LIPSCHITZ REGULARITY FOR ORTHOTROPIC FUNCTIONALS WITH NONSTANDARD GROWTH CONDITIONS

Transcription

1 LIPSCHITZ REGULARITY FOR ORTHOTROPIC FUCTIOALS WITH OSTADARD GROWTH CODITIOS PIERRE BOUSQUET AD LOREZO BRASCO Abstract We consider a model convex functional with orthotropic structure and super-quadratic nonstandard growth conditions We prove that bounded local imizers are locally Lipschitz, with no restrictions on the ratio between the highest and the lowest growth rate Contents Introduction Overview Main result 3 3 Structure of the proof 4 4 Plan of the paper 9 Preliaries 9 3 Caccioppoli-type inequalities 4 Local energy estimates for the regularized problem 4 4 Towards an iterative Moser s scheme 4 4 Towards higher integrability 6 5 A Lipschitz estimate 8 6 A recursive gain of integrability for the gradient 3 7 Proof of Theorem 9 Appendix A Calculus lemmas 30 A Tools for the Lipschitz estimate 3 A Tools for the higher integrability 34 References 37 Introduction Overview We pursue our study of the gradient regularity for local imizers of functionals from the Calculus of Variations, having a structure that we called orthotropic We refer to our previous contributions [, 3, 4, 5] and [6], for an introduction to the subject More precisely, we want to expand the investigation carried on in [6], by studying functionals of the form f u dx, f : R R convex, which couple the following two features orthotropic structure and nonstandard growth conditions 00 Mathematics Subject Classification 35J70, 35B65, 49K0 Key words and phrases onstandard growth conditions, degenerate elliptic equations, Lipschitz regularity, orthotropic problems

2 BOUSQUET AD BRASCO The first one means that we require fz = f i z i, i= while the second one means that with f i : R R convex, z p fz z q +, with < p < q As we will see, these two features give rise to one of the most challenging type of functionals, at least if one is interested in higher order regularity of local imizers, ie regularity of their gradients Let us be more specific on the type of functionals we want to study We tae a vector p = p,, p with p p Let Ω R be an open set For every u W,p loc Ω and every Ω Ω, we consider the orthotropic functional with nonstandard growth F p u, Ω = p i= i We say that u W,p loc Ω is a local imizer of F p in Ω if Ω u xi pi dx F p u, Ω F p v, Ω, for every v u W,p 0 Ω and every Ω Ω Here W,p and W,p 0 are the classical anisotropic Sobolev spaces, defined for an open set E R by and W,p E = u L E : u xi L pi E, i =,,, W,p 0 E = W,p E W, 0 E It is easy to see that a local imizer of F p is a local wea solution of the following quasilinear equation with orthotropic structure u xi pi u xi i= x i = 0 It is well-nown that local imizers of functionals lie F p above can be unbounded if the ratio p p, is too large, see the celebrated counter-examples by Giaquinta [] and Marcellini [8, 9, 30] see also Hong s paper [3] In Western countries, the regularity theory for non degenerate functionals with nonstandard growth was initiated in the seal papers [8, 9] by Marcellini For strongly degenerate functionals, including the orthotropic functional with nonstandard growth F p introduced above, the question has been addressed in the Russian litterature, see for example the papers [4] by Kolodīĭ, [5] by Koralev and [34] by Uralt seva and Urdaletova However, in spite of a large number of papers and contributions on the subject, a satisfactory gradient regularity theory for these problems is still missing Some higher integrability results for the gradient have been obtained, for example in [8, Theorem ] and [7, Theorem 5] In any case, we point out that even the case of basic regularity ie C 0,α estimates and Harnac inequalities for u itself is still not completely well-understood, we refer to the recent paper [] and the references therein

3 ORTHOTROPIC WITH OSTADARD GROWTH 3 Main result In this paper, we are going to prove that bounded local imizers of our orthotropic functional F p are locally Lipschitz continuous We point out that no upper bounds on the ratio p p, are needed for the result to hold Theorem Let p = p,, p be such that p p Let U W,p loc Ω be a local imizer of F p such that U L locω Then U L loc Ω Remar L assumption Sharp conditions in order to get U L loc can be found in [0, Theorem ] by Fusco and Sbordone, see also [9, Theorem 3] and the papers [, ] by Cupini, Marcellini and Mascolo for the case of more general functionals Pioneering results are due to Kolodīĭ, see [4] We also mention the recent paper [6] by DiBenedetto, Gianazza and Vespri, where some precise a priori L estimates on the solution are proved, see Section 6 there Remar 3 Comparison with previous results Some particular cases of our Theorem can be traced bac in the literature We try to give a complete picture of the subject The first one is [34, Theorem ] by Uralt seva and Urdaletova, which proves local Lipschitz regularity for bounded imizers, under the restrictions p p 4 and < p The method of proof of [34] is completely different from ours and is based on the so-called Bernstein s technique We refer to [3] for a detailed description of their proof More recently, Theorem has been proved in the two-dimensional case = by the second author in collaboration with Leone, Pisante and Verde, see [6, Theorem 4] In this case as well, the proof is different from the one we give here, the former being based on a two-dimensional tric introduced by the authors and Julin in [3, Theorem A] Still in dimension =, Lindqvist and Ricciotti in [7, Theorem ] proved C regularity for solutions of, by extending to the case of nonstandard growth conditions a result of the authors, see [, Main Theorem] In the standard growth case, ie when p = = p, local Lipschitz regularity has been obtained in any dimension in in [4, Theorem ] As we will explain later, the result of [4] is the true ancestor of Theorem, since the latter is partly based on a generalization of the method of proof of the former An alternative proof, based on viscosity methods, has been given by Demengel, see [4] In [3], the same author extended her result to cover the case p < p, under the assumptions p < p + The result of [3, Corollary ] still requires p and applies to continuous local imizers Finally, Lipschitz regularity for solutions of has been claimed in the abstract of [8] However, a closer inspection of the assumptions of Theorem there see [8, equation ] shows that their result does not cover the case of Remar 4 A paper by Lieberman The reader who is familiar with this subject may observe that apparently our Theorem is already contained in Lieberman s paper [6] Indeed, [6, Example, page 794] deals with exactly the same result for bounded imizers, by even dropping the requirement p However, Lieberman s proof seems to be affected by a crucial flaw This is a delicate issue, thus we prefer to explain in a clean way the doubtful point of [6]

4 4 BOUSQUET AD BRASCO We first recall that the proof by Lieberman is inspired by Simon s paper [3], dealing with L gradient estimates for solutions of non-uniformly elliptic equations One of the crucial tool used by Simon is a generalized version of the Sobolev inequality for functions on manifolds This is a celebrated result by Michael and Simon himself [3, Theorem ], which in turn generalizes the idea of the cornerstone paper [9] by Bombieri, De Giorgi and Miranda on the imal surface equation The idea of [6] is to enlarge the space dimension and identify the set Ω with the flat dimensional submanifold M := Ω 0,, 0 contained in R Then the author introduces: a suitable gradient operator ϕ δϕ := γ,j ϕ xj,, γ,j ϕ xj j= Here γ = γ i,j is a measurable map with values into the set of positive definite symmetric matrices; a suitable non-negative measure µ defined on sets of the form M E for all Borel sets E R ; a mean curvature type operator H = H,, H, H +,, H defined on M The ey point of [6, Section 4] is to apply the Sobolev type inequality of Michael and Simon in conjunction with a Caccioppoli inequality for the gradient, in order to circumvent the strong degeneracy of the equation However, in order to apply the result by Michael and Simon, some conditions lining the three objects above are needed amely, the crucial condition [ ] δ i ϕ + H i ϕ dµ = 0, for every ϕ C0 U, for every i =,,, M must be verified, where U M is an open set of R This is condition in [3], which is stated to hold true within the framewor of Lieberman s paper, see the proof of [6, Proposition ] However, with the definitions of µ, δ and H taen in [6], one can see that this crucial condition fails to be verified Indeed, with the definitions of [6, Proposition ], for + i, it holds and j= γ i,i = and γ i,j = 0 for j i, thus δ i ϕ = ϕ xi, H i = 0, while µ coincides with the dimensional Lebesgue measure on Ω Thus condition for + i becomes ϕ xi x, 0,, 0 dx = 0, for every ϕ C0 U, Ω which in general is false Thus the proof of [6, Proposition ] does not appear to be correct, leaving in doubt the whole proof of [6, Lemma 4], which contains the L gradient estimate On the other hand, we observe that other results contained in [6] are not affected by the above problem, in particular the higher integrability result on the gradient, see [6, Lemma 4] Further details on this question are given below 3 Structure of the proof The proof of Theorem is quite involved, thus we prefer spending a large part of this introduction in order to neatly introduce the main ideas and novelties As usual when dealing with higher order regularity, the first issue to be tacled is that the imizer U lacs the smoothness needed to perform all the necessary manipulations However, this is a or issue, which can be easily fixed by approximating our local imizer U with solutions u ε of uniformly elliptic problems, see Section The solutions u ε are as smooth as needed basically, C regularity is enough and

5 ORTHOTROPIC WITH OSTADARD GROWTH 5 they converge to our original local imizer U, as the small regularization parameter ε > 0 converges to 0 Thus it is sufficient to prove good a priori estimates on u ε which are stable when ε goes to 0 For this reason, in the rest of this subsection we will pretend that our imizer U is smooth and explain how to get the needed a priori estimates The building blocs of Theorem are the following two estimates: A a local L L γ a priori estimate on the gradient, ie an estimate of the type Θ 3 U L B r C U γ γ dx, where γ p + and Θ > are two suitable exponents This is the content of Proposition 5; B a local higher integrability estimate of arbitrary order on the gradient, ie an estimate of the type U q dx C q, where < q < + is arbitrary and C q > 0 is a constant depending only on q, the data of the problem and the local L norm of U This is proved in Proposition 6 It is straightforward to see that once A and B are established, then our main result easily follows We explain how to get both of them: in order to obtain A we employ the same method that we successfully applied in [4, Theorem ], for the standard growth case p = = p = p This is based on a new class of Caccioppoli-type inequalities for U, which have been first introduced by the two authors in [] and then generalized and exploited in its full generality in [4] In a nutshell, the idea is to tae the equation satisfied by U U xi pi U xi = 0, x i i= differentiate it with respect to x j and then insert weird test functions of the form ΦU xj ΨU x, with, j,, With these new Caccioppoli-type inequalities at hand, we can follow the same scheme as in [4, Proposition 5] and obtain 3 We point out that, apart from a number of technical complications lined to the fact that p p, in the present setting there is a crucial difference with the case treated in [4] Indeed, after a Moser type iteration, there we obtained an a priori estimate of the type 4 U L B r C U p+ p+ dx Apparently, in that case as well we needed the further higher integrability information U L p+ loc However, thans to the homogeneity of the estimate 4, one can use a standard interpolation tric see the Step 4 of the proof of [4, Proposition 5] and upgrade 4 to the following U L B r C U p p dx This does not require any prior integrability information on U beyond the natural growth exponent Thus in the standard growth case, we are dispensed with point B, ie point A is enough to conclude

6 6 BOUSQUET AD BRASCO On the contrary, in our case, the same tric does not apply to estimate 3, because of the presence of the exponent Θ > For this reason, we need a higher integrability information on U In order to have a better understanding of the proof described above, we refer the interested reader to the Introduction of [4], where the whole strategy for point A is explained in details; part B is the really involved point of the whole proof At first, we point out that we do not have a good control on the exponent γ appearing in 3 unless some restrictions on the ratio p /p are imposed For this reason, we need to gain as much integrability on U as possible This is a classical subject in the regularity theory for functionals with nonstandard growth conditions, ie integrability gain on the gradients of imizers Since in general imizers of this ind of functionals may be very irregular when p and p are too far apart, usually one needs to impose some restrictions on the ratio p /p to gain some regularity These restrictions are typically of the type p < c, for some constant c > 0 such that lim p c = If one further supposes local imizers to be bounded, then the previous restriction can be relaxed to conditions of the type p p < C or p < p + C, with a universal constant C > 0 In any case, to the best of our nowledge, almost all the results appearing in the literature require some upper bound on the ratio p /p More precisely, all the results except two The first one is contained in the aforementioned paper [6] by Lieberman and the second one is due to Bildhauer, Fuchs and Zhong [7] In [6], the higher integrability result is stated in a rather implicit formulation, but a close inspection of the proof of [6, Lemma 4] shows that it implies the higher integrability result B that we need However, in this paper we prefer to present a new proof, that we find more elementary As a matter of fact, our strategy is inspired by the second article [7] where the authors consider a functional with nonstandard growth of the type 5 u, Ω Ω i= u xi p dx + u x p dx, Ω with p p, and prove that any local imizer u L loc is such that u Lq loc for every < q < +, no matter how large the ratio p /p is, see [7, Theorem ] The idea of [7] is partially inspired from Choe s result [0, Theorem 3], which in turn seems to find its roots in DiBenedetto s paper [5] see [5, Proposition 3] It relies on a suitable integration by parts in conjunction with the Caccioppoli inequality for u For functionals as in 5, this leads to an iterative scheme of the type gain of integrability on u x and viceversa = gain of integrability on u x,, u x gain of integrability on u x,, u x = gain of integrability on u x By means of a doubly recursive scheme which is quite difficult to handle, [7] exploits the full power of the above approach to avoid any unnecessary restriction on the exponents In [0] instead, this scheme was extremely simplified, at the price of taing the assumption p < p + Incidentally, we point out that this is the same assumption as in the aforementioned paper [3], which uses however different techniques

7 ORTHOTROPIC WITH OSTADARD GROWTH 7 6 We will try to detail the main difficulties of this method in a while Before this, we point out that 5 is only concerned with two growth exponents Moreover, the type of degeneracy of the functional 5 is much lighter than that of our functional F p For these reasons, even if our strategy is greatly inspired by that of [7], all the estimates have to be recast and the resulting iterative scheme becomes of far reaching complexity We now come to explain such a scheme By proceeding as in [0] and [7], we get an estimate of the type see Proposition 43 B r0 U x p ++α dx C + C i 0 U xi pi p p ++α dx, for =,,, where C > 0 depends on the data and on the local L norm of U, as well Here the free parameter α 0 has to be carefully chosen, in order to improve the gradient summability We observe that, technically speaing, this scheme is not of Moser-type Indeed, the ey point of 6 is that it entails estimates on a fixed component U x, in terms of all the others First step We start by using 6 as follows: we tae = in 6 and choose α 0 in such a way that p i p + + α p i, for i =,, p It is possible to mae such a choice without imposing restrictions on p /p, the optimal choice being p + + α 0 = p p i i p i This permits to upgrade the integrability of U x = p p p =: p q to L p q loc This is the end of the first step Second step Once we gain this property on U x, we shift to U x : we tae = in 6, that we write in the following form B r0 U x p ++α dx C + C + C i= BR0 U xi p i p p ++α dx BR0 U x p p p ++α dx Then by using that U xi L pi loc, for i =,, and U x L p q loc, we choose α in such a way that p i p + + α p i, for i =,,, p If we set as above p p p + + α p q q i = i p p = p i, i =,,, p i For ease of presentation, in what follows we assume that pi > for every i =,,

8 8 BOUSQUET AD BRASCO the optimal choice is now p + + α 0 = p q q, q i i However, this is not the end of the second step Indeed, rather than applying 6 directly to the other components U x,, U x as above, we come bac to U x More precisely, we apply 6 to U x taing into account the new information on U x This gives higher integrability for U x We next apply alternatively 6 to U x and U x, taing into account the higher integrability gain at each step After a finite number of iterations, it can be established that U x L p q loc and U x L p q loc This is the end of the second step j th step for j When we land on this step, we have iteratively acquired the following nowledge U xi L pi loc, for i =,, j and U x i L p q j+ loc, for i = j +,, We then start to get into play the component U x j+ We tae = j + in 6 and we choose α in such a way that p i p j+ + + α p i, for i =,, j, p j+ p i p j+ p j+ + + α p i q j+, for i = j +,, This permits to infer that where U x j+ L p j++α 0 j+ loc, p j+ + + α 0 j+ = p j+ q i, q j+ i=,, j i= j+,, q i As illustrated in the second step, we now use this information and start to cyclically use 6 on U x, U x,, U x j+, in order to improve their integrability After a finite number of iterations of this algorithm, we obtain U xi This is the end of the j th step L pi q j+ loc, for i = j +,, Last step We fix q 0 q arbitrary We finally consider the last component U x, as well By using the starting information and proceeding as above, we finally get U xi L pi q, for i =,,, U xi This yields the desired conclusion L pi q0 loc, for i The main difficulty of B is to prove that this algorithm does not require any restriction on the exponents p i, and that each step ends up after a finite number of loops

9 ORTHOTROPIC WITH OSTADARD GROWTH 9 4 Plan of the paper In Section, the reader will find the approximating scheme and all the basic material needed to understand the sequel of the paper Section 3 contains the crucial Caccioppoli-type inequalities for the gradient, needed to build up the Moser s scheme for point A of the strategy presented above Then in Section 4, we prove integral estimates for the gradient: the first one is a Caccioppoli inequality for power functions of the gradient Proposition 4, while the second one is the self-improving scheme à la Bildhauer-Fuchs-Zhong Proposition 43 With Sections 5 and 6, we enter into the core of the paper: they contain the L L γ gradient estimate and the higher integrability estimate for the gradient, respectively Then in the short Section 7, we eventually prove our main result There we also show how Theorem implies a Uhlenbec type higher differentiability result on local bounded imizers see Corollary 7 A technical appendix concludes the paper: it contains the study of all the intricate sequences of real numbers needed in this paper Acnowledgments Part of this wor has been done during two visits of P B to Bologna & Ferrara in February 08 and October 08, as well as during a visit of L B to Toulouse in June 08 The latter has been financed by the AR project Entropies, Flots, Inégalités, we wish to than Max Fathi Hosting institutions are indly acnowledged Preliaries We will use the same approximation scheme as in [3, Section ] and [6, Section 5] We recall that we are interested in local imizers of the following variational integral F p u; Ω = u xi pi dx, u W,p loc p Ω, Ω Ω, i Ω i= where p = p,, p and p p In the rest of the paper, we fix U W,p loc Ω a local imizer of F p We also fix a ball B Ω such that B Ω as well We use the usual notation λ B to denote the ball concentric with B, scaled by a factor λ > 0 continuous inclusion W,p B W,p B and the Poincaré inequality, it holds We set ε 0 =, U L p B radius of B > 0, By the then for every 0 < ε ε 0 and every x B, we set U ε x = U ϱ ε x, where ϱ ε is the usual family of Friedrichs mollifiers, supported in a ball of radius ε centered at the origin We also set g i,ε t = p i t pi + ε t, t R, i =,, Finally, we define the regularized functional F p,ε v; B = i= B g i,ε v xi dx The following preliary result is standard, see [3, Lemma 5 and Lemma 8] Lemma Basic energy estimate For every 0 < ε ε 0, the problem F p,ε v; B : v U ε W,p 0 B,

10 0 BOUSQUET AD BRASCO admits a unique solution u ε Moreover, the following uniform estimate holds u ε xi pi dx U xi pi dx + ε 0 U dx Finally, u ε C B p i= i B p i= i Proof The only difference with respect to [3] is on the uniform energy estimate, due to the nonstandard growth conditions We show how to obtain this: it is sufficient to test the imality of u ε against U ε, this gives u ε xi pi dx U ϱ ε xi pi dx + ε U ϱ ε dx p i= i B p i= i B B ϱ ε L R U xi pi dx + ε U dx B p i= i By using the scaling properties of the family of mollifiers, we get the conclusion As usual, we will also rely on the following convergence result Lemma Convergence to a imizer With the same notation as above, we have [ ] 3 lim u ε U L p B + u ε U xi L p i B = 0 ε 0 i= Proof We observe that u ε U ε W,p 0 B and the set B is bounded in every direction Thus by the Poincaré inequality, we have 4 u ε U ε pi C i B p i u ε U ε xi pi dx, i =,,, B for some C i = C i, p i > 0 For i =, this in turn gives for every 0 < ε ε 0 B B u ε L p B u ε U ε L p B + U ε L p B B C u ε U ε x L p B + U ε L p B C u ε x L p B + C U W,p B, for a constant C = C, p By Lemma, the last term is uniformly bounded for 0 < ε ε 0 Thus the family u ε 0<ε ε0 is bounded in W,p B We can infer the wea convergence in W,p B of a subsequence u ε to a function u W,p B This convergence is strong in L p B, by the Rellich- Kondrašov Theorem For every ϕ W,p 0 B, we test the imality of u ε against ϕ + U ε Thus, by lower semicontinuity of the L pi norms on L p B, we can infer u xi pi dx lim inf u ε xi pi dx + 5 p i= i B = lim p i= i + p i= i p i= i B B B ϕ + U ε xi pi dx + ε ϕ + U xi pi dx B B ϕ + U ε dx

11 ORTHOTROPIC WITH OSTADARD GROWTH This shows that u xi L pi B for i =,, and u solves F p v; B : v U W,p 0 B By strict convexity of the functional F p, we thus obtain u = U We can now tae ϕ 0 in 5 Since we obtained that u = U, we have equality everywhere in 5, thus in particular 6 lim u ε xi pi dx = U xi pi dx, i =,, + p i= i B p i= i We next observe that the wea convergence of u ε xi to U xi and the lower semicontinuity of the L pi norm on L p imply that 7 lim inf u ε xi + U xi + dx U pi xi pi dx, i =,, B Moreover, by Clarson s inequality for p i, one has p u ε xi + U xi i p + u ε xi U xi i L p i B L p i B We divide by p i, sum over i =,, and rely on 6 and 7 to obtain lim u ε xi U xi pi + L p p i B = 0 i i= B B u ε xi pi L p i B + U x i pi L p i B By using this into 4 with i = and using the strong convergence of U ε to U, we get [ ] u ε U L p B + u ε xi U xi L p i B = 0 lim + i= Finally, we observe that we can repeat this argument with any subsequence of the original family u ε ε>0 Thus the above limit holds true for the whole family u ε 0<ε ε0 instead of u ε and 3 follows We recall that our main result Theorem is valid for bounded local imizers Thus, the following simple uniform L estimate will be crucial Proposition 3 Uniform L estimate With the notation above, let us further assume that U L B Then for every 0 < ε ε 0 we have u ε L B U L B Proof By the maximum principle, see for example [33, Theorem ], we have max B u ε = max B u ε = max B U ε max U ε B By recalling the construction of U ε and using the hypothesis on U, we get the desired conclusion As in [4], the following standard technical result will be useful The proof can be found in [, Lemma 6], for example Lemma 4 Let 0 < r < R and let Zt : [r, R] [0, be a bounded function r t < s R we have A Zt s t + B + C + ϑ Zs, α0 β0 s t Assume that for

12 BOUSQUET AD BRASCO with A, B, C 0, α 0 β 0 > 0 and 0 ϑ < Then we have [ λ α0 Zr λ α0 λ α0 ϑ A R r + B α0 R r + C β0 ], where λ is any number such that ϑ α 0 < λ < 3 3 Caccioppoli-type inequalities The solution u ε of the problem satisfies the Euler-Lagrange equation i= g i,εu ε xi ϕ xi dx = 0, for every ϕ W,p 0 B From now on we will systematically suppress the subscript ε on u ε and simply write u In order to prove higher regularity, we need the equation satisfied by the gradient u Thus we insert a test function of the form ϕ = ψ xj W,p 0 B in 3, compactly supported in B After an integration by parts, we get 3 i= i,εu xi u xi x j ψ xi dx = 0, for j =,, We thus found the equation solved by u xj Observe that we are legitimate to integrate by parts, since u C B by Lemma The following Caccioppoli inequality can be proved exactly as [3, Lemma 3], we omit the details Lemma 3 Let Φ : R R + be a C convex function Then for every function η C 0 B and every j =,,, we have 33 i= g i,εu xi Φu xj x i η dx 4 i= i,εu xi Φu xj η xi dx Actually, we can drop the requirement that Φ has to be convex, under some circumstances The resulting Caccioppoli inequality is of interest, even if it will not be used in this paper Lemma 3 Let < α 0 For every function η C 0 B and every j =,,, we have i= g i,εu xi u x i x j u xj α η 4 dx + α i= i,εu xi u xj α+ η xi dx When α < 0, in the left hand side of the above inequality, the quantity u x i x j u xj α is defined to be 0 on the set where u xj vanishes Proof Let κ > 0 We tae in 3 the test function ψ = u xj κ + u xj α η,

13 where η is as in the statement We get We observe that i= ORTHOTROPIC WITH OSTADARD GROWTH 3 i,εu xi u x i x j κ + u xj α η dx + α = i= i= i,εu xi u x i x j κ + u xj α u xj η dx i,εu xi u xi x j κ + u xj α uxj η η xi dx κ + u xj α u xj κ + u xj α and κ + u xj α uxj κ + u xj α+ From the previous identity, we get remember that α is non-positive + α i= i,εu xi u x i x j κ + u xj α η dx i= i,εu xi u xi x j κ + u xj α+ η η xi dx By using Young s inequality, we can absorb the Hessian term in the right-hand side, to get i= g i,εu xi u x i x j κ + u xj α η 4 dx + α i= i,εu xi κ + u xj α+ η xi dx By taing the limit as κ goes to 0 on both sides, and using Fatou Lemma on the left-hand side and the Doated Convergence Theorem in the right-hand side, we get the conclusion As in the standard growth case p = = p, a ey role is played by the following sophisticated Caccioppoli-type inequality for the gradient The proof is the same as that of [4, Proposition 3] and we omit it It is sufficient to observe that the proof in [4] does not depend on the particular form of the functions g i,ε Proposition 33 Weird Caccioppoli inequality Let Φ, Ψ : [0, + [0, + be two non-decreasing continuous functions We further assume that Ψ is convex and C Then for every η C 0 B, 0 θ and every, j =,,, we have 34 i= i,εu xi u x ix j Φu x j Ψu x η dx 4 i= + 4 i= i= i,εu xi u x j Φu x j Ψu x η dx i,εu xi u x ix j u x j Φu x j Ψ u x θ η dx i,εu xi u x θ Ψu x θ η dx

14 4 BOUSQUET AD BRASCO 4 Local energy estimates for the regularized problem 4 Towards an iterative Moser s scheme We recall that 4 i,εt = p i t pi + ε, i =,, We use Proposition 33 with the following choices 4 Φt = t s and Ψt = t m, for t 0, with s m The proof of the following result is exactly the same as that of [4, Proposition 4] Proposition 4 Staircase to the full Caccioppoli Let p p p and let η C 0 B Then for every, j =,, and s m, 43 i= i,εu xi u x ix j u xj s u x m η dx 4 i= + 4 m + + i= i,εu xi u xj s+ m η dx i= i,εu xi u x s+ m η dx i,εu xi u x ix j u xj 4 s u x m s η dx By iterating a finite number of times the previous estimate, we get the following Proposition 4 Caccioppoli for power functions Tae an exponent q of the form q = l0, for a given l 0 \ 0 Let p p p and let η C 0 B Then for every =,,, we have 44 for some C = C, p > 0 u x q+ p u x η dx C q 5 i,j= + C q 5 i= i,εu xi u xj q+ η dx i,εu xi u x q+ η dx, Proof The proof is essentially the same as that of [4, Proposition 4] We define the two finite families of indices s l and m l through s l = l, m l = q + l, l 0,, l 0 By definition, we have s l m l, l 0,, l 0, s l + m l = q +, l 0,, l 0, and 4 s l = s l+, m l s l = m l+, s 0 =, m 0 = q, s l0 = l0, m l0 = 0

15 ORTHOTROPIC WITH OSTADARD GROWTH 5 From inequality 43, we get for every l 0,, l 0, i= i,εu xi u x ix j u xj s l u x m l η d 4 i= + 4 m l + + i= i,εu xi u xj q+ η dx i= i,εu xi u x q+ η dx i,εu xi u x ix j u xj s l+ u x m l+ η dx By starting from l = 0 and iterating the previous estimate up to l = l 0, we then get i= i,εu xi u x ix j u x q η dx C q i= + C q + i= i= i,εu xi u xj q+ η dx i,εu xi u x q+ η dx i,εu xi u x ix j u xj q η dx, for a universal constant C > 0 For the last term, we apply the Caccioppoli inequality 33 with thus we get Φt = t q+ q +, t R, i= i,εu xi u x ix j u x q η dx C q i= + C q + i= C q + i,εu xi u xj q+ η dx i,εu xi u x q+ η dx i= i,εu xi u xj q+ η dx; that is, 45 i= i,εu xi u x ix j u x q η dx C q i= + C q i= i,εu xi u xj q+ η dx i,εu xi u x q+ η dx, possibly for a different constant C > 0, still universal

16 6 BOUSQUET AD BRASCO We now recall 4, thus we get g i,εu xi u x ix j u x q η dx u x p u x x j u x q η dx i= i,j= = u x q+ p u x q + p x j η dx We can sum over j =,, to obtain g i,εu xi u x ix j u x q η dx u x q+ p u x η dx q + p This proves the desired inequality 4 Towards higher integrability In order to prove the higher integrability of the gradient, we will need the following self-improving estimate This is analogous to the estimate at the basis of [7, Theorem ], which deals with the case p = = p < p only As pointed out in the Introduction, our case will be much more involved Proposition 43 For every α > and every =,,, there exists a constant C = C, p, α > 0 such that for every 0 < ε ε 0 and every pair of concentric balls B r0 0 B, we have p ++α u L B 46 B r0 u x p ++α dx C R 0 R 0 r 0 u L B + C R 0 r 0 p p ++α 0 i u xi p i p p ++α dx + ε 0 R Proof We fix,, and tae η C0 B a non-negative cut-off function For every α >, we estimate the quantity u x p++α η dx = u x u x u x p+α η dx By integration by parts recall that u C B, one gets u x p++α η dx = u u x u x p+α η dx x = p + α + u u x x u x p+α η dx u u x u x p +α η η x dx Hence, we have 47 u x p++α η dx p +α+ u L B u x x u x p+α η dx + u x p+α+ η η dx We now use the Young s inequality for the two terms in the right-hand side: for every τ > 0, u x x u x p +α τ u x p +α+ + 4 τ u x p +α u x x, and u x p +α+ η η τ u x p +α+ η + 4 τ u x p +α η In the first inequality, when p + α < 0, the quantity u x p +α u x x is defined to be 0 on the set where u x = 0

17 ORTHOTROPIC WITH OSTADARD GROWTH 7 Inserting these two inequalities into 47 and choosing τ =, 4 p + α + u L B we can absorb the two terms multiplied by τ in the left-hand side This leads to η u x p++α dx C u L B u x x u x p+α η dx + u x p+α η dx, for a constant C = Cp, α > 0 Observe that u x x u x p +α = u x x u x p u x α,εu x u x x u x α Thus, if α > 0, we can apply the Caccioppoli inequality of Lemma 3 with the convex function Φt = t α + Otherwise, if < α 0, we can apply Lemma 3 This gives u x x u x p+α η dx C g i,εu xi u x α+ η xi dx, and thus we obtain η u x p++α dx C u L B C u L B i= u x α+ i= u x α+ i B i,εu xi + u x p +α η dx u xi pi + u x p+α + ε u x α+ η dx, where in the second inequality the constant C may differ from the previous one There we used 4 We now fix a pair concentric balls B r B Applying the above estimate to a non-negative cut-off function η C 0 such that one gets 48 η on B r and η L C R r, u x Br p++α dx C u L B u R r x α+ i u xi pi + u x p+α + ε u x α+ dx We now want to absorb all the terms containing u x from the right-hand side Thus, we apply again Young s inequality For every τ > 0, there exists C 0 > 0 which depends only on, p and α such that u x α+ u xi pi τ u x p+α+ + C 0 u i τ α+ xi pi p +α+ p, p i and u x p +α τ u x p +α+ + C 0 τ p +α Moreover, we use that ε u x α+ ε 0 + τ u τ α+ x p+α+ p thans to the fact that ε ε 0 < Inserting these inequalities into 48 and choosing R r τ = 6 C u, L

18 8 BOUSQUET AD BRASCO one obtains B r u x p ++α dx u x p++α dx + C u L R r R τ p +α By recalling the choice of τ above, this is the same as + τ α+ p u x p++α dx u x p++α dx + C R B r α++p p u L B + C R r i u xi p i p ++α p p +α+ u L B R r i u xi pi p ++α p dx + ε 0 R τ α+ p dx + ε 0 R We now fix r 0 < R 0 as in the statement and use the previous estimate for r 0 r < R R 0 By applying Lemma 4, one finally obtains that p +α+ u L B B r0 u x p ++α dx C R 0 R 0 r 0 u L B + C R 0 r 0 p +α+ p 0 Here, the constant C depends on, p and α This concludes the proof i u xi pi p ++α p dx + ε 0 R 5 A Lipschitz estimate Proposition 5 Let p p and 0 < ε ε 0 There exist an exponent γ p +, two exponents Θ, β > and a constant C > 0 such that for every pair of concentric balls B r0 0 B with 0 < r 0 < R 0, Θ γ C 5 u L B r0 R 0 r 0 β u γ dx + 0 The parameters γ, β, Θ and the constant C are independent of ε Proof The proof is very similar to that of [4, Theorem 5], though some important technical modifications have to be taen into account For simplicity, we assume throughout the proof that 3, so in this case the Sobolev exponent is finite Observe that the case =, which could be treated with or modifications, is already contained in [6, Theorem 4] the proof there is different As in [4], we divide the proof into four steps Step : a first iterative scheme By starting from 44, we can proceed as in [4, Proposition 5, Step ] by replacing the term η u x q+p dx, there, with the following one η u x q+p dx

19 Then the relevant outcome is now u x q+p 5 = ORTHOTROPIC WITH OSTADARD GROWTH 9 η dx C q 5 i,= + C η for some C = C, p > We now introduce the function and observe that This in turn gives Also, we have that By further observing that Ux := i,εu xi u x q+ η dx = max u x x, =,, u x q+p dx, u x q+p u x q+p u x q+p +, for =,, U q+p u x q+p U q+p + = i,εu xi u x q+ + ε 0 U q+p + + ε 0, for every i, U q+p + u x q+p = + u x q+p =, and recalling that ε 0 <, we obtain from 5 U q+p η C q 5 U q+p η dx + C q 5 η dx + C q 5 η dx, for a possibly different C = C, p > By using the Sobolev embedding W, 0 B L B η dx C η dx, the previous estimate leads to 53 U q+p η dx C q 5 η U q+p + dx We fix two concentric balls B r B, with 0 < r < R Then for every pair of radius r t < s R we tae in 53 a standard cut-off function 54 η C0 B s, η on B t, 0 η, η L C s t This yields 55 U q+p dx B t C q 5 s t B s U q+p + dx

20 We use the convention that log t = for t 0 0 BOUSQUET AD BRASCO We define the sequence of exponents B t U γ j = p + j+, j 0, and tae in 55 q = j+ This gives for every j 0, γj+p p dx C 5 j 56 s t B s U γj + dx, for a possibly different constant C = C, p > Observe that we always have thans to the definition of γ j γ j + p p j+, j, Step : filling the gaps By using the definition of γ j, it is not difficult to see that γ j < γ j + p p j > log p p Thus we introduce the starting index j 0 = j : j > log p p By definition, this entails that γ j < γ j < γ j + p p, for every j j 0 + By interpolation in Lebesgue spaces, we obtain τ j γ j γ j U γj dx U γj dx U B t B t Bt γj+p p dx where the interpolation exponent 0 < τ j < is given by We now rely on 56 to get τ j = γ j γ j τ j γ j γ j U γj dx U γj dx C 5 j B t B t s t = C 5 j s t τ j τ j We also introduce the second index γ j γ j +p p γ j + p p γ j γ j + p p γ j B s τ j γ j γ j +p p U γj + dx B t U γj dx γ j γ j τ j τ j γ j γ j +p p, j = j : j > log p p, B s τ j γ j γ j +p p U γj + dx Observe that j 0 = 0 whenever p p < 4 i e p < + p + 8

21 ORTHOTROPIC WITH OSTADARD GROWTH and set Then by Lemma A we now that J = + maxj 0, j 57 0 < C τ j γ j γ j + p p C <, for every j J By Young s inequality with exponents for every j J, we get 58 B t U γj dx τ j γ j γ j + p p + γj + p p τ j γ j γ j + p p τ j γ j and B s U γj + dx C By 57, we get in turn that for every j J Thus from 58 we obtain 59 B t U γj dx C γj + p p, τ j γ j 5 j τ j γ j γj +p p γ j +p γ j γj +p p τ j γ j γ τ p j j τ j γ j s t U γj dx B t γj + p p B s + C τ j γ j U γj + dx C 5 j C = C β γ j γj +p γ τ p j j τ j γ j s t U γj dx, B t for some < β <, depending on, p and p In the last inequality we also used that s R and C >, together with 57 Finally we set γj + p p ε j = τ j, for j J, τ j γ j and rewrite 59 as 50 B t U γj dx C + C B s U γj dx C 5 j s t β B t U γj dx γ j γ j +ε j which holds for every r s < t R By applying Lemma 4 with Zt = U γj dx, α 0 = β, and ϑ = C, B t + C,

22 BOUSQUET AD BRASCO we finally obtain for every j J, 5 U γj dx C 5 j β R r β U γj dx B r for some C = C, p, p > γ j γ j +ε j Step 3: Moser s iteration We now iterate the previous estimate on a sequence of shrining balls We fix two radii 0 < r < R and define the sequence R j = r + R r, j J j J We use 5 with R j+ < R j in place of r < R Thus we get 5 U γj dx C 7 j β R r β U j+ B γj dx Rj where the constant C > depends on, p and p only We introduce the notation Y j = U γj dx, j + γ j γ j +ε j thus 5 reads γ j Y j+ C 7 j β R r β γ +ε Y j j j + C 7 β R r β j Yj +, + γ j γ +ε j j Here, we have used again that C > and R, so that the term multiplying Y j is larger than By iterating the previous estimate starting from j = J and using some standard manipulations, we obtain Y n+ C 7 β R r β n Yn + γn γ n +ε n C 7 β R r β n C 7 β R r β γn γ n n γ +ε γ +ε Yn + n n n n + C 7 β R r β n C 7 β R r β n Yn + n C 7 β R r β j=j j γn γ j n +ε =j+ [ ] γn γ J Y J + γ n γ +ε n n n +ε j j=j, γn γ n +ε n where we used that C 7 β R r β > We now simply write C in place of C 7 β and tae the power /γ n on both sides: In the previous estimate, we set n Y γn n+ C R r β j=j C R r β Θ Θ = lim j γ j n j=j n j=j n +ε [ =j+ ] Y J + j [ ] Θ γ j γ Y J + J n + ε j, n +ε j j=j γ J

23 ORTHOTROPIC WITH OSTADARD GROWTH 3 which is a finite number, thans to Lemma A We observe that γ j j+ as j goes to This implies the convergence of the series above and we thus get γn Θ U L B r = lim U γn dx C R r β U n n+ B γ γ J J dx +, R for some C = C, p, p > and β = β, p, p > 0 By recalling the definition of U, we finally obtain Θ u L B r C R r B β u γ γ J J dx + R This concludes the proof 6 A recursive gain of integrability for the gradient The main outcome of estimate 5 is the following: assume that our local imizer U has a gradient with a sufficiently high integrability, then one would be able to conclude that U has to be bounded We notice that since the explicit deteration of the exponent γ in 5 is actually very intricate unless some upper bounds on p /p are imposed, we essentially need to prove that U L q loc for every q < +, in order to be on the safe side Thus, in order to infer the desired local Lipschitz regularity on U, we are going to prove a higher integrability estimate on u ε, which is uniform with respect to 0 < ε ε 0 This is the content of the result of this section Remember that we simplify the notation u ε and replace it by u Proposition 6 Let p p < + and 0 < ε ε 0 For every q 0 < + and every 0 B, there exists a constant C > 0 such that u xi pi q0 dx C 0 i= The constant C depends on, p, q 0, R 0, dist0, B, u L B and i= B u xi pi dx Proof We proceed to exploit the scheme of Proposition 43 In what follows, we use the convention that p p = +, when p = We can also assume without loss of generality that p >, otherwise p = = p = and in this case the regularity theory for our problem is well-established U would be a harmonic function in such a situation We fix q 0 as in the statement and introduce the exponents pj pj 6 q j = p j, q 0 =, q0, j =,, Since p p, we get that q q q and thus 6 i=,, q i = q, for every =,,

24 4 BOUSQUET AD BRASCO We now prove by downward induction on j =,, that the following fact holds: 63 for every j,, and every B, In particular, for j =, 63 implies for a uniform constant C > 0 computations below i=j i= u xi pi qj dx C, with C > 0 independent of ε u xi pi q0 dx C, The statement on the quality of the constant C will be clear from the Initialization step We start from j = We observe that in this case the right-hand side of 46 written for = is uniformly bounded with respect to ε > 0, provided that α > is chosen in such a way that for every i,,, one has p i p + + α p p i If p i =, this is automatically satisfied Otherwise, this is equivalent to p i p + + α p p i We define α by By definition of q, p + + α = p q p p + + α = p p, q 0 p as desired We need to chec that α >, or equivalently p 64 p p, q 0 > p + i p i p i, Since q 0, one has p q 0 > p + Moreover, using that p p, one gets p > p which in turn is equivalent to p p p > p + This proves 64 Thus for every B, it follows from 46 that u x p q dx C R u L B R R q u L + C B R R Since q i q for i, one has p q i= p i q p i u xi p i p p q dx + ε 0 R

25 ORTHOTROPIC WITH OSTADARD GROWTH 5 Using Hölder s inequality for each term of the sum of the right-hand side, with the exponent p i /q p i and its conjugate, one gets u x p q dx C R + C u L B R R p q u L B R R q i= u xi pi dx p i q p i + ε 0 R By using Lemma and Proposition 3 in order to control the two terms on the right-hand side, we get a uniform in ε control on the L p q norm of u x This finally establishes the initialization step j =, ie for every B, we have u x p q dx C, with C > 0 independent of ε Inductive step We then assume that the assertion 63 is true for some j,, and we prove it for j By the induction assumption, we thus now that for some j,, 65 u and every B, xi pi qj dx C, with C > 0 independent of ε i=j In the rest of the proof, we establish that 65 implies 66 for every B, u xi pi qj dx C, with C > 0 independent of ε i=j In order to prove this, as explained in the Introduction, we need to employ a multiply iterative scheme based on Proposition 43 More specifically, we start by relying on 46 with the choices = j and p j + + α = p j q i, q j q i i j j i We first justify the fact that such a choice for α is feasible Observe that hence the definition of α is equivalent to q i = q j and q i = q, 0 i j j i 67 p j + + α = p j q j, q j q Since pj q j = p j, q 0 with q 0, one has p j + p j q j By recalling that the exponents q j are non increasing and larger than, this implies that p j + p j q j and p j + < p j q j q, and thus This implies that α 0 as desired p j + + α = p j q j, q j q p j +

26 6 BOUSQUET AD BRASCO We next rely on the fact that by Lemma and Proposition 3, we have u L B + u xi pi dx C, i= with a constant C > 0 independent of ε > 0 We also use the induction assumption 65 Both facts give a local uniform in ε control on u xi pi qj dx, i=j for B Hence, the definition 67 of α ensures that the right-hand side of 46 for = j is uniformly bounded Thus from Proposition 43, we get that for every B we have 68 u xj β 0 j dx C, with C > 0 independent of ε Here, the exponent β 0 j is given by β 0 j = p j + + α = p j q j, q j q We can summarize the previous integrability information as the following estimate: for every B, 69 u xi β 0 i dx C, i=j with C > 0 independent of ε > 0 and β 0 j = p j q j, q j q, 60 β 0 i = p i q j, for i = j,, We proceed to define by induction a vector sequence 6 β l j,, βl q j, B, l, as follows: for l = 0 this is given by 60 and then we use the following multiply recursive scheme 3 β l+ β l = p j p β l+ = p q j, β l+ = p q j, j j 3 β l p, β l+ p β l p, β l+ p = β l+ j = p j q j, j β l+ p 3 We use the convention that β l p = +, if p =

27 ORTHOTROPIC WITH OSTADARD GROWTH 7 We first observe that this scheme is well-defined, since each β l+ i is detered either by β l j,, βl or by an updated information on the β l+, with i + Moreover, thans to Lemma A3 and Lemma A4 below, we have the two crucial features β l i l and there exists l 0 such that is non-decreasing, for every j i, 6 for every l l 0, β l i = p i q j, for i = j,, With these definitions at hand, we now prove that 63 for every B, u xi β l i dx C, for every l, with C > 0 independent of ε i=j By taing into account 6, this will eventually establish 66, thus concluding the proof In turn, the proof of 63 relies on an induction argument The assertion 63 is true for l = 0, thans to 69 We now assume 63 to hold for some l and establish the same for l +, ie for every B, i=j l+ u xi β i dx C, with C independent of ε Actually, by a downward induction on m =,, j, we prove that u xi β l+ i dx C, with C independent of ε i=m For the initialization step m =, we apply 46 with = and for the following choice of α: β l p + + α = p q j, = β l+ j p In order to justify that α defined as such is non-negative, we rely on the fact that for every i j,,, the sequence β l i l is non-decreasing This implies that α β 0 p + = p q j p + p q p + 0 Hence, such a choice of α is feasible We get that for every B, u x β l+ β l+ dx C R u L B R R p β l+ u L B + C R R + C u L p β l+ B R R j u xi pi β l+ p dx + ε 0 R i= i=j u xi p i p β l+ dx + ε 0 R

28 8 BOUSQUET AD BRASCO For the terms in the first sum of the right hand side, we use Hölder s inequality with the exponent p i p, p i β l+ and its conjugate 4 In the second sum, we use Hölder s inequality with the exponent 5 One gets u x β l+ p p i l+ β dx C R u L B R R β l+ u L + C p B j R R + C β l+ p u L B R R β l i β l+ i= i=j u xi pi dx u xi βl i p i p i dx p i p l+ β p + ε 0 R l+ β β l i + ε 0 R By using the induction assumption 63 to control the last term, Lemma and Proposition 3 in order to control the other two, uniformly in ε, we get the desired estimate for u x We now assume that for some m j,,, we have 64 for every B, and prove that this entails for every B, u xi β l+ i dx C, with C > 0 independent of ε, i=m i=m l+ u xi β i dx C, with C > 0 independent of ε Obviously, we only need to improve the control on the last component of the gradient, ie on u xm We still rely on Proposition 43, this time with the choices β l i = m and p m + + α = p m q j, j i m p i, β l+ i = β l+ m m i p i 4 Observe that for pi = there is no need of Hölder s inequality If p i >, one can easily chec that these exponents are not lower than by observing that β l+ p q j p q i, for i j 5 As before, there is no need of Hölder s inequality for p i = For p i >, we rely on the fact that by definition β l+ β l i This justifies that the exponent is not lower than p, for j i p i

29 The fact that β l+ m u xm β ORTHOTROPIC WITH OSTADARD GROWTH 9 β0 m p m + ensures that α 0 Hence, for every B, l+ m dx C R u L B R R + C + C + C β l+ m p β l+ u L B m m R R u L p β l+ B m m R R j u xi pi β l+ m p m dx + ε0 R i= p β l+ u L B m m R R m i=j i=m u xi pi β l+ m p m dx + ε0 R u xi pi β l+ m p m dx + ε0 R We now proceed as above: we control the last term by using the induction assumption 64 and the fact that β l+ m p i β l+ i, if m i p m The third term is estimated thans to the induction assumption 63 and the inequality 6 β l+ m p m p i β l i, if j i m Finally, on the two first terms, we use Lemma and Proposition 3, and also that 7 This finally establishes that β l+ m p m p i p i, if i j for every B, i=j l+ u xi β i dx C, with C independent of ε As already explained, this is enough to safely conclude the proof 7 Proof of Theorem The cornerstones of the proof of Theorem are the uniform L estimate for the gradient of Proposition 5 and the uniform higher integrability estimate of Proposition 6 Indeed, by using Proposition 6 with the choice q 0 = γ ie the exponent in 5, we get that for every B r0 B with r 0 < u ε L B r0 C, with C > 0 independent of ε Observe that to infer that C is independent of ε, we use Lemma and Proposition 3 Once we have this uniform estimate at our disposal, the Lipschitz regularity of U follows with a standard covering argument, by taing into account that u ε converges to U see Lemma We refer to the proof of [3, Theorem A] for details 6 This part of the discussion is void when m = j 7 This part of the discussion is void when j =

30 30 BOUSQUET AD BRASCO Once we have Theorem at our disposal, we can prove a higher differentiability result à la Uhlenbec For the model case of the functional u xi pi dx, p i= i the following result considerably improves [6, Theorem ] Corollary 7 Let p = p,, p be such that p p Let U W,p loc Ω be a local imizer of F p such that Then U L locω U xi p i U xi W, Ω, for i =,, Proof The proof is the same as the one in [6, Proposition 3] incremental quotients, which aims at differentiating the equation loc U xi pi U xi xi = 0, i= It is based on irenberg s method of in a discrete sense By proceeding as in [6], we get for every j =,, and every pair of concentric balls B r0 0 Ω i= B r0 δ hej U xi p i h s j + U xi dx C R 0 r 0 see [6, equation 36] Here we used the notation i= 0 U xi pi dx p i p i e i = 0,, 0,, 0,, 0 and δ hei ψx = ψx + he i ψx By using that U L loc, we can choose s j = so that s j + =, to control the last term on the right-hand side and obtain an estimate on δ hej U xi p i U xi B r0 h dx, j =,,, i= 0 δ hej U h s j + p i p i, which is uniform in h By appealing to the difference quotient characterization of Sobolev spaces, we get the conclusion Appendix A Calculus lemmas In this section, we separately present some proofs on the elementary facts for the sequences needed in the proof of our main result

31 ORTHOTROPIC WITH OSTADARD GROWTH 3 A Tools for the Lipschitz estimate In what follows, we denote as usual Lemma A Let p p We define =, for 3 j 0 = j : j > log p p, j = j : j > log p p and J = + maxj 0, j We set and τ j = γ j γ j γ j = j+ + p, γ j + p p γ j γ j + p p γ j Then there exist two constants 0 < C < C < depending on, p and p such that C τ j γ j γ j + p p C, for every j J Proof It is easily seen that the sequence γ j j is increasing Moreover, by definition of j 0, we have that γ j < γ j + p p, for every j j 0, thus τ j is well-defined and positive for j j 0 The definition of τ j entails that A = τ j τ j + γ j γ j γ j + p p Thus the previous discussion implies that 0 < τ j <, for every j j 0 +