Fine properties of the subdifferential for a class of one-homogeneous functionals

Transcription

1 Fine properties of the subdifferential for a class of one-homogeneous functionals A. Chambolle, M. Goldman M. Novaga Abstract We collect here some known results on the subdifferential of one-homogeneous functionals, which are anisotropic and nonhomogeneous variants of the total variation, and establish a new relationship between Lebesgue points of the calibrating field and regular points of the level lines of the corresponding calibrated function. Introduction In this note we recall some classical results on the structure of the subdifferential of first order one-homogeneous functionals, and we give new regularity results which extend and precise previous work by G. Anzellotti [5, 6, 7]. Given an open set R d with Lipschitz boundary, and a function u C ) BV ), we consider the functional Ju) := F x, Du) where F : R d [0, + ) is continuous in x and F x, ) is a smooth and uniformly convex norm on R d, for all x. Since BV ) L d/d ) ), it is natural to consider J as a convex, l.s.c. function on the whole of L d/d ) ), with value + when u BV ) see [2]). In this framework, for any u L d/d ) ) we can define the subgradient of a u in the duality L d/d ), L d ) as { } Ju) = g L d ) : Jv) Ju) + gx)vx) ux)) dx v L d/d ) ). CMAP, Ecole Polytechnique, CNRS, Palaiseau, France, antonin.chambolle@cmap.polytechnique.fr Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22, 0403 Leipzig, Germany, goldman@mis.mpg.de, funded by a Von Humboldt PostDoc fellowship Dipartimento di Matematica, Università di Padova, via Trieste 63, 352 Padova, Italy, novaga@math.unipd.it

2 The goal of this paper is to investigate the particular structure of the functions u and g Ju), when the subgradient is nonempty. Since J can be defined by duality as { } Ju) = sup ux)div zx) dx : z Cc ; R d ), F x, zx)) = 0 x where F x, ) is the Legendre-Fenchel transform of F x, ) it is equivalent to require that F x, zx)), F x, ) being the convex polar of F defined in 4)), it is easy to see that such a g has necessarily the form g = div z, for some field z L ; R d ) with F x, zx)) = 0 a.e. in. Since by a formal integration by parts one gets z Du = F x, Du), Du -a.e., natural questions are: in what sense can this relation be true? can one assign a precise value to z on the support of the measure Du? The first question has been answered by Anzelotti in the series of papers [5, 6, 7]. However, for the particular vector fields we are interested in, we can be more precise and obtain pointwise properties of z on the level sets of the function u. Indeed, we shall show that z has a pointwise meaning on all level sets of u, up to H d -negligible sets which is much more than Du -a.e., as illustrated by the function u = + n= 2 n χ 0,xn), defined in the interval 0, ), with x n ) a dense sequence in that interval). We will therefore focus on the properties of the vector fields z L, R d ) such that F x, zx)) = 0 a.e. in and g = div z L d ), and such that there exists a function u such that for any φ Cc ), div zx)ux)φx) dx = ux) zx) φx) dx + φx)f x, Du). In particular, one checks easily that u minimizes the functional F x, Du) gx)ux) dx ) among perturbations with compact support in. Conversely, given g L d ) with g L d sufficiently small, there exist functions u which minimize ) under various types of boundary conditions, and corresponding fields z. This kind of functionals appears in many contexts including image processing and plasticity [4, 7]. Notice also that, by the Coarea Formula [2], it holds F x, Du) gu dx = R F x, ν) {u>s} {u>s} where ν is the unit normal to {u > s}, and one can show see for instance [0]) that the characteristic function of any level set of the form {u > s} or {u s} is a minimizer of the geometric functional E g dx ) ds, F x, ν) gx) dx. 2) E 2

3 The canonical example of such functionals is given by the total variation, corresponding to F x, Du) = Du. In this case, 2) boils down to P E) gx) dx. 3) E In [8], it is shown that every set with finite perimeter in is a minimizer of 3) for some g L ). However, if g L p ) with p > d, and E is a minimizer of 2), then E is locally C,α for some α > 0, out of a closed singular set of zero H d 3 -measure []. When g L d ), the boundary E is only of class C α out of the singular set see [3]). Since the Euler-Lagrange equation of 2) relates z to the normal to E, understanding the regularity of z is closely related to understanding the regularity of E. Our main result is that the Lebesgue points of z correspond to regular points of {u > s} or {u s} Theorem 3.7), and that the converse is true in dimension d 3 Theorem 3.8). 2 Preliminaries 2. BV functions We briefly recall the definition of function of bounded variation and set of finite perimeter. For a complete presentation we refer to [2]. Definition 2.. Let be an open set of R d, we say that a function u L ) is a function of bounded variation if Du := sup u div z dx < +. z C c ) z We denote by BV ) the set of functions of bounded variation in when = R d we simply write BV instead of BV R d )). We say that a set E R d is of finite perimeter if its characteristic function χ E is of bounded variation and denote its perimeter in an open set by P E, ) := Dχ E, and write simply P E) when = R d. Definition 2.2. Let E be a set of finite perimeter and let t [0; ]. We define { } E t) := x R d E B r x) : lim = t. r 0 B r x) We denote by E := E 0) E )) c the measure theoretical boundary of E. We define the reduced boundary of E by: { } E := x Spt Dχ E ) : ν E Dχ E B r x)) x) := lim r 0 Dχ E B r x)) exists and νe x) = E 2). 3

4 The vector ν E x) is the measure theoretical inward normal to the set E. Proposition 2.3. If E is a set of finite perimeter then Dχ E = ν E H d E, P E) = H d E) and H d E \ E) = 0. Definition 2.4. We say that x is an approximate jump point of u BV ) if there exist ξ S d and distinct a, b R such that lim ρ 0 B ρ + uy) a dy = 0 and lim x, ξ) ρ 0 Bρ uy) b dy = 0, x, ξ) B + ρ x,ξ) B ρ x,ξ) where B ± ρ x, ξ) := {y B ρ x) : ±y x) ξ > 0}. Up to a permutation of a and b and a change of sign of ξ, this characterize the triplet a, b, ξ) which is then denoted by u +, u, ν u ). The set of approximate jump points is denoted by J u. The following proposition can be found in [2, Proposition 3.92]. Proposition 2.5. Let u BV ). Then, defining there holds J u Θ u and H d Θ u \J u ) = 0. Θ u := {x / lim inf ρ 0 ρ d Du B ρ x)) > 0}, 2.2 Anisotropies Let F x, p) : R d R d R be a convex one-homogeneous function in the second variable such that there exists c 0 with c 0 p F x, p) c 0 p x, p) R d R d. We say that F is uniformly elliptic if for some δ > 0, the function p F p) δ p is still a convex function. We define the polar function of F by F x, z) := sup z p 4) {F x,p) } so that F ) = F. It is easy to check that denoting the Legendre-Fenchel convex conjugate) [F x, ) 2 /2] = F x, ) 2 /2, in particular if differentiable), F x, ) p F x, ) and F x, ) z F x, ) are inverse monotone operators. If we denote by F the convex conjugate of F with respect to the second variable, then F x, z) = 0 if and only if F x, z). If F x, ) is differentiable then, for every p R d, F x, p) = p p F x, p) Euler s identity) and z {F x, ) } with p z = F x, p) z = p F x, p). 4

5 If F is elliptic and of class C 2 R d R d \ {0}), then F is also elliptic and C 2 R d R d \ {0}). We will then say that F is a smooth elliptic anisotropy. Observe that, in this case, the function F 2 /2 is also uniformly δ 2 -convex this follows from the inequalities D 2 F x, p) δ/ p I p p/ p 2 ) and F x, p) δ p ). In particular, for every x, y, z R d, there holds F 2 x, y) F 2 x, z) 2 F x, z) p F x, z)) y z) + δ 2 y z 2, 5) and a similar inequality holds for F. We refer to [6] for general results on convex norms and convex bodies. 2.3 Pairings between measures and bounded functions Following [5] we define a generalized trace [z, Du] for functions u with bounded variation and bounded vector fields z with divergence in L d. Definition 2.6. Let be an open set with Lipschitz boundary, u BV ) and z L, R d ) with div z L d ). We define the distribution [z, Du] by [z, Du], ψ = u ψ div z u z ψ ψ Cc ). Proposition 2.7. The distribution [z, Du] is a bounded Radon measure on and if ν is the inward unit normal to, there exists a function [z, ν] L ) such that the generalized Green s formula holds, [z, Du] = udiv z [z, ν]u dh d. The function [z, ν] is the generalized inward) normal trace of z on. Given z L, R d ), with div z L d ), we can also define the generalized trace of z on E, where E is a set of locally finite perimeter. Indeed, for every bounded open set with Lipschitz boundary, we can define as above the measure [z, Dχ E ] on. Since this measure is absolutely continuous with respect to Dχ E = H d E we have [z, Dχ E ] = ψ z x)h d E with ψ z L E) independent of. We denote by [z, ν E ] := ψ z the generalized inward) normal trace of z on E. If E is a bounded set of finite perimeter, by taking strictly containing E, we have the generalized Gauss-Green Formula div z = [z, ν E ]dh d. E Anzellotti proved the following alternative definition of [z, ν E ] [6, 7] E 5

6 Proposition 2.8. Let x, α) R d R d \{0}. For any r > 0, ρ > 0 we let There holds C r,ρ x, α) := {ξ R d : ξ x) α < r, ξ x) [ξ x) α]α < ρ}. [z, α]x) = lim lim ρ 0 r 0 2rω d ρ d z α C r,ρx,α) where ω d is the volume of the unit ball in R d. 3 The subdifferential of anisotropic total variations 3. Characterization of the subdifferential The following characterization of the subdifferential of J is classical and readily follows for example from the representation formula [9, 4.9)]. Proposition 3.. Let F be a smooth elliptic anisotropy and g L d ) then u is a local minimizer of ) if and only if there exists z L ) with div z = g, F x, zx)) = 0 a.e. and [z, Du] = F x, Du). Moreover, for every t R, for the set E = {u > t} there holds [z, ν E ] = F x, ν E ) H d -a.e. on E. We will say that such a vector field is a calibration of the set E for the minimum problem 2). Remark 3.2. In [5], it is proven that if z ρ x) := B ρx) B zy) dy, then z ρx) ρ ν E weakly* converges to [z, ν E ] in L d loc H E). Using 5) it is then possible to prove that if z calibrates E then z ρ converges to p F x, ν E ) in L 2 H d E) yielding that up to a subsequence, z φρ) converges H d -a.e. to p F x, ν E ). Unfortunately this is still a very weak statement since it is a priori impossible to recover from this the convergence of the full sequence z ρ. The main question we want to investigate now is whether we can give a classical meaning to [z, ν E ] that is understand if [z, ν E ] = z ν E ). We observe that a priori the value of z is not well defined on E which has zero Lebesgue measure since z has Lebesgue points only a.e.). We let S := suppdu) be the smallest closed set in such that Du \ S) = 0. The next result is classical. Lemma 3.3 Density estimate). There exists ρ 0 > 0 depending on g) and a constant γ > 0 which depends only on d), such that for any B ρ x) with ρ ρ 0, and any level set E of u that is, E {{u > s}, {u s}, {u < s}, {u s}, s R}), if B ρ x) E < γ B ρ x) then B ρ/2 x) E = 0. As a consequence, E 0 and E are open, E is the topological boundary of E, and possibly changing slightly γ) if x E, then H d E B ρ x)) γρ d. 6

7 For a proof we refer to [3, 2]. This is not true anymore if g L d ) [2]. If is Lipschitz, it is true up to the boundary. Corollary 3.4. It follows that u L loc ) and u C\Θ u). Proof. For any ball B ρ x) and inf Bρ/2 x) u < a < b < sup Bρ/2 x) u, one has + > Du B ρ x)) b a ρ ) d P {u > s}, B ρ x)) ds b a)γ, 2 so that osc Bρ/2 x)u) must be bounded and thus u L loc ). Moreover, if x \Θ u we find that lim ρ 0 osc Bρx)u) = 0 so that u is continuous at the point x. It also follows from Lemma 3.3 that all points in the support of Du must be on the boundary of a level set of u: Proposition 3.5. For any x S, there exists s R such that either x {u > s} or x {u s}. Proof. First, if x S then Du B ρ x)) = 0 for some ρ > 0 and clearly x cannot be on the boundary of a level set of u. On the other hand, if x S, then for any ball B /n x) n large) there is a level s n uniformly bounded) with {u > s n } B /n x) and by Hausdorff convergence, we deduce that either x {u > s} or x {u s} where s is the limit of the sequence s n ) n which must actually converge). The following stability property is classical see e.g. []). Proposition 3.6. Let E n be local minimizers of 2), with a function g = g n L d ), and converging in the L -topology to a set E. Assume that the sets E n are calibrated by z n, that z n z weakly- in L and g n g = div z L d ), in L ) as n. Then z calibrates E, which is thus also a minimizer of 2). In particular, one must notice that when z n z and F x, z n ) a.e., then in the limit one still has F x, z) a.e. thanks to the convexity, and continuity w.r. the variable x). 3.2 The Lebesgue points of the calibration. The next result shows that the regularity of the calibration z implies some regularity of the calibrated set. Theorem 3.7. Let x E be a Lebesgue point of z, with E = {u > t} or E = {u t}. Then, x E and z x) = p F x, ν E x)). 6) 7

8 Proof. We follow [, Th. 4.5] and let z ρ y) := z x + ρy). Since x is a Lebesgue point of z, we have that z ρ z in L B R ), hence also weakly- in L B R ) for any R > 0, where z R d is a constant vector. We let E ρ := E x)/ρ and g ρ y) = g x + ρy) so that div z ρ = ρg ρ ). Observe that E ρ minimizes F x + ρy, ν Eρ y)) dh d y) + ρ E ρ B R g ρ y) dy, E ρ B R with respect to compactly supported perturbations of the set in the fixed ball B R ). Also, ρg ρ Ld B R ) = g Ld B ρr ) ρ 0 0. By Lemma 3.3, the sets E ρ and the boundaries E ρ ) satisfy uniform density bounds, and hence are compact with respect to both local L and Hausdorff convergence. Hence, up to extracting a subsequence, we can assume that E ρ Ē, with 0 Ē. Proposition 3.6 shows that z is a calibration for the energy Ē B F x, νēy)) dh d y), and that R Ē is a minimizer calibrated by z. It follows that [ z, νē] = F x, νēy)) for H d -a.e. y in Ē, but since z is a constant, we deduce that Ē = {y ν 0} with ν/f x, ν) = pf x, z). In particular the limit Ē is unique, hence we obtain the global convergence of E ρ Ē, without passing to a subsequence. We want to deduce that x E, with ν E x) = F x, ν E x)) p F x, z), which is equivalent to 6). The last identity is obvious from the arguments above, so that we only need to show that Assume we can show that lim ρ 0 Dχ Eρ B ) Dχ Eρ B ) = ν. 7) lim Dχ E ρ B R ) = Dχ Ē B R ) = ω d R d ) 8) ρ 0 for any R > 0, then for any ψ Cc B R ; R d ) we would get ψ Dχ Eρ = div ψx) dx Dχ Eρ B R ) B R Dχ Eρ B R ) B R E ρ div ψx) dx = ψ Dχ Dχ Ē B R ) B R Ē Dχ Ē B R ) Ē B R and deduce that the measure Dχ Eρ / Dχ Eρ B R )) weakly- converges to Dχ Ē / Dχ Ē B R )). Using again 8)), we then obtain that lim ρ 0 Dχ Eρ B R ) Dχ Eρ B R ) = ν 9) We use here that F x, ) F x, ) = [F x, ) F x, )], so that z = F x, νēy)) implies both F x, z) = and νēy)/f x, νē)y) = F x, z) 8

9 for almost every R > 0. Since Dχ Eρ B µr )/ Dχ Eρ B µr )) = Dχ Eρ/µ B R )/ Dχ Eρ/µ B R )) for any µ > 0, 9) holds in fact for any R > 0 and 7) follows, so that x E. It remains to show 8). First, we observe that, by minimality of E ρ and Ē plus the Hausdorff convergence of E ρ in balls, we can easily show the convergence of the energies lim F x + ρy, ν Eρ y)) dh d y) + ρ g ρ y) dy ρ 0 E ρ B R E ρ B R = Ē B R F x, νēy)) dh d y) and, by the continuity of F, lim ρ 0 E ρ B R F x, ν Eρ y)) dh d y) = Ē B R F x, νēy)) dh d y). 0) Then, 7) follows from Reshetnyak s continuity theorem where, instead of using the Euclidean norm as reference norm, we use the uniformly convex norm F x, ) and the convergence of the measures F x, Dχ Eρ ) to F x, Dχ Ē ) see [5, ]). In dimension 2 and 3 we can also show the reverse implication, proving that regular points of the boundary corresponds to Lebesgue points of the calibration. The idea is to show that the parameters r, ρ in Proposition 2.8 can be taken of the same order. Theorem 3.8. Assume the dimension is d = 2 or d = 3. Let x, s be as in Proposition 3.5, E be a minimizer of 2) and assume x E. Then x is a Lebesgue point of z and zx) = p F x, ν E ). Proof. We divide the proof into two steps. Step. We first consider anisotropies F which are not depending on the x variable. Without loss of generality we assume x = 0. By assumption, there exists the limit ν = lim ρ 0 Dχ E B ρ 0)) Dχ E B ρ 0)) ) and, without loss of generality, we assume that it coincides with the vector e d corresponding to the last coordinate of y R d. Also, if we let E ρ = E/ρ, the sets E ρ, E c ρ, E ρ converge in B 0), in the Hausdorff sense thanks to the uniform density estimates), respectively to {y d 0}, {y d = 0}, {y d 0}. We also let z ρ y) = zρy) and g ρ y) = gρy), in particular div z ρ = ρg ρ. We let ωρ) = sup g L d B ρx) ) 2) x 9

10 which is continuously increasing and goes to 0 as ρ 0, since g d is equi-integrable. We introduce the following notation: a point in R d is denoted by y = y, y d ), with y R d. We let D s := { y s}, z := F ν) and Ds t = {D s + λ z : λ t} and denote with D s the relative boundary of D s in {y d = 0}. We choose s, 0 < t s, t is chosen small enough so that Ds t B 0), that is t < / z ) s 2 ). We integrate in Ds t the divergence ρg ρ = div z ρ = div z z ρ ) against the function 2χ E )t ν y F ν), which vanishes for y d = ±tf ν) if ρ is small enough given t > 0), so that E ρ B 0) { y d tf ν)}. For y on the lateral boundary of the cylinder Ds, t let ξy) be the internal normal to D s + t, t) z at the point y. Using the fact that z ρ is a calibration for E ρ, we easily get that for almost all s, D t s ρg ρ 2χ E )t ν y F ν) = D s+ t,t) z 2t E ρ D t s ) dy 2χ E )t ν y ) [ z z ρ ), ξy)] dh d F ν) z ν E ρ F ν Eρ ) ) dh d + D t s z ) ρ ν dy. 3) F ν) Now since F F ν)) =, there holds z ν Eρ F ν Eρ ) 0 and using that z ξy) = 0 on D s + t, t) z, we get D t s z ) ρ ν dy F ν) D t s ρg ρ 2χ E )t ν y F ν) D s+ t,t) z We claim that for ξ with ξ z = 0, there holds ) dy 2χ E )t ν y F ν) ) z ρ ξy) dh d. 4) ξ z ρ ) 2 CF ν) ν z ρ ) 5) Since it is enough to prove ξ z ρ ) 2 z ρ 2 [z ρ z/ z )] 2 z ρ 2 [z ρ z/ z )] 2 CF ν) ν z ρ ). Using that ν/f ν) = F z), from 5) applied to F together with F z) = F z ρ ), we find F ν) ν z ρ ) = F ν) z ρ F z)) C z ρ z 2. 0

11 which readily implies 5). We thus have D s+ t,t) z Now, we also have 2χ Eρ )t ν y ) z ρ ξ) dh d F ν) 2C F ν)t z ρ ν d dh F ν) 2CF ν)t t D s+ t,t) z D s+ t,t) z ρ 2χ Eρ )t ν y ) g ρ y) dy 2tρ d gx) dx D F ν) s t Dρs ρt z ) ) 2 ρ ν dh d H F ν) d 2 D s ). 6) 2tρ d g L d B ρs0)) D ρt ρs /d ct 2 /d s d 2+/d ωρs) 7) where here, c = 2H d D ) /d, and ω is defined in 2). We choose a <, close to, and choose t 0, / z ) a 2 ). If ρ > 0 is small enough so that E ρ B is in { y d tf ν)}), letting fs) := ) zρ ν Ds t F ν) dy, we deduce from 4), 6) and 7) that for a.e. s with t s a, one has possibly increasing the constant c) fs) 2 c s d 2 t 3 f s) + t 4 2/d s 2d 4+2/d ωρs) 2). 8) Unfortunately, this estimate does not give much information for d > 3. It seems it allows to conclude only whenever d {2, 3}. Since the case d = 2 is simpler, we focus on d = 3. Estimate 8) becomes fs) 2 c st 3 f s) + t 0/3 s 8/3 ωρs) 2). 9) Given M > 0, we fix a value t > 0 such that loga/t) cm. If ρ is chosen small enough, then E ρ B 0) { y d < tf ν)}, and 9) holds. It yields assuming ft) > 0, but if not, then the Proposition is proved) f s) fs) 2 + ct 3 s ct/3 5/3 ωρs)2 s fs) 2 ct /3 5/3 ωaρ)2 s ft) 2 20) where we have used the fact that t s a and f, ω are nondecreasing. Integrating 20) from t to a, after multiplication by t 3 we obtain Hence we get t 3 fa) t3 ft) + loga/t) c t 3 ft) + ca8/3 t 8/3 ωaρ) 2 t 6 3c 8 t0/3 a 8/3 t 8/3 ) ωaρ)2 ft) 2. M. 2) ft) 2

12 Eventually, we observe that ft) = zρy) ν ) dy = F ν) ρ d D t t D ρt ρt zx) ν ) dx, F ν) so that 2) can be rewritten D ρt ρt ) zx) ν F ν) dx ρt) Mca 8/3 t 8/3 ωaρ) 2 2ca 8/3 t 8/3 ωaρ) 2 22) The value of t being fixed, we can choose the value of ρ small enough in order to have 4Mca 8/3 t 8/3 ωaρ) 2 <, and using + X + X/2 X 2 /8 if X 0, )), 22) yields It follows that D ρt ρt ) zx) ν F ν) dx ρt) 3 lim sup ε 0 D ε ε M M 2 ca 8/3 t 8/3 ωaρ) M. 23) ) zx) ν F ν) dx ε M 24) and since M is arbitrary, 0 is indeed a Lebesgue point of z, with value z = F ν) recall that zx) ν F ν) C/F ν)) zx) z 2 ). Step 2. When F depends also on the x variable, the proof follows along the same lines as in Step, taking into account the errors terms in 4) and 6). Keeping the same notations as in Step and setting z := p F 0, ν) we find that since F 0, z), there holds z ν Eρ F 0, ν Eρ ) and thus z ν Eρ F ρx, ν Eρ )dh d F 0, ν Eρ ) F ρx, ν Eρ ) dh d Cρs d E ρ D t s E ρ D t s where the last inequality follows from t s and the minimality of E ρ inside D t s. Now since F ) 2 0, z ρ ) F ) 2 ρx, z ρ ) F ) 2 ν 0, z ρ ) 2 F 0, ν) z ρ z) + δ 2 z ρ z 2 we find that 5) transforms into, [ ] ξ z ρ ) 2 C F 0, ν) ν z ρ ) + F ) 2 0, z ρ ) F ) 2 ρx, z ρ )) 2

13 for every ξ and ξ z = 0, from which we get D s+ t,t) z 2χ Eρ )t ν y ) z ρ ξ) dh d F ν) 2CF 0, ν)t t CF 0, ν)t t + 2Ct D s+ t,t) z D s+ t,t) z D s+ t,t) z z ) ) 2 ρ ν dh d H F 0, ν) d 2 D s ) F ) 2 0, z ρ ) F ) 2 ρx, z ρ ) /2 dh d z ) ρ ν F 0, ν) dh d ) 2 H d 2 D s ) + Ctρ /2 s d t. Using these estimates, we finally get that, setting as before fs) := ) zρ ν Ds t F 0,ν) dy, there holds fs) 2 c s d 2 t 3 f s) + t 4 2/d s 2d 4+2/d ωρs) 2 + ρts d + ρ /2 t 2 s d ). From this inequality, the proof can be concluded exactly as in Step. Eventually, we can also give a locally uniform convergence result. Proposition 3.9. For all x we let z ρ x) := B ρ 0) Then, F x, z ρ x)) locally uniformly on S. B ρx) zy) dy. Proof. Given K a compact set, we can check that for any t > 0, there exists ρ 0 > 0 such that for any x K S, if E x is the level set of u through x, then for any ρ ρ 0, the boundary of E x x)/ρ B 0) lies in a strip of width 2t, that is, there is ν x S d with E x x)/ρ) B 0) { y ν x t}). Indeed, if this is not the case, one can find t > 0, ρ k 0, x k K S, such that E x k x k )/ρ k ) B 0) is not contained in any strip of width 2t. Up to a subsequence we may assume that x k x K S, and from the bound on the perimeter, that E x k x k )/ρ k B 0) converges to a local minimizer of E F 0, νe )dh d and is thus a halfspace. 2 Moreover, E x k x k )/ρ k ) B 0) converges in the Hausdorff sense thanks to the density estimates) to a hyperplane. We easily obtain a contradiction. The thesis follows when we observe that the proof of Proposition 3.8 can be reproduced by replacingthe direction ν Ex x) which exists only if x lies in the reduced boundary of E x ) with the direction ν x given above. 2 If d = 2, this Bernstein result readily follows from the strict convexity of F, see [, Prop 3.6] whereas for d = 3, see [8]. In the case of the area i.e when F x, Du) = Du and d 7, see also [2, Rem 3.2]. 3

14 3.3 A counterexample. We provide an example where g L d ε ), with ε > 0 arbitrarily small, and Theorem 3.8 does not hold. Let = B 0) be the unit ball of R d and let E = {x d 0}. We shall construct a vector field z : R d such that z = ν E on E, z everywhere in, divz L d ε ), but 0 is not a Lebesgue point of z. Notice that E minimizes the functional 3) with g = divz. Letting r n 0 be a decreasing sequence to be determined later, and let B n = B rn x n ) with x n = 2r n e d. Without loss of generality, we may assume r n+ < r n /4 so that the balls B n are all disjoint. We define the vector field z as follows: zx) = e d if x \ n B n, and zx) = x x n e d if x B n. It follows that divz = 0 in \ n B n and divz /r n in B n, so that divz d ε dx = divz d ε dx ω d rn ε < + n B n if we choose r n converging to zero sufficiently fast, so that g = div z L d ε ). However, since z e d /2 in B rn/2x n ), we also have so that B 3rn 0) z e d dx B 3rn 0) 2 B 3rn 0) B 3rn 0) n Brn/2x n ) z e d dx 6 d <. On the other hand, for δ 0, /6 d ) we have z e d dx B rn 0) B rn 0) B rn 0) B rn 0) i=n+ B ri x i ) ) δ, if we take the sequence r n converging to 0 sufficiently fast. It follows that 0 is not a Lebesgue point of z. References [] F.J. Almgren, R. Schoen, L. Simon, Regularity and singularity estimates on hypersurfaces minimizing elliptic variational integrals, Acta Math., ), [2] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Science Publications, [3] L. Ambrosio, E. Paolini, Partial regularity for quasi minimizers of perimeter, Ricerche Mat ), suppl.,

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