ) = e(ρ)dρ + µ, where e L (0, d Ω (0)) is a nonnegative function and µ is a postive singular measure. precisely, More 0 < a < b < d Ω (0), µ(a, b) = H d ((B b \ B a ) J u {x : ν u (x) x = x }). As a consequence, from the Leibniz formula for BV functions, it is enough to estimate the absolutely continuous part of the derivative of E. At a.e. point ρ where E is derivable and at which we have ρ E(ρ) = (d )c α ρ α 0. For a.e. point r at which we have E(ρ) r d c dλ d d, E(ρ) ρ d < c dλ d d, (7) ρ E(ρ) = ρe (ρ) (d )E(ρ) + (d )c α ρ d +α ρ d. (8) Assuming for contradiction that for some point ρ we have ρ E(ρ) < 0, then from (8) we get and using (7) we get Since E (ρ) < d ρ E(ρ) (d )c α ρ d +α (9) ρ E (ρ) < c d ρ d Λ d. ε(u) = τ u dx + H d (J u ) E (ρ) we can use Lemma.4 to get the existence of a function w such that w dx + ΛH d ({u w} ) r d E (ρ). Using (9), we get w dx + ΛH d ({u w} ) E(ρ) c α ρ d +α. 4

5 Consequently w dx + ΛH d ({u w} ) + c α ρ d +α E(ρ), (0) and using the almost-quasi minimality of u we get that equality holds in (0) and thus in all preceding inequalities. This contradicts the hypothesis ρ E(ρ) < 0. Step 4. Proof of the implication: Theorem. in R d = Lemma.3 in R d+, for every d. It is enough to prove Lemma.3 only for ρ =, and to rescale. To simplify our computations, we shall assume without restricting generality that c d (d )ω d, where ω d is the volume of the unit ball in R d. We consider the unit ball B R d+ and the d-dimensional unit sphere S d R d+, S d = B. Let u SBV (S d ) be fixed, such that ε(u) := τ u dx + H d (J u ) cλ d. () S d The value of c will be specified below. We introduce an auxiliary problem min τ w dh d + H d (J w ) + ΛdH d ({u w}), () w SBV (S d ) S d and prove that for a sufficiently small constant c in (), depending only on the dimension of the space, a minimizer w of () will satisfy H d (J w ) = 0. Note that here the hypothesis Λ can be relaxed. Consequently, for this minimizer w we have w H (S d ) and τ w dh d + ΛdH d ({u w}) τ u dx + H d (J u ) S d S d so that we achieve the proof of Lemma.3 in R d+. Let us consider a minimizer w for the auxiliary problem () and assume for contradiction that H d (J w ) 0, for some value of the constant c. We can further assume that the point e d+ = (0, 0,.., 0, ) is a regular point of the jump set J w and consider the orthogonal projection Π : S d B (e d+ ) R d {0}. Let us denote in the sequel the (d-dimensional) ball B d (0) R d and introduce the functions w, u SBV (B d (0)) defined by Then w is a minimizer for the functional w(x) = w Π (x), u(x) = u Π (x). B d w dx + H d (J w ) + ΛdH d ({u w}) + R(w, Λ, ), where 0 R(w, Λ, r) C d ( + Λ)r d, for some constant C d depending only on the dimension of the space. Consequently, w is a (,, c ) almost-quasi minimizer of a free discontinuity problem for Λ =, α = and c = (Λ + )(d + C d ), so that we can use the monotonicity formula of Theorem.. 5

6 On the one hand, since 0 is a regular point of J w(x) we have lim E(ρ) ρ 0 + c d d ω d = c d d. Meanwhile, from the monotonicity formula we get for every ρ (0, /) [ ( )] ρ d w dx + H d (J w ) Assume that for some ρ we have that c d d + (d )(Λ + )(d + C d)ρ cλ d ρ d < c d d. c d d. (3) If ρ is fixed (its value will be given below), this assumption will be satisfied as soon as c will be chosen small enough. Indeed, combining hypothesis () with (3) and the assumption above, we get that c (Λρ) d + (d )(Λ + )(d + C d)ρ or introducing suitable dimensional constants, for ρ Let us fix ρ := C d c/d Λ c (Λρ) d + C d Λρ c d (d ) (d+c d ) c d (d ).. It is enough to take c small enough, e.g. C d c/d Λ c d d, we get < C d C d (d )Λ c d (d ) (d + C d ), to obtain a contradiction and finish the proof. 3 Regularity properties for free discontinuity problems Relaying on the monotonicity formula, in this section we prove the primary regularity property for almost-quasi minimizers of free discontinuity problems: their jump set is closed and satisfies density estimates from above and below (i.e. it is Ahlfors regular) and the function is smooth in the complement of the this set. 3. Closure of the jump set Theorem 3. Let u be an almost-quasi minimizer of a free discontinuity problem and denote J u the family its jump points. Then H d (J u \ J u ) = 0. Proof Since J u is rectifiable, it is contained in a countable union of C hypersurfaces, up to a set of zero H d -measure. We denote Ju r the set of regular points of J u (i.e. which have density in some of these C manifolds). Let us consider x n Ju r such that x n x J u. For every such point x n, we have H d (J u (x n )) lim ρ 0 ρ d ω d, 6

7 so that relaying on the monotonicity formula and comparing E(ρ) to the behaviour of E near 0, we get for ρ (0, d Ω (x n )) ( )] ρ B d u dx + H d (J u (x n )) + (d ) c α ρ(x n) α ρα ω d c dλ d d There exists ρ 0, C 0 > 0 independent on n such that, provided ρ ρ 0 d Ω (x n ), u dx + H d (J u (x n )) C 0 ρ d. (5) (x n) By continuity, inequality (5) holds also at the point x u dx + H d (J u (x)) C 0 ρ d. (6) (x) In general, for every function u SBV (Ω), it is well known (see for instance [5, Theorem 3.6]) that for H d -almost every point y Ω \ J u we have (4) B ] lim ρ d[ u dx + H d (J u (y)) = 0. (7) ρ 0 ρ(y) Consequently, there exists a set A such that H d (A) = 0 and such that every point y satisfying inequality (6) belongs to A J u, and so the point x. Consequently J r u J u A. Since H d (J u \ Ju) r = 0, we get that H d (J u A \ J r u) = 0. As the following argument shows, the set J u \ J r u is empty, which concludes the proof since this implies J u J r u and so H d (J u \ J u ) = 0. Indeed, since J r u is closed and H d (J u \ J r u) = 0, we get that u H (Ω \ J r u). Since u is a local almost minimizer for the Dirichlet energy in the open set Ω \ J r α 0, u, we get that u C, so that u has no jump points outside J r u (see for instance [, Remark 7.0]). Remark 3. The proof of the result above shows in fact a slightly stronger result, precisely H d (S u \ S u ) = 0 where S u stands for the family of singular points of u. The set S u consists on all the points where the approximate limit of u does not exist (see [, Definition 3.63]), and may be slightly larger than the set of jump points J u. Nevertheless H d (S u \ J u ) = Ahlfors regularity The Ahlfors regularity of the jump set can be proved as a consequence of the monotonicity theorem. and of the decay lemma [, Lemma 7.4], as we show below. In the next paragraph we shall give a more technical monotonicity type lemma, which not only gives in a direct way the Ahlfors regularity, but also contains implicit estimates of the density constants. We start with the following density result for the energy which is a consequence of almost-quasiminimality and of the monotonicity formula. Corollary 3.3 There exists constants c = c(d, Λ), ρ 0 = ρ 0 (d, α, c α, Λ) such that for every almost quasi-minimizer u, for every x J u and for every 0 < ρ < ρ 0 such that (x) Ω we have c ( ) ρ d u dx + H d (J u ) c. (8) 7

8 Remark 3.4 There exists ρ 0 > 0, c > 0 such that for every x J u and for every ρ < ρ 0 satisfying (x) Ω, we have c ρ d H d (J u (x)) c ρd. (9) Indeed, the right hand side of the inequality is a direct consequence of the minimality of u. To prove the inequality of the left hand side, one can relay on [, Lemma 7.4], which roughly speaking asserts that if H d (J u ) ερ d for ε small enough, then the decay of the energy is like ρ d which is in contradiction with the monotonicity Theorem. which gives a density estimate of the energy. Assume for contradiction that there exist sequences x n J u, ρ n 0, C n 0 such that H d (J u n (x n )) C n ρ d n. (0) Inequalities (4)-(9)-(0) imply that there exist ε n 0 such that ε n c ρd n ε n u dx H d (J u n (x n )). n (x n) Consequently, the deviation from minimality (see [, Definition 7.] for its definition) of u for ψ Λ (u, (x)) := (x) u dx + ΛH d (J u (x)) in n (x n ) satisfies Dev (u, n (x n )) [ (Λ ) ε n c + c αρ α ] n ρ d n θ n ψ Λ (u, n (x n )), for some θ n 0. For every τ > 0, we can apply [, Lemma 7.4] so that there exists C > 0 such that for n large enough ψ Λ (u, B τρn (x n )) Cτ d ψ Λ (u, n (x n )). which contradicts the lower density estimate in (4), provided τ is chosen small enough. 3.3 Density estimates In the sequel, we give a refined monotonicity type formula which allows a direct proof of the Ahlfors regularity and moreover gives explicit estimates for the density constants, depending on the dimension of the space, Λ, α, c α. We start with a technical result consisting in a refined inequality for harmonic extensions of H -functions. Lemma 3.5 There exists a constant α d > 0 such that for every harmonic function w in B, such that w B H ( B ), we have τ w dx (d ) B w dx + α d B τ (w w) dx, B () where w(y) = B wdx + ( B wdx ) y. Proof The proof is a consequence of the identity τ w dx = (d ) w dx + ( x ) D w dx, B B B 8

9 and of the following inequality ( x ) D w dx α d τ (w w) dx, B B which can be proved, for instance, by contradiction. Clearly, equality holds in the previous inequality if and only if w = w. In the sequel, we assume that u is an almost quasi minimizer of a free discontinuity problem, satisfying the hypotheses of Theorem.. We shall denote u ρ (x) = u(ρx) and, as before, E(ρ) = u dx + H d (J u ). Lemma 3.6 There exist constants ρ 0 = ρ 0 (d, α, c α, Λ)and δ, ε, β depending on d only, such that x Ω, 0 < ρ < ρ 0 with (x) Ω at least one of the following assertions holds: a) H d (J uρ S d ) δ; b) H d (J uρ S d ) < δ and E(ρ) < ρd d S d τ u ρ dx+βλρ d H d (J uρ S d )+c α ρ d +α, and ε d < E(ρ) ρ d S d τ u ρ dx < + ε d, and a ρ R, ξ ρ R d such that ρ / H d ({x S d : u ρ (x) a ρ ξ ρ x > δ ξ ρ }) < δ and ξ ρ S d τ u ρ. c) E(ρ) < ε d ρd S d τ u ρ dx + βλρ d H d (J uρ S d ) Proof Notice that by almost quasi minimality of u, condition a) implies condition c) for some suitable constant β. Without restricting the generality, we assume x = 0. Step. From the minimality of u, we get for every admissible ρ and for a suitable constant K E(ρ) Λρ d dω d + c α ρ d +α K ρ d. () For some constant c > d, if then S d τ u ρ dx ck, (3) E(ρ) c ρd S d τ u ρ dx. Introducing ε, given by c = ε d, assertion c) holds. Step. We assume the contrary of (3), i.e. S d τ u ρ dx < ck. For some constants c, c, we introduce the following auxiliary problem on S d : c min τ w dx + c H d ({w u ρ } S d ) + H d (J w S d ). (4) w SBV (S d ) Λ S d 9

10 Following Lemma.3, there exists a constant c such that if c τ u ρ dx + H d (J uρ S d ) < Λ S d c ( c (5) d )d then problem (4) has a solution w such that w H (S d )), so H d (J w S d ) = 0. We introduce δ < c ( c, d )d and fix c such that Consequently, either assertion a) holds with c Λ ck < c ( c (6) d )d H d (J uρ S d ) δ, or condition (5) is satisfied. Step 3. Assume in the sequel that H d (J uρ S d ) < δ and condition (5) is satisfied. We know that c Λ S d τ w dx + c H d ({w u ρ } S d ) c Λ S d τ u ρ dx + H d (J uρ S d ). Using Lemma 3.5 and multiplying by Λ c we get: (d ) w dx + α d τ (w w) dx + Λc H d ({w u ρ } S d ) (7) B S d c τ u ρ dx + Λ H d (J uρ S d ). (8) S d c We introduce some ε > 0 (which will be fixed later) and assume that c) does not hold for this ε. Then E(ρ) ε d ρd τ u ρ dx + βλρ d H d (J uρ S d ). (9) S d We fix β = c (d ). Then, by simple computation and using (7) (d )E(ρ) + ερ d τ u ρ dx ρ d S d ρ d [ (d ) w dx + α d B τ u ρ dx + Λ ρ d H d (J uρ S d ) S d c τ (w w) dx + Λc ] H d ({w u ρ } S d ). S d c We chose c = c (d ) and use the almost quasi minimality of u. Consequently, (d )E(ρ) + ερ d S d τ u ρ dx (d )E(ρ)+α d ρ d S d τ (w w) dx+(d )ρ d ΛH d ({w u ρ } S d ) (d )c α ρ d +α, 0

11 thus ερ d S d τ u ρ dx + (d )c α ρ d +α (30) α d ρ d S d τ (w w) dx + (d )ρ d ΛH d ({w u ρ } S d ) (3) We notice in the same time (we successively apply the quasi-minimality, inequality (6) and the auxiliary problem) E(ρ) ρ d B w dx + Λρ d H d ({w u ρ } S d ) + c α ρ d +α ρd [ ] τ w dx + Λ(d )H d ({w u ρ } S d ) + c α ρ d +α d S d ρd [ τ u ρ dx + Λ ] H d (J uρ S d ) + c α ρ d +α. d S d c Finally E(ρ) ρd [ τ u ρ dx + Λ ] H d (J uρ S d ) + c α ρ d +α. (3) d S d c Assume that ρ 0 is such that ε cλdω d (d )c α ρ α 0. Then, from (30) ε cλdω d α d S d τ (w w) dx + ΛH d ({w u ρ } S d ). For ε small enough, using the Poincaré inequality on S d we get From (9) H d ({x S d : u ρ (x) w > δ ξ ρ }) < δ. E(ρ) ε d ρd τ u ρ dx. S d Since H d (J uρ S d ) < δ for a suitable ε we get E(ρ) + ε d ρd τ u ρ dx. S d Assume that Then τ w dx ( ε ) τu ρ dx. S d S d B w dx ε d S d τ u ρ dx. Using quasi-minimality and the auxiliary problem, from (30) one reduces this case to c). Assume now that τ w dx > ( ε) S d τu ρ dx S d

12 and τ (w w) dx > ε τ w dx. S d S d Relying on () we arrive again in situation c). If τ w dx > ( ε) S d τu ρ dx S d and τ (w w) dx ε S d τ w dx, S d then by direct computation ( ε) τ w dx S d τ w dx, S d so that ( ε) S τ u ρ dx τ w dx = S d ξ 0. d S d Lemma 3.7 (Decay Lemma) There exists constants c = c d, ε = ε d, δ = δ d and ρ 0 = ρ 0 (d, α, c α, Λ) such that for every jump point x and min{ρ 0, Ω c(x)} > ρ > ρ > 0, one of the following situations holds: where d = d ε, or ρ d E(ρ ) ρ d E(ρ ) ( ln ρ c exp cλ e σ H d (e σ S d J u ) ln ρ E(e σ ) ) dσ, (33) ρ <ρ<ρ H d (S d J uρ ) > δ. (34) Proof As noticed, case a) in Lemma 3.6 can be seen as a particular case of c), possibly considering a slightly greater β. As a consequence, only issues b) and c) shall be considered in the sequel after a possible modification of β. We shall work for simplicity in logarithmic co-ordinates by setting σ = ln ρ. We get after rescaling and denoting χ b, χ c the characteristic functions associated to cases b) and c) in the previous lemma σ ln E(e σ ) (d )χ b (σ) + d χ c (σ) + ρ ε d E(e σ u n dx βλ ρ ) ρs d E(e σ ) Hd (J u ρs d ). (35) If χ b (σ 0 ) =, let a 0 = a ρ0 R and ξ 0 = ξ = ξ ρ0 R d be given by point b). We introduce the set M 0 = {x S d : u ρ a 0 x ξ 0 δ ξ 0 }. One can choose the set M 0 to be symmetric with respect to every axes hyperplane, so that i j, x i x j dh d = 0, M 0

13 and H d (S d \ M 0 ) δ. We have S d δ ξ 0 u ρ0 a 0 x ξ 0 dx = M 0 u ρ0 a 0 dx+ M 0 x ξ 0 dx M 0 (u ρ0 a 0 )(x ξ 0 )dx. M 0 Using the symmetry property of M 0, we get that Provided δ is small enough, we get Let us introduce M 0 x ξ 0 dx = C d ξ 0. M 0 (u ρ0 a 0 )(x ξ 0 )dx C d ξ 0. (36) f(ρ) = ρ d ( ) M 0 u ρ a 0 x ξ 0 dx ( ). M0 x ξ 0 From (36) and the density of the energy, if condition b) holds, we have for a suitable constant c > 0 f(ρ) c E(ρ) if χ b (ρ) =. (37) If the second assertion of the Lemma does not hold, so that we can chose M 0, possibly decreasing its measure, such that M 0 {J uρ S d } =, then for a suitable constant c and ρ,h d (J uρ S d ) 0 σ [ln f(e σ ) (ln(e(e σ ) c )]χ b = σ ln f(e σ )χ b, σ ln f(e σ ) (d ) c ρ ρm 0 u n dx f(e σ. (38) ) This inequality relies on the derivative of the boundary integral in the expression of f with respect to the parameter ρ. In view of (38), inequality (35) becomes σ ln E(e σ ) (d )χ b (σ)+ d χ c (σ)+ f(eσ ) ε d c E(e σ ) σ ln f(e σ ) (d ) βλ ρ E(e σ ) Hd (J u ρs d ). Let c be a sufficiently small constant and γ > 0 a constant that will be fixed later. Then if ( σ ln E(e σ )[( ( E(e σ ) f(e σ ) ) γ ) ) c ] = σ ln E(e σ ), c ( ) E(e σ γ ) f(e σ ) [ c, c ], so that we are not in situation b). Being in c) we get σ ln E(e σ ) d ε d βλ ρ E(e σ ) Hd (J u ρs d ) 3

14 Otherwise, ( σ ln E(e σ )[( ( E(e σ ) f(e σ ) ) γ ) ) c ] = σ ln(e(e σ ) γ+ ) σ ln(f(e σ ) γ ) c ( + γ)(d ) + + γ c σ ln f(e σ ) (d ) ( + γ)βλ ρ c E(e σ ) Hd (J u ρs d ) γ σ ln f(e σ ). Since + γ c σ ln f(e σ ) (d ) γ( σ ln f(e σ ) (d )) γ, c c 4 for a constant c 4 small enough and independent on γ, we get ( ( E(e σ ln E(e σ σ ) ) γ ) )[( f(e σ ) c ] ) c ( + γ)(d ) γ(d + γ ( + γ)βλ ρ c 4 E(e σ ) Hd (J u ρs d ) γ σ ln f(e σ ). For γ small enough, we get for some d > d ( ( E(e σ ln E(e σ σ ) ) γ ) )[( f(e σ ) c ] ) c d ( + γ)βλ ρ E(e σ ) Hd (J u ρs d ). Summing between ρ < ρ, we conclude the proof. Corollary 3.8 There exists d = d d > d, c = c(d, Λ) and ρ 0 = ρ 0 (d, α, c α, Λ) such that x J u, 0 < ρ < ρ 0, B τρ (x) Ω and for every τ > we have either ch d ((B τρ (x) \ (x)) J u ) E(ρ), (39) or ce(τρ) τ de(ρ). (40) Proof For some τ >, we consider 0 < ρ < ρ such that ρ = τρ. Assume that assertion (34) in Lemma 3.7 holds. Then δ < H d (S d J uρ ) = Hd (S ρ J u ) ρd Hd ((B \ B ) J ρ ρ u ) ρ <ρ<ρ ρ <ρ<ρ ρ d. From the density estimate of the energy (4), relation (39) holds. Assume that (39) does not hold. Consequently, assertion (33) in Lemma 3.7 holds so we get where C depends on d and Λ. Thus (40) holds. ρ d E(ρ ) ρ d E(ρ ) C, We conclude with the following corollary, as a consequence of the Corollaries 3.3 and 3.8. Corollary 3.9 There exist constants c = c(d, Λ), ρ 0 = ρ 0 (d, α, c α, Λ) such that for every almost quasi-minimizer u, for every x J u and for every 0 < ρ < ρ 0 such that (x) Ω we have c ρd H d (J u ) cρ d. (4) 4

15 4 A free boundary problem with Robin conditions Let D be a smooth, bounded open subset of R d, d such that B D. Let 0 < α < α, 0 < β and g H (B ) such that a.e. x B α g(x) α. We consider the following free boundary problem, the unknown being a (sufficiently smooth) open set Ω and a function u H (Ω): min min u dx + β u dh d + {u > 0}. (4) B Ω D u H (Ω),u=g on B Ω Formally, if β = +, this is a free boundary problem of Bernoulli type (see for instance the seminal paper of Alt-Caffarelli []). If β is finite, (4) becomes a free discontinuity problem, with Robin boundary conditions on the unknown jump set. Indeed, if Ω is fixed and smooth enough, the boundary trace term in the formulation above is well defined and the minimizer u H (Ω) solves in Ω \ B the following elliptic equation with Robin conditions on the free boundary Ω: Ω u = 0 in Ω \ B, u n + βu = 0 on Ω \ B u = g on B. In order to deal with problem (4), we write the relaxed formin the SBV framework (see [3]) min u dx + β D [(u + ) + (u ) ]dh d + {u > 0}. J u (44) u SBV (D),u=g on B Above, SBV (D) stands for all nonnegative measurable functions u such that u SBV (D). The existence of a solution of the relaxed problem (44) is a direct consequence of [3, Theorem 3.3], once we have remarked that the minimization process has to be carried in the class of functions satisfying a.e. u(x) α. In order to apply the regularity result for almost-quasi minimizers, the key point is to prove that any solution u is minorated a.e. on its positivity region by a positive constant. Notice, that if the set {u > 0} were a priori smooth, the fact that u has a strictly positive lower bound on {u > 0} is a consequence of the Hopf s lemma and of the pointwise equality u n (x) + βu(x) = 0 on Ω, given by the Robin boundary condition. Theorem 4. For every solution u of the free discontinuity problem (44), there exists α > 0 such that u(x) α, a.e. x {u > 0}. In particular u SBV (D) and H d (J u ) < +. (43) Proof We introduce the following notations: f(s) := H d ( {u > s} \ J u ) Ω(ε, δ) := {δ < u < ε}, Ω(ε) = Ω(ε, 0) E(ε, δ) = u dx, E(ε) = E(ε, 0) Ω(ε,δ) γ(ε, δ) = H d ( Ω(ε, δ) J u ) 5

16 For 0 < δ < ε < α, we get estimates for f(s), Ω(ε, δ), E(ε, δ), γ(ε, δ) relying on the co-area formula, the isoperimetric inequality and the optimality of u. All the estimates hold for almost every ε and δ. For the simplicity of the exposition, up to the end of the proof we omit the words almost everywhere and work only with ε and δ in the complement of a zero measure set. We notice first that u Ω(ε,δ) SBV (D). By the co-area formula we have Ω(ε,δ) u dx = and by the Cauchy inequality ε f(s)ds = δ Ω(ε,δ) ε δ H d ( {u > s} \ J u )ds, u dx Ω(ε, δ) E(ε, δ). (45) Denoting by γ d the isoperimetric constant in R d, from the isoperimetric inequality we get Ω(ε, δ) d d γ d ( f(ε) + f(δ) + H d ( Ω(ε, δ) J u ) ), so Ω(ε, δ) d d γ d ( f(ε) + f(δ) + γ(ε, δ) ). (46) For almost every ε > δ > 0, the optimality condition written for u and the test function u {u ε} gives E(ε) + δ γ(ε, δ) + Ω(ε) ε f(ε). (47) The main idea of the proof relies on a sort of reverse asymptotic method and can be summarized as follows: prove that if for some interval I (0, α ) the sum I f(s)ds is small enough, then there exists a smaller non trivial interval I I such that I f(s)ds = 0. The optimality of u will directly give that u(x) sup I a.e. {u > 0}. More precisely, let η > 0. We construct two sequences (ε n ) n and (δ n ) n such that ε 0 = η, δ 0 = η, ε i = ( α i )ε i, δ i = ( + α i )δ i, where α i > 0 is chosen such that ε i ε, δ i δ, with δ < ε. We shall prove that for η small enough, there exists a suitable choice of α i such that ε δ f(s)ds = 0. (48) For simplicity, we denote a i = εi δ i f(s)ds and b i = Ω(ε i, δ i ). We start with the following estimates: for η ε > δ η, from (47) we get δ γ(ε, δ) ε f(ε) so γ(ε, δ) 4f(ε). From (46) we get Ω(ε, δ) d d γ d (5f(ε) + f(δ)) 5γ d (f(ε) + f(δ)). 6

17 Summing between [ε i, ε i ] and taking into account the range of ε i, δ i we get and further summing on [δ i, δ i ] Consequently Ω(ε i, δ i ) d [ ε i ] d αi ε i 5γ d f(s)ds + f(δ)α i ε i, ε i Ω(ε i, δ i ) d [ εi δi d α i ε i δ i 5γ d αi δ i f(s)ds + α i ε i f(s)ds ] 5γ d εα i (a i a i ). ε i δ i b d d i On the other hand, from (45) and (47) we get Finally, we have ε δ 0γ d a i a i ηα i. (49) f(s)ds Ω(ε, δ) d εf(ε) Ω(ε, δ) d Ω(ε, δ) d εf(ε) [ Ω(ε, δ) d 5γd (f(ε) + f(δ)) ] εf(ε) Ω(ε, δ) d ε(5γd ) (f(ε) + f(δ)). ε δ f(s)ds (5γ d ) Ω(ε, δ) d ε(f(ε) + f(δ)). (50) We sum again (50) between [ε i, ε i ] and [δ i, δ i ] with respect to ε and δ, respectively, and using the monotonicity of a i, b i and the range of ε and δ so α i η 4 a i (5γ d ) b d i η(a i a i ), a i 4(5γ d) b d i α i (a i a i ). (5) Assume that the choice α i = 4(5γ d ) b d i is valid, in the sense that ε > δ. Then we get that a i a i a i = a i a 0 i, (5) which leads to the conclusion (48). It remains to prove that the choice of α i is valid, i.e. ε > δ. This will be ensured by a suitable choice of η. In an equivalent way, this can be re-written as ( α i ) > ( α i ) ( + α i ) or > +. α i i= If α i 3, it is enough to have since i= i= > exp ( (3α i ) ) i= i= i= exp ( (3α i ) ) ( α i ) > ( + 3α i ) > +. α i 7 i=

18 We need that i= α i < ln 3 or, equivalently, by the definition of α i : i= b d i < ln. (5γ d ) From (49) and (5) we get so b d a d 0 i 0γ d i η b i (5γ d ) 4(5γ d ) a 0 η i d bi b d i But b i b i and if η is small enough, b i, so that we have Consequently, we get b i (5γ d ) a 0 η i.. b d i, α i i= 4(5γ d ) i= ( (5γ d ) a0 ) ( d ) d 4d η i ( a0 ) d = C d, η where C d is a constant depending only on the dimension of the space. It remains to prove that we can find some η such that From (47) we get ( a0 ) d C d η < ln 3. E(ε) + Ω(ε) ε f(ε). Summing from ε to ε and using the monotonicity of E and Ω, we get E(ε) + Ω(ε) ε ε ε s f(s)ds 4ε ε ε f(s)ds 4ε [ E(ε) + Ω(ε) ]. The last inequality above comes from (45) on [ε, ε] and the arithmetic/geometric inequality. So we get E(ε) + Ω(ε) 4ε [ E(ε) + Ω(ε) ]. In the same way hence We choose η such that ε ε ε ε f(s)ds [ E(ε) + Ω(ε) ], f(s)ds ε 4 [ E(ε) + Ω(ε) ]. 4 [ E(η) + Ω(η) ] 8 Cd d ( ln ) d 3

19 so that and the proof is complete. a 0 η = η η f(s)ds η ( ln ) d, Cd d 3 Theorem 4. Every solution u of the free discontinuity problem (44) satisfies H d (J u \ J u ) = 0 and the density property (4). Proof After a suitable rescaling in space, u turns out to be an almost-quasi minimizer, as a consequence of its optimality and of the inequality proved in Theorem 4.. The conclusion then follows is a direct consequence of Theorem 3. and Corollary 3.9. Acknowledgements. The authors wish to thank the anonymous referee for pointing out an error in the original manuscript. References [] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 35 (98), pp [] L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 000. [3] D. Bucur and A. Giacomini, A variational approach to the isoperimetric inequality for the Robin eigenvalue problem, Arch. Ration. Mech. Anal., 98 (00), pp [4] G. Dal Maso, J.-M. Morel, and S. Solimini, A variational method in image segmentation: existence and approximation results, Acta Math., 68 (99), pp [5] E. De Giorgi, M. Carriero, and A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Rational Mech. Anal., 08 (989), pp