Gradient regularity in mixed local and nonlocal problems

Minimizers of functionals of mixed local and nonlocal type are locally $C^{1,\beta}$-regular.


Introduction
Mixed local and nonlocal problems are a subject of recent, emerging interest and intensive investigation.Essentially, the main object in question is an elliptic operator that combines two different orders of differentiation, the simplest model case being −∆ + (−∆) s , for s ∈ (0, 1).Here, the simultaneous presence of a leading local operator, and a lower order fractional one, constitutes the essence of the matter.In this special case, from a variational viewpoint, one is considering energies of the type |x − y| n+2s dx dy , 0 < s < 1 .
Here, as in all the rest of the paper, Ω ⊂ R n denotes at a bounded, Lipschitz regular domain and n ≥ 2. First results in this direction have been obtained in [18][19][20]37], via probabilistic methods.More recently, in a series of interesting papers, Biagi, Dipierro, Valdinoci, and Vecchi [5][6][7][8] have started a systematic investigation of problems involving mixed operators, proving a number of results concerning regularity and qualitative behaviour for solutions, maximum principles, and related variational principles.Up to now, the literature is mainly devoted to the study of linear operators.As for nonlinear cases, for instance those arising from functionals as the study of regularity of solutions has been confined to L ∞ loc (Ω) and C 0,α loc (Ω) estimates (for small α), that is, the classical De Giorgi-Nash-Moser theory.In this paper, our aim is to propose a different approach, aimed at proving maximal regularity of solutions to variational mixed problems in nonlinear, possibly degenerate cases as in (1.1).Specifically, we shall prove the local Hölder continuity of the gradient of minimizers.A sample of our results is indeed Theorem 1.Let u ∈ W 1,p 0 (Ω) ∩ W s,γ (R n ) be a minimizer of (1.1), with p, γ > 1 > s > 0 and p > sγ, and such that u ≡ 0 on R n \ Ω.If ∂Ω ∈ C 1,α b for some α b ∈ (0, 1), and f ∈ L d (Ω) for some d > n, then Du is locally Hölder continuous in Ω and u ∈ C 0,α (R n ) for every α < 1.
Moreover, considering the more familiar case of the sum of two p-Laplaceans, we have Theorem 2. Let u ∈ W 1,p loc (Ω) ∩ W s,p (R n ) be a solution to −∆pu + (−∆p) s u = f in Ω, with f ∈ L d loc (Ω) for some d > n.Then Du is locally Hölder continuous in Ω.
Our approach is flexible and allows us to consider general functionals of the type Φ(w(x) − w(y))K(x, y) dx dy modelled on the one in (1.1), i.e.F (Dw) ≈ |Dw| p in the C 2 -sense, Φ(t) ≈ t γ in the C 1 -sense and K(x, y) ≈ |x − y| −n−sγ .Note that, although we specialize to the variational setting, the regularity estimates we are presenting here actually work for general mixed equations almost verbatim, as our analysis is essentially based on the use of the Euler-Lagrange equation of functionals as in (1.2); for this, see Section 1.2.For the correct notion of minimality, and the related functional setting, as well as for results in full generality, see Section 1.1.Theorem 1 achieves the maximal regularity of minima, namely, the local Hölder continuity of the gradient of minimizers in Ω.This is the best possible result already in the purely local case given by the p-Laplacean equation −∆pu = 0, which is covered by Uraltseva-Uhlenbeck theory and related counterexamples [51,54,55,61,62].In addition, the case p = γ is here considered for the first time, thereby allowing a full mixing between local and nonlocal terms.In this respect, the central assumption is that says, roughly speaking, that the fractional W s,γ -capacity generated by the nonlocal term in (1.1) can be controlled by the W 1,p -capacity (the standard p-capacity) generated by w → |Dw| p dx.This is exactly the point ensuring that the nonlocal term in (1.1) has less regularizing effects that the local one, as it happens in the basic case −∆ + (−∆) s , when p = γ = 2, and also in the nonlinear models of the type −∆p + (−∆) s p , where the fractional p-Laplacean operator appears [23,35,36,38,[44][45][46][47].We also note that, as far as we known, allowing the condition p = γ is a new, non-trivial feature already when p = 2 and that even the basic De Giorgi-Nash-Moser theory is not available when p = γ.As a matter of fact, all our estimates simplify in the case p = γ.
We have reported Theorem 1 for the sake of exposition but it is actually a very special case of more general results, i.e., Theorems 3-5, whose statements are necessarily more involved due to their greater generality.Before stating the precise assumptions and the results in full generality, we spend a few words about the techniques we are going to use, and on some relevant connections.Up to now, the methods proposed in the literature to deal with mixed operators are, in a sense, direct.More precisely, both the local terms and the nonlocal ones stemming from the equations interact simultaneously via energy methods.These techniques ultimately rely on those used in the nonlocal case [9-11, 30, 31, 44-47] for purely nonlocal operators.This approach does not allow to prove regularity of solutions beyond that allowed by nonlocal operators techniques, which is not the best one can hope for, as, in mixed operators, the leading regularizing term is the local one.In this paper we reverse the approach, relying more on the methods, and, especially, on the estimates available in regularity theory of local operators.In a sense, we separate the local and nonlocal part combining energy estimates of Caccioppoli type with a perturbative like approach.The crucial point is to fit the terms stemming from the nonlocal term in the iteration procedures that would naturally come up from considering the local part only.For this we have to consider a complex scheme of quantities, interacting with each other, and controlling simultaneously both the oscillations of the solution on small balls, and those averaging the oscillations over their complement (such quantities are detailed in Section 3).This first leads to Hölder regularity of solutions with every exponent (Theorem 3) and then to the same kind of estimates globally (Theorem 4); combining these ingredients with a priori regularity estimates from the classical local theory, leads to Theorem 5. We mention that, due to the assumption p = γ, functionals as in (1.1)-(1.2) connect to a large family of problems featuring anisotropic operators and integrands with so-called nonstandard growth conditions [22,24,26,27,34,43,52], and to some other classes of anisotropic nonlocal problems [14-17, 28, 53].We mention that a further connection has been established in [25], where a class of mixed functionals has been used to approximate local functionals with (p, q)-growth in order to prove higher integrability of minimizers.Further approximations via mixed operators occur in the interesting paper [59].
1.1.Assumptions and results.When considering the functional F in (1.2), the integrand F : R n → R is assumed to be C 2 (R n \ {0}) ∩ C 1 (R n )-regular and to satisfy the following standard p-growth and coercivity assumptions (see [50,54,55]) for all z ∈ R n \ {0}, ξ ∈ R n , where µ ∈ [0, 1] and Λ ≥ 1 are fixed constants.The function Φ : R → R is assumed to satisfy for all x, y ∈ R n , x = y.As already mentioned, unless otherwise stated, p, s, γ are such that p, γ > 1 > s > 0, with p > sγ.We shall consider a boundary datum g ∈ W 1,p (Ω) ∩ W s,γ (R n ).In order to get global continuity of minimizers, we consider the following requirements on the boundary ∂Ω: In particular, this implies that q, aχ > n.Interior Hölder estimates, both for minima and their gradients, need less, and essentially no boundary assumption; for this, we shall replace (1.7) by the weaker that in fact will only be needed in when γ > p.
) lead to consider the following natural functional setting: Note that some ambiguity arises in the definition of Xg; in fact, this is actually meant as the subspace of functions w ∈ W s,γ (R n ) whose restriction on Ω belongs to g + W 1,p 0 (Ω).Compare for instance with the discussion made in [5,8], where related functional settings are considered.Under assumptions (1.4)- (1.6) and (1.8), there exists a unique solution u ∈ Xg(Ω) to (1.9) F(w) .

Moreover
(1.10) holds for every ϕ ∈ X0(Ω).The proof of these facts is quite standard, and relies on the application of Direct Methods of the Calculus of Variations.The details can be found for instance in [25,, where actually a more delicate case of mixed operators is considered.As for the derivation of the Euler-Lagrange equation, this is standard once (1.4)-(1.6)are assumed, and, for the nonlocal part, proceeds as in [25,30].Assumptions (1.4)-(1.8)come along with two different lists of parameters (the data of the problem) that we shall use to simplify the dependence on the various constants.These are For the sake of brevity we shall sometimes indicate a dependence of a constant c on one of the lists in (1.11), also when it will actually occur on a subset of the parameters involved.For example, a constant c depending only on n, p, s, γ might be still indicated as c ≡ c(data h ).
Let us briefly comment on the assumptions considered in Theorems 3-5.These are essentially sharp.For instance, the statement of Theorem 3 does not hold when only assuming that f ∈ L t , for any t < n.As for Theorem 5, one cannot obtain in general the gradient Hölder continuity only assuming that f ∈ L n ; counterexamples arise already in the purely local (and linear) case −∆u = f [21].In Theorem 4 the assumptions on g guarantee that g ∈ C 0,κ (R n ) with κ = min{1 − n/q, a − n/χ}; indeed, note that W a,χ ⊂ C 0,κ (see [32,Theorem 8.2]).This is the natural assumption in this setting in order to guarantee that the boundary regularity of solutions obtained matches with the one of the boundary data.Note that, accordingly, Morrey-Sobolev embedding gives W 1,q ⊂ C 0,κ .In other words, assumption (1.7)3 encodes the necessary Hölder continuity of the boundary data g both with respect to the Sobolev space related to the local part of the functional in (1.2), and with respect to the nonlocal one.In the following, letting f ≡ 0 in R n \ Ω, we can always take f ∈ L d (R n ) and f ∈ L n (R n ) in Theorems 3-4 and 5, respectively.Remark 1.1.When considering the case γ ≤ p, in Theorems 3 and 5 no assumption is put on boundary datum g, and, in fact, our results are purely local.See Remark 4.1 and Theorem 7. Note that, in the case γ ≤ p, on the contrary of other papers devoted to the subject, we dot not need to prove that u is bounded to get its Hölder continuity.Theorems 3-5 come along with explicit a priori estimates.These can be directly inferred from the proofs and whose shape reflects the optimal approach used here.For brevity we confine ourselves to report the a priori estimate related to Theorem 3.This is in the next Theorem 6 (Campanato type estimate for Theorem 3).Under assumptions (1.4)-(1.6)and (1.8), let u ∈ Xg(Ω) be as in (1.9).For every α < 1, there exist r * > 0 and c ≥ 1 such that holds whenever B̺ ≡ B̺(x0) ⊂ Br(x0) ≡ Br ⊂ Ω are concentric balls with r ≤ r * .Both r * and c depend on n, p, s, γ, Λ, α if γ ≤ p, and on data h , α when γ > p. Assumption (1.8) can be dropped when γ ≤ p.
When γ ≤ p the constant c in (1.12)only depends on n, p, s, γ, Λ. Therefore in this case (1.12), if reduced to the content of the first line, gives back the classical Campanato type decay estimate for solutions to local non-homogeneous equations (see for instance [40,Theorem 7.7]).As it is well-known, such decay estimates on the integral average of u − (u)B r imply the local C 0,α -regularity of solutions.Instead, the second line of (1.12) encodes the long-range interactions due to the presence of the nonlocal term in the functional.In this respect, the average u − (u)B r is performed with respect to a suitable measure, on the complement of Br; the resulting term is often called snail, it is essentially the nonlocal counterpart of the integrals appearing in the first line and some variations of it are of common use in nonlocal problems (see Section 3 for more).In the range γ > p, the nonlocal term exhibits a growth larger than the local one, and a careful analysis of the proofs, actually reveals that the constant c appearing in (1.12), depends on n, p, s, γ, Λ and u L ∞ (see Remark 4.1 for details).This typically happens in all those situations when anisotropic operators are considered, especially in the setting of nonuniformly elliptic problems (see for instance the a priori estimates in [22,[25][26][27]).Apart from this unavoidable detail, the shape of (1.12) still neatly reproduces the one known for the classical local case.We note that estimate (1.12) can be further improved including the decay rate of the last term appearing in (1.12); this follows from the estimate on certain (fractional) sharp maximal operators considered in Section 4.3, estimate (4.36), eventually implying (1.12).1.2.Possible extensions, local solutions.The results in this paper can be extended in several directions.For instance, one can consider more general functionals of the type where this time we assume that z → F (x, z) satisfies (1.4) uniformly with respect to x ∈ Ω.The assumption regulating coefficients is to hold for every choice x, y ∈ Ω and z ∈ R n .Here ω : [0, ∞) → [0, 1) is a modulus of continuity, that is, a continuous and non-decreasing function, such that ω(0) = 0.Under assumption (1.13), it is then easy to see that Theorems 3-4 continue to hold.In order to get an analog of Theorem 5 we assume in addition that ω(t) ≤ t σ holds for some σ ∈ (0, 1), this condition being necessary; then the Hölder exponent of Du does not exceed σ.We note the proof of these assertions is in fact implicit in the proof of boundary regularity provided in Proposition 5.1 below.Another extension, already mentioned above, is about general solutions to nonlinear integroredifferential operators, not necessarily coming from integral functionals.Moreover, a purely local regularity approach can be considered.For this, we consider a general vector field A : with the same meaning of (1.4)- (1.5).Note that the classical p-Laplacean operator given by A(z) ≡ |z| p−2 z is covered by (1.14).We consider functions u ∈ W 1,p (Ω) ∩ W s,γ (R n ), where Ω ⊂ R n is as usual a bounded and Lipschitz-regular domain, such that y) dx dy = 0 holds for every ϕ ∈ X0(Ω).Note that here no boundary datum g appears.The definition of solution is instead purely local.In this case we have Theorem 7.Under assumptions (1.6) and (1.14), let u ∈ W 1,p (Ω) ∩ W s,γ (R n ) be a solution to (1.15).
The proof of Theorem 7 follows verbatim the ones for Theorems 3-5.Again, the assumption u ∈ L ∞ loc is only needed when γ > p.Note that Theorem 2 follows as a corollary of Theorem 7.

Preliminaries
2.1.Notation.Unless otherwise specified, we denote by c a general constant larger or equal than 1.Different occurrences from line to line will be still denoted by c. Special occurrences will be denoted by c * , c1 or likewise.Relevant dependencies on parameters will be as usual emphasized by putting them in parentheses.In the following, given a ∈ R, we denote a+ := max{a, 0}.We denote by Br 2.2.Fractional spaces.For γ ≥ 1 and s ∈ (0, 1), the space W s,γ (R n ) is defined via |x − y| n+sγ dx dy < ∞ , and it is endowed with the norm With w ∈ W s,γ (R n ), we also denote whenever A ⊂ R n is measurable.In a similar way, by replacing R n by Ω in the domain of integration, it is possible to define the fractional Sobolev space W s,γ (Ω) in an open domain Ω ⊂ R n .Good general references for fractional Sobolev spaces are [1,32].For the next result, see also [2] and related references.Proof.By standard rescaling -i.e., passing to B1 ∋ x → w(x0 + ̺x), with x0 being the center of B̺we can reduce to the case B̺ ≡ B1(0).The assertion then follows by [32, Proposition 2.2] and standard Poincaré's inequality, as w ∈ W 1,p 0 (B1).Using interpolation from [13] (see also [12]), we can also prove the following improved imbedding: holds with c ≡ c(n, p, s, γ).
We find it useful to have a unified reformulation of Lemmas 2.2-2.3.For this, we introduce, with reference to the exponents p, s, γ considered in Theorems 1-5, the following quantities: and we set Note that Aγ + Bγ + Cγ = 1.With this definition we note that (1.3) translates into (2.6)p = γ =⇒ p > ϑγ and p ≥ ϑγ .
We can now summarize the parts we need of Lemmas 2.2 and 2.3 in the following: Lemma 2.4.Let w ∈ W 1,p 0 (B̺), with p, γ > 1, s ∈ (0, 1) be such that sγ ≤ p; assume also that w ∈ L ∞ (B̺), when γ > p. Then w ∈ W s,γ (B̺) and holds with c ≡ c(n, p, s, γ).In (2.7) we interpret w 1−ϑ L ∞ (B̺) = 1 when γ ≤ p and therefore ϑ = 1.2.3.Miscellanea.We shall often use the auxiliary vector field Vµ : R n → R n , defined by where the equivalence holds up to constants depending only on n, p.A standard consequence of (1.4)3 is the following strict monotonicity inequality: (2.10) holds whenever z1, z2 ∈ R n , where c ≡ c(n, p, Λ).The two inequalities in the last two displays are in turn based in on the following one (2.11) that holds whenever t > −1 and z1, z2 ∈ R n are such that |z1| + |z2| + µ > 0. As a consequence of (2.9) and (2.10), it also follows that (2.12) holds for every z ∈ R n , where, again, it is c ≡ c(n, p, Λ); for the facts in the last four displays see for instance [2,25,41] and related references.Finally, three classical iteration lemmas.The first one can be obtained by [40,Lemma 6.1] after a straightforward adapatation.Lemma 2.6 comes via a reading of the proof of (the very similar) [ holds too, where c ≡ c(θ, γi).
Lemma 2.6.Let h : [0, r0] → R be a non-negative and non-decreasing function such that the inequality , where a > 0 and 0 < β < n.For every positive b < n, there exists ε0 then ṽi ≤ a −i/t ṽ0 holds for every i ≥ 0 and hence ṽi → 0.

Integral quantities measuring oscillations
In this section we fix two generic functions w and f , such that, unless otherwise specified, w ∈ W 1,p (Ω) ∩ W s,γ (R n ) and f ∈ L n (R n ), and an arbitrary ball B̺(x0) ⊂ R n .We are going to list a number of basic quantities that will play an important role in this paper.In most of the times, such quantities give an integral measure of the oscillations of a function w in B̺(x0) or in its complement.A fundamental tool in the regularity theory of fractional problems is the nonlocal tail, first introduced in [30], which, in some sense, keeps track of long range interactions.In [9], a related nonlocal quantity, called snail, was considered, namely The snail can be essentially seen as the L γ -average of |w| on R n \ B̺(x0) with respect to the measure defined by dλx 0 := |x − x0| −n−sγ dx.We refer to [9-11, 30, 45, 46, 58] for extra details on this matter.In this paper we use a Campanato-type variation of (3.1), that is Note that This clearly involves the oscillations of u and it is a nonlocal version of the more classical object The right notion of excess functional now combines the previous two quantities, i.e., (3.4) With θ ∈ (0, 1) and δ ≥ sγ, we further define L n (B̺) + 1 Abbreviations above such as avp(̺) ≡ avp(w, B̺(x0)), ccp δ, * (̺) ≡ ccp δ, * (w, B̺(x0)), and the like, will be made in the following whenever there will be no ambiguity on what w and B̺(x0) are.Of course all the quantities defined above also depend on f , but this dependence will be omitted as it will be clear from the context.The motivation for the notation above is that terms of the type rhs θ (•) appear as right-sides quantities of certain inequalities related to equations as in (1.10).Terms of the type ccp δ (•) will instead occur in certain Caccioppoli type inequalities.
Remark 4.1.The constant c and the radius r * appearing in (1.12), depend on n, p, s, γ, Λ and u 1−ϑ L ∞ , see the definition in (4.1).In turn, in the case γ > p, via Proposition 2.1, u L ∞ can be bounded via a constant depending on data b and this justifies the final dependence of c, r * described in the statement of Theorem 6. Accordingly, by (2.5), when γ ≤ p no dependence on u L ∞ occurs in the estimates as ϑ = 1 and this explains the peculiar definition of data h in (1.11)2.3.In fact, when γ ≤ p, we are directly proving Hölder estimates on u without using any bound on u L ∞ and this justifies the claim in Theorem 3 that we can avoid using assumption (1.8).
Remark 4.2.When neglecting the presence of the snail δ and rhs θ in the definition of gl θ,δ in (3.8), that is, when considering the purely local, homogenous setting, we have that (4.35) turns into

This is nothing but the classical local and fractional variant of the Feffermain-Stein Sharp Maximal
Operator widely used in [29].Moreover, note that a bound of the type in (4.37) immediately implies the local Hölder continuity of u as holds whenever x, y ∈ B ̺/4 , for every ball B̺ ⊂ R n (see [29]).

Proof of Theorem 4
In this section we permanently work under the assumptions of Theorem 4, that is (1.4)-(1.6)and (1.7).The proof goes in seven different steps.

5.1.
Step 1: Flattening of the boundary and global diffeomorphisms.The classical flattening-of-theboundary procedure needs to be upgraded here, as we are in a nonlocal setting.We first recall the standard local procedure, as for instance described in [3,4,48,49], and summarize its main points.Let us consider x0 ∈ ∂Ω; without loss of generality (by translation) we can assume that x0 ∈ {xn = 0} and that Ω touches {xn = 0} tangentially, so that its normal at x0 is en, where {ei} i≤n is the standard basis of R n .By the assumption ∂Ω ∈ C 1,α b , there exists a radius r0 ≡ rx 0 , depending on x0, and a , where c * ∈ (1, 10/9) can be chosen close to 1 at will taking a smaller r0.Moreover, it is where J T and J T −1 denote the Jacobian determinants of T and T −1 , respectively.We refer for instance to [3, Section 3.2] and [4, pages 306 and 318] for the full details and for the explicit expression of the map T considered here; see also [48,49].We next extend T to a C 1 -regular global diffeomorphism of R n into itself, following a discussion we found in math stackexchange 1 .With η ∈ C ∞ 0 (B4r 0 (x0)) being such that 1B 3r 0 ≤ η ≤ 1B 4r 0 and |Dη| 1/r0, we define It follows that Tx 0 is C 1,α b -regular and, being DT(x0) invertible, that Tx 0 is a smooth global diffeomorphism of R n .We now use that the set of [42,Chapter 2,Theorem 1.6], also for the relevant definitions).For this, we take rx 0 > 0, such that if By using (5.4) and mean value theorem, it now easily follows that with c depending again on x0, so that, by taking r0 such that cr α b 0 < rx 0 , we obtain that Tx 0 (from now on also denoted by T) is C 1 -regular global diffeomorphism.Summarizing, and recalling the explicit expression of Tx 0 in (5.2), we have that for every x0 ∈ ∂Ω, there exists a global C 1 -regular diffeomorphism (here we are further enlarging c0) and which is C 1,α b -regular diffeomorphism on B2r 0 .A comment needs perhaps to be made here, on the inequalities in (5.4).Since Tx 0 is a C 1 -regular diffeomorphism, then (5.4) holds when replacing R n by B4r 0 (x0) by compactness, for a suitable constant c0; on the other hand Tx 0 is affine on R n \ B4r 0 (x0) and it is D Tx 0 = DT(x0), which is invertible as T is a local diffeomorphism in B2r 0 .Therefore (5.4) holds as stated, by eventually enlarging c0.Note that, at this stage, the constant c0 appearing in (5.4) is still depending on the point x0 via the diffeomorphism T. As we are going to flatten the entire boundary ∂Ω with maps as T, by compactness we can assume that r0 and c0 are independent of x0 ∈ ∂Ω; see also Remark 5.2 below for more on this aspect.
By the very definition of ũ, Proposition 2.1, and directly from (5.5), we also find . From now on, any dependence of the various constants from T, that is T C 1,α b (Br 0 (x 0 )) , T C 1 (R n ) and the like, will be incorporated in the dependence on Ω, and therefore on data (compare with (1.11)4).It 1 https://math.stackexchange.com/questions/148808/the-extension-of-diffeomorphismfollows from the very definitions given, (1.6) and (5.4) that c(•) is continuous and Again by (1.4) and (5.4), as for the new integrand F (•), we have (5.9) for all ξ ∈ R n , z ∈ R n \{0}, x, y ∈ Br 0 (x0).In (5.8) and (5.9) it is Λ ≡ Λ(data) ≥ 1.The Euler-Lagrange equation corresponding to (5.6) is now and holds for all φ ∈ X0( Ω).Performing the same transformation described in Section 2.5, we can use with the new kernel Ks(•) that can be obtained by K(•) as explained in (2.20) and satisfies (5.12) Ks(x, y) = Ks(y, x) and Ks(x, y) ≈Λ 1 |x − y| n+sγ for every x, y ∈ R n , x = y.
Remark 5.1.The various constants generically appealed to as Λ, c0 and c ≡ c(data) from Sections 5.1 and 5.2, actually depend on the point x0 via the features of the map T considered; this dependence has been omitted above, and we will continue to do so.Indeed, by a standard compactness argument, we can cover and flatten the whole boundary ∂Ω by using a finite number of such diffeomorphisms {Ti} i≤k (and points {xi} ≤k ), generating the corresponding constants in the estimates.Eventually, we take the largest constants/lowest and make all the resulting constants independent of the specific point xi considered.We note that all such dependences will be incorporated in data, since this last one also depends on Ω.Similarly, we can assume that the size of the radius r0, that can be decreased at will, is independent of the point x0; we remark that such reasoning is standard [3,4,48,49].
For the proof of Proposition 5.1, from now on we shall consider points x0 ∈ Γ r 0 /2 (x0), radii ̺ ≤ r0/4 ≤ 1/4, and upper balls B̺ ≡ B + ̺ (x0) ⊂ B + r 0 (x0).Unless otherwise stated, all the upper balls will be centred at x0, and x0 will be a fixed, but generic point as just specified.In analogy to the interior case, with δ being such that sγ < δ < p (such a choice is allowed by (1.3)), we define the boundary analog of the quantities introduced in Section 3 as follows: (5.13) where ϑ has been defined in (2.5), and, finally where Aγ , Bγ are defined in (2.5).Thanks to (2.6), by Young's inequality, δ < p and ̺ ≤ 1, we find The above definitions, and the content of the last display, yield (5.17) ] p with c ≡ c(s, γ, p).We shall often use the inequality , that follows by a simple application of Hölder's inequality.

5.4.
Step 4: Boundary Caccioppoli type inequality.We begin the proof of Proposition 5.1 with Lemma 5.1.The inequality holds with c ≡ c(data).