Regularity of minimizers for free-discontinuity problems with $p(\cdot)$-growth

A regularity result for free-discontinuity energies defined on the space $SBV^{p(\cdot)}$ of special functions of bounded variation with variable exponent is proved, under the assumption of a log-H\"older continuity for the variable exponent $p(x)$. Our analysis expand on the regularity theory for minimizers of a class of free-discontinuity problems in the nonstandard growth case. This may be seen as a follow-up of the paper Fusco, Mingione and Trombetti (2001), dealing with a constant exponent.


Introduction
Integral functionals with non-standard growth, introduced by Zhikov [35,36], are customary in the modeling of composite materials which exhibit a strongly anisotropic behavior.In such a setting, a point-dependent integrability of the deformation gradient is usually assumed, which may be captured, for instance, in terms of variable exponents spaces (see [30,34]).Over the years, the regularity properties of minimizers (in the Sobolev space W 1,p(•) (Ω; R m ), where Ω is a reference configuration) of variational integrals of the form ˆΩ f (x, ∇u(x)) dx , under a p(x)-growth condition has been the subject of many contributions.Among them, we may mention [2,3,11,18,19,29,37].The common key assumptions to these papers are superlinearity of p(•) (meaning min Ω p(•) > 1), and the (possibly strong) log-Hölder continuity of the exponent.This condition on the modulus of continuity of the variable exponent was firstly considered by Zhikov in [37] to prevent the Lavrentiev phenomenon.Roughly speaking, it allows one to freeze the exponent on small balls around a point, as pointed out in [15,Lemma 3.2] and is particularly suitable for blow-up methods.Besides regularity issues, we may point out for instance its usage in [1], in order to show that the singular part of the measure representation of relaxed functionals with growth (1.2) disappears.The focus of our paper is, instead, on the regularity of minimizers of free-discontinuity functionals in variable exponent spaces.In such a setting, which is rather natural to describe failure phenomena such as fracture and damage, singularities may appear in the form of jump discontinuities.These problems are characterized by the competition between a "bulk" energy of the form (1.1) and a "surface" energy accounting for the energy spent to produce a crack ( [26,22]).A prototypical functional is then taking the form ˆΩ f (x, ∇u(x)) dx + ˆJu g x, u + (x), u − (x), ν u (x) dH d−1 (x) . ( Above, J u is the set of jump discontinuities of u with normal ν u , which, exactly like the gradient ∇u and the one-sided traces u + (x), u − (x), have in general to be understood in an approximate measure-theoretical sense (see Section 2.2).The p(•)-growth condition (1.2) is assumed on f , while g is bounded from above and from below by positive constants.
The above problem is usually complemented with lower order fidelity terms or boundary data, which, whenever p(•) is superlinear, allow one to apply the results of [5,6] (see also [23] for the case of boundary data) and obtain sequential coercivity of the functional in the space of Special functions of Bounded Variation (SBV ), see [7].Under the BV -ellipticity of g, which provides lower semicontinuity of the surface integral, the well-posedness of the minimum problem (1.3) in the subspace SBV p(•) of SBV functions with p(•)-integrable gradients can be then inferred from the results of [13], whenever f is convex in the gradient variable, or more in general of [12], whenever f is quasiconvex and and the exponent is log-Hölder continuous.We also refer the reader to the recent [4], where mere continuity of p is shown to be sufficient for lower semicontinuity of the bulk energy.We however warn the reader that log-Hölder continuity is again going to play a central role when coming to regularity issues.It is also worth mentioning that, besides Materials Science, applications of a variable exponent in the setting of functions of bounded variation already appeared in image reconstruction [10,27,28,31].This is the setting where free-discontinuity problems were originally introduced [32].Now, for g ≡ c, with c > 0, (1.3) can be seen as a weak formulation of the problem ˆΩ\K f (x, ∇u(x)) dx + cH d−1 (K) (1.4) for a closed set K, not prescribed a-priori, and a deformation u which is smooth outside of K.
One is then willing to show that, given a minimizer u of (1.3), the pair (u, J u ) indeed provides a minimizer of (1.4).This is clearly the case whenever H d−1 (J u \ J u ) = 0; notice that, once this is achieved, setting K = J u the smoothness of u in Ω \ K can be then deduced from the regularity theory for variational integrals in Sobolev spaces.For the model case with p a constant exponent, this difficult task was first accomplished in the seminal paper [14].There, it was namely shown that H d−1 (S u \ S u ) = 0, where S u is the singular set of u, a superset of J u which, in a BV setting, differs therefrom only by a H d−1 -null set.A crucial estimate in order to get their result was the so-called density lower bound for x 0 ∈ S u and sufficiently small balls B ρ (x 0 ), with θ 0 independent of x 0 and ρ.This requires a fine decay analysis for the energy on small balls, which is partially simplified by the homogeneity of the bulk energy, ensuring that minimality is invariant under suitable rescaling.
If one renounces to homogeneity, regularity results for minimizers of a class of nonhomogenous bulk integrands with p-growth in (1.3) have been obtained in [24], whose results we extend to the variable growth setting.Description of our results.In this paper, we focus on the scalar-valued case u : Ω → R and consider f (x, ξ) = |ξ| p(x) + h(x, ξ) in (1.3), where h is a continuous function, convex in ξ and has p(x)-growth.We assume that the variable exponent is strongly log-Hölder continuous (see (2.4)) and show that minimizers of the weak formulation (1.3) are strong minimizers in the sense clarified above.
The proof of the crucial density lower bound goes, exactly as in [14,24], through a decay Lemma (see Lemma 4.4).One assumes by contradiction that the energy is decaying faster than ρ d−1 around a jump point x 0 .Setting p = p(x 0 ), one may exploit smallness of the energy to show that a scaled copy of blown-up sequences converges to a smooth minimizer of a variational integral of the type for which decay estimates independent of p are available by [3], and provide a contradiction.The function h ∞ is recovered as locally uniform limit of a scaled version of h acting on a scaled deformation gradient.The strong log-Hölder continuity assumption is crucial for a proper choice of the scaling constants, despite the presence of a variable exponent, in order to ensure minimality of the limit function.This also requires the proof of a Γ-liminf type inequality (see Step Outline of the paper.The paper is structured as follows.In Section 2 we fix the basic notation and recall some definitions in variable exponent spaces (Subsection 2.1) and in SBV (Subsection 2.2), while Subsection 2.3 deals with the space SBV p(•) , the Poincaré inequality and some of its consequences useful in the sequel.In Subsection 2.4 we recall a Lusin-type approximation result in SBV p(•) , while Subsection 2.5 contains some definitions and results for free-discontinuity problems in the variable exponent setting.Section 3 is entirely devoted to the proof of a technical tool concerning the asymptotic behavior of almost minimizers with small jump sets.The main result of the paper is contained in Section 4: in Subsection 4.1 we state the problem and list the main assumptions on the energies, in Subsection 4.2 we establish a crucial decay estimate for our functionals while in Subsection 4.3 we prove a density lower bound and then the main result with Theorem 4.7.

Basic notation and preliminaries
We start with some basic notation.2.1.Variable exponent Lebesgue spaces.We briefly recall the notions of variable exponents and variable exponent Lebesgue spaces.We refer the reader to [16] for a comprehensive treatment of the topic.
A measurable function p : Ω → [1, +∞) will be called a variable exponent.Correspondingly, for every A ⊂ Ω we define The variable exponent Lebesgue space L p(•) (Ω) is defined as the set of measurable functions u such that ̺ p(•) (u/λ) < +∞ for some λ > 0. In the case p + < +∞, L p(•) (Ω) coincides with the set of functions such that ̺ p(•) (u) is finite.It can be checked that while an analogous inequality holds by exchanging the role of p − and p + if 0 ≤ u L p(•) (Ω) ≤ 1.Another useful property of the modular, in the case p + < +∞, is the following one: for all λ > 0.
We say that a function p : Ω → R is log-Hölder continuous on Ω if the modulus of continuity for p(x) satisfies with C a positive constant.In other words lim sup To prove our regularity result the previous condition will be reinforced into the strong log-Hölder continuity in complete accordance with the theory of regularity in the variable Sobolev framework (see [3]).
The following lemma provides an extension to the variable exponent setting of the well-known embedding property of classical Lebesgue spaces (see, e.g., [16,Corollary 3.3.4]).
Lemma 2.1.Let p, q be measurable variable exponents on Ω, and assume that L d (Ω) < +∞.Then L p(•) (Ω) ֒→ L q(•) (Ω) if and only if q(x) ≤ p(x) for L d -a.e.x in Ω.The embedding constant is less or equal to 2(1 + L d (Ω)) and 2 max{L d (Ω) 2.2.BV and SBV functions.For a general survey on the spaces of BV and SBV functions we refer for instance to [7].Below, we just recall some basic definitions useful in the sequel.If u ∈ L 1 loc (Ω) and x ∈ Ω, the precise representative of u at x is defined as the unique value The set of points in Ω where the precise representative of x is not defined is called the approximate singular set of u and denoted by S u .We say that a point x ∈ Ω is an approximate jump point of u if there exist a, b ∈ R and ν ∈ S ).The Borel functions u + and u − are called the upper and lower approximate limit of u at the point x ∈ Ω.The set of approximate jump points of u is denoted by J u ⊆ S u .
The space BV (Ω) of functions of bounded variation is defined as the set of all u ∈ L 1 (Ω) whose distributional gradient Du is a bounded Radon measure on Ω with values in R d .Moreover, the usual decomposition holds, where ∇u is the Radon-Nikodým derivative of Du with respect to the Lebesgue measure and D c u is the Cantor part of Du.If u ∈ BV (Ω), then ∇u(x) is the approximate gradient of u for a.e.x ∈ Ω: For the sake of simplicity, we denote by ; so in the sequel we shall essentially identify the two sets.We recall that the space SBV (Ω) of special functions of bounded variation is defined as the set of all u ∈ BV (Ω) such that D s u is concentrated on S u ; i.e., |D s u|(Ω \ S u ) = 0. Finally, for p > 1 the space SBV p (Ω) is the set of u ∈ SBV (Ω) with ∇u ∈ L p (Ω; R d ) and In order to state a Poincaré-Wirtinger inequality in SBV p(•) , we first fix some notation, following [9].With given a, b ∈ R, we denote a ∧ b := min(a, b) and a ∨ b := max(a, b).Let B be a ball in R d .For every measurable function u : B → R, we set we define and the truncation operator where γ iso is the dimensional constant in the relative isoperimetric inequality.
For any M > 0, we also define We recall the following Poincaré-Wirtinger inequality for SBV functions with small jump set in a ball, which was first proven in the scalar setting in [14, Theorem 3.1], and then extended to vector-valued functions in [9, Theorem 2.5].
Theorem 2.2.Let u ∈ SBV (B) and assume that and where p * := dp d−p .If p ≥ d, then, for any q ≥ 1, As a first application of Theorem 2.2 one can obtain the following sufficient condition for the existence of the approximate limit at a given point (see [7,Theorem 7.8]).
Another consequence of Theorem 2.2 is the following compactness result, which is a slight extension of the result established in [33,Theorem 2.8] (see also [12,Theorem 4.1] for a related result under the additional stronger assumption (2.3)).Motivated by the blow-up analysis of Lemma 4.4, we will prove the result for a fixed ball and a uniformly convergent sequence of continuous variable exponents p h : B → (1, +∞) satisfying: Then there exist a function u 0 ∈ W 1,p(•) (B) and a subsequence (not relabeled) of {u h } such that (2.14) Proof.The extension with respect to Theorem 2.8 in [33] regards only the lower semicontinuity inequality in (2.14).We repeat the argument, since this gives us the occasion to develop some details.First, observe that p(•) also satisfies (2.11) with the same constants.We set for brevity ūh := T B u h − med(u h ; B) and we distinguish two cases, according to the values of p − and p + . Step meaning that u M 0 ∈ W 1,1 (B).Now we apply De Giorgi's semicontinuity Theorem ( [13]) to the integral functional ´B f (p(y), w(y)) dy defined by f : R × R d → [0, +∞) as f (p, ξ) = |ξ| p∨1 .The continuous function f is convex in ξ, thus ensuring the sequential lower semicontinuity of the functional whenever p j strongly converges in L 1 (B) and w j weakly converges in L 1 (B, R d ).This implies that and Therefore, letting M → +∞, we obtain that u 0 ∈ W 1,p(•) (B) and (2.14)(iii) follows.Thanks to (2.15), since p + < (p − ) * , we get that |ū h | p + is equiintegrable, hence ūh strongly converges to u 0 in L p + (B).Now, (2.14)(i) follows from Lemma 2.1, while (2.14)(ii) and (iv) can be inferred from (2.9) and (2.13).
Step 2.Here we assume p − ≥ d.In this case, the proof goes exactly as in Step 1 using (2.10) of Theorem 2.2 instead of (2.8).
Remark 2.5.We observe that the previous assumptions (2.11) and (2.12) are always satisfied if the sequence p h (•) converges uniformly to a constant function p.In fact, if p ≤ d, then we can find η > 0 such that, for h large, p − η < p h (•) < p + η and p + η < (p − η) * .
If p > d, then we can find η > 0 such that, for h large, and these are the two cases in the previous theorem.

2.4.
Lusin approximation in SBV p(•) .Let µ be a positive, finite Radon measure in R d .The maximal function is defined as As a consequence of the Besicovitch covering theorem (see, e.g., [20]), it can be shown that for a constant c depending only on d.
We recall the Lipschitz truncation result for SBV p(•) functions, proved in [12, Theorem 3.1] (see also [17]), which we state here in a slightly modified version suitable for our purposes.
Theorem 2.6.Let Ω ⊂ R d be an open bounded set with Lipschitz boundary, and let p : Ω → (1, +∞) be log-Hölder continuous on Ω and 1 < p − ≤ p(x) ≤ p + < ∞, for every x ∈ Ω.Let {v h } h∈N ⊂ SBV p(•) (Ω) be a sequence of functions with compact support in Ω, such that Let {θ h } h∈N be a sequence of strictly positive numbers such that θ h → 0 and Then for every h there exist sequences µ j , λ h,j > 1 such that for every h, j ∈ N and there exists a sequence {v h,j } ⊂ W 1,∞ (Ω) such that for every h, j for some constant C depending on d, p − , p + , the log-Hölder constant of p(•), and also on v h in terms of v h W 1,p(•) (Ω) .Moreover, up to a null set, for some constant K depending on d, p − , p + , the log-Hölder constant of p(•), and also on Finally, there exists a sequence ε j > 0 with ε j → 0 such that for every h, j ∈ N, 2.5.Free-discontinuity functionals with p(•)-growth.In this paragraph we consider integral functionals of the form defined on SBV loc (Ω), where c > 0 and A ⊂ Ω is an open set.The Carathéodory function f : Ω × R d → R will be supposed to satisfy the following growth condition: for any ξ ∈ R d , a.e.x ∈ Ω, where L ≥ 1 and the variable exponent p : Ω → (1, +∞) is a bounded function.We will write F (u, A) for F (u, 1, A).
We recall the classical definition of deviation from minimality (see, e.g., [7]), which gives an estimate of how far is u from being a minimizer of F in Ω. (2.20)

Asymptotic behavior of almost minimizers with small jump sets
This section is devoted to the proof of an auxiliary Γ-convergence type result for sequences of functionals with variable growth.We will namely consider a sequence of continuous functions for any ξ ∈ R d , x ∈ Ω, p h : Ω → (1, +∞) uniformly log-Hölder continuous satisfying (2.3) and (2.11), and and we write Dev h (v, c, B ρ ) for the corresponding deviation from minimality.The next theorem describes the behavior of a sequence u h of "almost minimizers" of functionals F h when the functions f h converge to f uniformly on compact sets and the measures of the discontinuity sets S u h are infinitesimal.The proof follows the lines of the proof of Theorem 2.6 in [24], taking advantage of a lower semicontinuity result proved in [12,Theorem 4.1] in the SBV setting of variable growth, here slightly modified.
+∞) be a sequence of continuous functions, convex with respect to the second variable, satisfying (3.1), with p h converging uniformly to some p ∈ (1, +∞) Proof.Replacing u h with u h − m h , we may always assume that m h = 0 for all h.By virtue of assumptions (3.2) and (3.3) we may appeal to Theorem 2.4 (see also Remark 2.5).Thanks to this and to an inspection of its proof, we have that where we still used the notation ūh := T B R u h .We divide the proof into three steps.
Step 1: a lower semicontinuity result.We claim that for a.e.ρ ∈ (0, R), it holds which in turn, taking into account (3.3), implies If we show the lower semicontinuity inequality (3.8) for ∇ū M h ⇀ ∇u M 0 weakly in L 1 , the claim will be proved, since f h (y, 0) → f (0) = 0 uniformly on B ρ .So, replacing ūh with ūM h we can suppose that ūh is bounded uniformly with respect to h.Let us denote v h := ūh − u 0 , so that v h → 0 in L q (B R ), for every q ≥ 1.Now we adapt the proof of [12,Theorem 4.1] to our setting.Even if only a few changes are significant, we will write it for the sake of completeness.
Possibly multiplying v h by a cut-off function ζ ∈ C ∞ (R d ), with compact support in B R and identically equal to 1 in B ρ , we can assume, without loss of generality, that the functions So we can apply Theorem 2.6 to v h : given θ h , with γ h /θ h → 0, there exist sequences ε j ∈ (0, 1), µ j , λ h,j > 1, with ε j → 0, µ j → +∞, and µ j ≤ λ h,j ≤ µ j+1 , and there exists a sequence with K and C two constants independent of h, j, η.Moreover, for every j ∈ N and for h → ∞, For any r > 0, we have where E r = {y ∈ B ρ : |∇u 0 | ≤ r}.Recalling the convergence of f h to f , we deduce that where the De Giorgi's semicontinuity Theorem ( [13]) and the growth assumption on f have been applied for the last inequality.We now treat the last term: Recalling (2.17), we have which is infinitesimal as h → +∞, since µ j ≤ λ h,j ≤ µ j+1 .Thus we are left to estimate ), for which we can use (3.10), so that In conclusion, we obtained lim inf h→+∞ ˆBρ f h (y, ∇ū h ) dy ≥ ˆEr f (∇u 0 ) dy, and letting r tend to +∞, we proved the lower semicontinuity result (3.8).
Step 2: asymptotics.Now we integrate H d−1 ({ ũh = ũh } ∩ ∂B ρ ) with respect to ρ, and using coarea formula and (2.9), we obtain We will prove that lim h→+∞ a h = 0. We need to distinguish two cases, according to the value of the limit c ∞ := lim h c h , which exists up to a subsequence (not relabeled).If c ∞ < +∞, the assertion immediately follows from (3.3).If c ∞ = +∞, from (3.2) we have and again the thesis follows.
Then, up to a subsequence, we may assume that, for almost every ρ ∈ (0, R), Since for any h and for L 1 -a.e.ρ ∈ (0, R), H d−1 (S ūh ∩ ∂B ρ ) = 0, we can apply Lemma 2.8, which gives and Moreover, from the growth assumption (3.1) on f h , taking into account that ūh is a truncation of u h , we also have Thus, if we set for all ρ < R, which exists since the function ρ → F h (u h , c h , B ρ ) is increasing and equibounded, thanks to (3.12), (3.14), (3.11), (3.4), and (3.7), we may conclude that for L 1 -a.e.ρ ∈ (0, R), From this and (3.13) we also have that

Now we observe that the sequence of Radon measures
2), so it weak * converges (up to a subsequence) to some Radon measure µ on B R .
Step 3: conclusion.To get the final result, let v ∈ W 1,p (B R ) be such that {v = u 0 } ⊂⊂ B R .We also consider a regularized function for a suitable constant c ≥ 1 depending only on L and p + , p − .Letting h → +∞ and using (3.7), we have Now we let ε → 0 and recalling that v = u 0 outside B ρ , we easily obtain Therefore, letting ρ ′ tend to ρ we finally get that for L 1 -a.e.ρ and any Choosing v = u 0 in the previous inequality and taking into account (3.9), we get that and that u 0 has the claimed minimizing property.This concludes the proof.

4.
Strong minimizers of free-discontinuity functionals with p(•)-growth 4.1.Assumptions on the energy.Consider a variable exponent p : Ω → (1, +∞) satisfying For K ⊂ R d be a closed set and u ∈ W 1,p(•) (Ω \ K) we define the functional where α > 0, q ≥ 1, and g ∈ L ∞ (Ω).Accordingly, we will denote with F the weak formulation of G, that is the free-discontinuity functional of the form defined for u ∈ SBV p(•) (Ω).We assume that the function where h is a continuous function, convex in ξ and such that for each (x, ξ) ∈ Ω × R d , and for any ξ ∈ R d , x, x 0 ∈ Ω and where L ≥ 1.Here ω : [0, +∞) → [0, +∞) is a nondecreasing continuous function such that ω(0) = 0, which represents the modulus of continuity of p(•).
From now on, we will assume ω to satisfy (2.4).

4.2.
A Decay lemma.We start by proving a crucial decay property of the energy F in small balls.The proof is the variable exponent counterpart of the well-known argument, based on a blow-up procedure, devised for energies with p-growth (see, e.g., [7,Lemma 7.14]).The adaptation to the variable exponent setting is nontrivial, since one of the major ingredientsthe homogeneity of the bulk densities -is missing.A first step in this direction, but still for a constant p, is provided by [24,Lemma 3.3] with the introduction of a inhomogeneous correction, namely a function h as above.Before stating and proving our decay Lemma, we recall a regularity result from [3] for Sobolev minimizers of autonomous variational integrals (see Theorem 3.2 therein), which will be used in our proof.
for all ξ ∈ R d , where 0 ≤ µ ≤ 1, L ≥ 1, 1 < p − ≤ p ≤ p + , and for all ξ ∈ R d , ϕ ∈ C 1 0 (Q), Q be the unit cube.Let B R be a ball in R d and let v ∈ W 1,p (B R ) be a local minimizer of the functional w → ´BR f (∇w) dx.Then there exists a constant Remark 4.2.Observe that the previous Theorem can be applied to where h is a convex function satisfying 0 ≤ h(ξ) ≤ C(µ The following result is a slight generalization of [24,Lemma 3.2] to sequences of functions. Lemma 4.3.Let (g j ) j∈N , g j : R d → [0, ∞), be a sequence of quasi-convex functions, and let (p j ) j∈N , with p j ≥ 1 for every j ∈ N, be a bounded sequence.Assume that and let (t j ) ⊂ (0, ∞) be a sequence such that lim j t j = +∞.Then, setting ĝj (ξ) := g j (t j ξ) t there exists a subsequence (t j k ) such that ĝj k converge to a quasi-convex function g ∞ uniformly on compact subsets of R d .If, in addition, p j → p for some p ≥ 1, then Proof.The assertion follows in a standard way noticing that each g j is continuous and complying with the estimate (see, e.g., [25,Lemma 5.2]) where C = C(L, sup j p j ).Then, the sequence ĝj is uniformly bounded and uniformly equicontinuous in any ball.
Extracting eventually a further subsequence (not relabeled for convenience), we may also assume that x j → x 0 , with x 0 ∈ Ω, so that, setting p := p(x 0 ), we have sup as j → +∞.Thus, p j (y) → p uniformly in B 1 .Taking into account Theorem 2.4 and Remark 2.5, we find a function v 0 ∈ W 1,p (B 1 ) such that the convergences stated in (2.14) hold.
We now prove that the function f j (y, ξ) converges to a convex function f ∞ (ξ) uniformly on compact subsets of we start by proving that, if for some ω j,R which is infinitesimal as j → +∞.In fact, we observe that + γ j ρ j h(x j + ρ j y, (γ j ρ j ) −1/p j ξ) − h(x j , (γ j ρ j ) −1/p j ξ) =: I j + J j .
We first estimate I j : by triangle inequality, we have As for the coefficient of I j,1 , we note that since lim j (γ j ρ j ) = 0 and ω satisfies (2.4).On the other hand it is easy to prove that |ξ| p j (y) −|ξ| pj converges to 0 uniformly on B 1 if |ξ| ≤ R. We now treat I j,2 : we have and the term in the right hand side is infinitesimal as j → +∞ by virtue of (4.16).We are only left with estimating J j : by (4.6) we have the last term being bounded again by (4.16).Therefore, Now, thanks to Lemma 4.3, applied to the sequences p j := pj , g j (ξ) := h(x j , ξ) with t j := (γ j ρ j ) −1/p j , since fj (ξ) = |ξ| pj + ĝj (ξ) we may conclude that up to another subsequence which provides the contradiction to (4.14).

Density lower bound.
In order to study the regularity of the jump set S u , a key tool will be an Ahlfors-type regularity result, ensuring that F (u, B ρ (x)), where B ρ (x) is any ball centred at a jump point x ∈ S u , is controlled from above and from below.We first recall the definition of quasi-minimizer (see [7,Definition 7.17]).The class of quasi-minimizers complying with (4.17) is denoted by M κ (Ω).
It is easy to check (see, e.g., [7,Remark 7.16]) that any minimizer u of the functional F in (4.3) belongs to M κ (Ω) with κ := 2 q αγ d g q ∞ .The following upper bound is quite immediate, as it follows from a standard comparison argument.Note that here the assumption x ∈ S u is, actually, not needed.where κ ′ = κ + Lγ d .
On the contrary, the following lower bound for F (u, B ρ (x)) requires that the small balls B ρ (x) be centred at x ∈ S u .The proof is based on the decay estimate of Lemma 4.4, and it follows along the lines of the proof of [7,Theorem 7.21] where p is constant.The difference is in the introduction of the set Σ where the L 1 loc -function |∇u| p(•) is locally large (see (4.29) below), which turns out to be H d−1 -negligible.For the sake of completeness, we prefer to provide all the details. for all balls B ρ (x) ⊂ Ω with centre x ∈ S u and radius ρ < ρ 0 κ ′ , where κ ′ is that of (4.18).Moreover, H d−1 ((S u \ S u ) ∩ Ω) = 0. (4.20) Proof.Without loss of generality, we may assume that x = 0. Let 0 < τ < 1 be fixed such that √ τ ≤ 1 Let Ω ⊂ R d be open and bounded.For every x ∈ R d and r > 0 we indicate by B r (x) ⊂ R d the open ball with center x and radius r.If x = 0, we will often use the shorthand B r .For x, y ∈ R d , we use the notation x • y for the scalar product and |x| for the Euclidean norm.Moreover, we let S d−1 := {x ∈ R d : |x| = 1} and we denote by R d 0 the set R d \ {0}.The m-dimensional Lebesgue measure of the unit ball in R m is indicated by γ m for every m ∈ N. We denote by L d and H k the d-dimensional Lebesgue measure and the k-dimensional Hausdorff measure, respectively.The closure of A is denoted by A. The diameter of A is indicated by diam(A).We write χ A for the characteristic function of any A ⊂ R d , which is 1 on A and 0 otherwise.If A is a set of finite perimeter, we denote its essential boundary by ∂ * A, see [7, Definition 3.60].

d− 1 ,
such that a = b and lim ) − b| dy = 0 where B ± ρ (x, ν) := {y ∈ B ρ (x) : y − x, ν ≷ 0}.The triplet (a, b, ν) is uniquely determined by the previous formulas, up to a permutation of a, b and a change of sign of ν, and it is denoted by

Proposition 4 . 1 .
Let f : R d → R be a continuous function satisfying