Regularity results for local solutions to some anisotropic elliptic equations

In this paper we study a class of anisotropic equations with a lower order term whose coefficients lay in Marcinkiewicz spaces. We prove some regularity results for local solutions requiring any control on the norm of the coefficients.


Introduction
Our aim is to obtain regularity results for the following class of anisotropic elliptic equations in Ω, (1.1) where Ω is a domain of R N , N > 2, p i > 1 for every i = 1, ..., N with 1 < p < N , denoting by p the harmonic mean of p 1 , • • • , p N , i.e.
Throughout this paper, we make the following assumptions for any i = 1, ..., N (H1) A i : Ω × R N → R is a Carathéodory function that satisfies for a.e.x ∈ Ω and for any vector ξ in R N , where 0 < α β i are constants; (H2) B i : Ω × R → R is a Carathéodory function such that for a.e.x ∈ Ω and for every s ∈ R, where b i is a non negative function in the Marcinkiewicz space L N p ′ i p ,∞ (Ω), where p ′ = p p−1 ; (H3) Our model operator is with b i as in (H2).The anisotropy of L is due to different power growths with respect to the partial derivatives of the unknown u and it coincides with the so-called pseudo-Laplacian operator when b i = 0 and p i = p for i = 1, ...., N .
Let us point out that term anisotropy is used in various scientific disciplines and could have a different meaning when it is related to equations as well.The interest in anisotropic problems has deeply increased in the last years and many results in different directions have been obtained.We quote a list of references that is obviously not exhaustive and we refer the reader to references therein to extend it: [1,2,3,4,7,9,10,12,13,14,15,18,19,25,24].
Several regularity results depending on the summability of datum are well-known in literature for isotropic counterpart of (1.1) div |∇u| p−2 ∇u + b(x)|u| p−2 u = div (|F| p−2 F) in Ω, where p > 1, b and F are vector fields with suitable summability.In the isotropic case for linear equations, assuming the coefficient of the lower order term in suitable Lebesgue spaces, Stampacchia in [28] proves that if the datum F belongs to (L r (Ω)) N with 2 < r < N , then the solution u belongs to L r * (Ω).Otherwise when r > N it follows that the solution u is bounded.Similar results have been obtained also for isotropic nonlinear operators taking the coefficient of the lower order term b in Lebesgue spaces in [5] and in the Marcinkiewicz spaces in [6], [16], [21] and [22].In this paper we prove such kind of regularity results for anisotropic equation (1.1) dealing with local solution, whose definition is recalled in what follows.
Definition 1.1 If F i ∈ L p i loc (Ω) for i = 1, .., N , we say that u ∈ W 1, p loc (Ω) is a local solution to (1.1) For the definition of the anisotropic Sobolev space W 1, p loc (Ω) we refer to Section 4. To give an idea of our result, let us consider, for simplicity, equation (1.1) without the lower order terms, i.e.B i ≡ 0. When the datum F i ∈ L p i loc (Ω) for i = 1, .., N with p < N , a local solution u belongs to L p * loc (Ω), where as suggested by the anisotropic imbedding (see Section 4).Otherwise if F i ∈ L r i loc (Ω) with r i > p i for i = 1, .., N , we expect that the summability of u improves.In order to analyze the higher summability of u we consider the following minimum first introduced in [8], and we are able to prove that u ∈ L (pµ) * loc (Ω) if pµ < N .Then the regularity of u depends on µ.
These regularity results are stated in Theorem 2.1 taking into account the lower order terms under the assumption that coefficients b i have a suitable distance to L ∞ sufficiently small for i = 1, • • • , N .
The principal difficulties are due to the anisotropy, to the managing of local solutions and to the presence of the lower order terms.In our proof, one of the key tool is a new anisotropic Sobolev inequality in Lorentz spaces that involves the product of different powers of two functions (see Proposition 4.1).This inequality naturally appears in the anisotropic framework, it is of independent interest and gives an estimate in terms of the norm of the geometric mean of the partial derivatives instead of the geometric mean of the norms of the partial derivatives as usual in literature (see (4.2)).
Moreover in Theorem 2.4 we give a sufficient condition in terms of µ for the boundedness of the solutions to (1.1).
We also make comments on the regularity of weak solutions of Dirichlet problems in a bounded open set Ω ⊂ R N with Lipschitz boundary.In this case when F i ∈ L p i (Ω) for i = 1, .., N with p < N a weak solution u belongs to L p∞ (Ω), where p ∞ = max{p * , p max } with p max = max{p 1 , • • • , p N } and p defined as in (1.7).It is evident that the regularity of u depends on how much the anisotropy is concentrated, so the situation is more diversified than the isotropic case.Otherwise if F i ∈ L r i (Ω) with r i > p i for i = 1, .., N and µp < N , we get u ∈ L s (Ω) with s = max{(pµ) * , µp max } and the regularity again depends on how much p i are spread out and on µ.
The paper is organized as follows.The main results are stated in Section 2. In Section 3 we recall same properties of Lorentz spaces and in Section 4 we introduce the anisotropic Sobolev spaces and the related inequalities.The proofs of main results (Theorem 2.1 and Theorem 2.4) are given in Section 5 and 6.We conclude the paper with a technical lemma contained in the Appendix.

Main results
The first result of this paper states the regularity of local solutions to (1.1) in terms of the summability of F i , replacing the usual smallness assumption on the norm of the coefficients of the lower order terms with a weaker one given in terms of the distance of a function f ∈ L p,∞ (Ω) to L ∞ (Ω), denoted by dist L p,∞ (Ω) (f, L ∞ (Ω)) and defined by (3.11) in Section 3.
where µ is defined in (1.8).There exists a positive constant d = d( r, N, α, p) such that if , then s → ∞ as expected.In the isotropic case condition p * > p max does not turn up and assumption (2.1) reads as isotropic assumption r < N considering the datum [22] and [16]).Some comments on the case p max ≥ p * are contained in Remark 5.2, Remark 5.3 and Remark 6.1, where weak solutions for Dirichlet problems are taking into account.
It is clear that in Theorem 2.1 regularity of u is related to µp 1 , • • • , µp N .Indeed if r and q are two different vectors such that min i , as in the case F i ∈ L µp i loc (Ω) for i = 1, .., N .
A standard approach to treat the presence of lower order terms is to require a smallness on the norm of b i , which is avoid using assumption (2.2), firstly introduced by [21] in the isotropic case.We stress that the value of d in (2.2) really depends on r (see Example 3.1 in [22]).
For example if Ω is the ball centered at the origin with radius R > 0, we can take Ω) and verifies (2.2) for suitable γ i .We emphasize that condition (2.2) is trivially satisfied whenever b i belongs to a Lebesgue space or to any Lorentz space contained in Then as a corollary of Theorem 2.1 we have immediately the following result.
Without lower order terms in [8] the authors have proved that the boundedness of a weak solution of Dirichlet problems is guaranteed under the assumption where µ is defined in (1.8).However if B i ≡ 0 for i = 1, • • • , N the boundedness is not assured assuming that (2.2) is in force as showed in Example 4.8 of [21] (when can be small as we want taking γ small enough. In order to obtain an L ∞ −regularity result we require extra summability on the coefficients of lower order terms.(2.4).Then any local solution u ∈ W 1, p loc (Ω) to equation (1.1) is locally bounded.

Some properties of Lorentz spaces
In this section we recall the definitions of Lorentz spaces and their properties (see [27] for more details).There are various definitions of Lorentz spaces but all of them manage the notion of rearrangement.
For 1 p < +∞ and 0 < q +∞ the Lorentz space L p,q (Ω) consists in all measurable functions v : Ω → R such that Ω) is a quasinorm, but replacing v * with v * * one obtains an equivalent norm and L p,q (Ω) becomes a Banach space.We observe that it holds The Lorentz spaces are a refinement of Lebesgue spaces.Indeed for p = q, L p,p (Ω) reduces to the standard Lebesgue space L p (Ω), and L p,∞ (Ω) is also known as Marcinkiewicz space M p (Ω), or weak-L p (Ω).If Ω is bounded, such inclusions follow where ω N stands for the Lebesgue measure of the unit ball of R N .Now we introduce a suitable characterization of Lorentz spaces that is useful to generalize Sobolev embedding theorems in anisotropic framework.
Let k > 1 fixed, for any measurable function v defined in Ω such that for every λ > 0 we choose the levels where v * is the decreasing rearrangement of v defined in (3.1) and where plus and minus denote the right and the left-hand limit respectively.Now we consider a particular sequence of functions ω n (v) defined by with the levels a v n defined for n ∈ Z as in (3.4).We observe that where χ is the characteristic function, so one finds for 0 < r < ∞, that Recalling that a sequence {a n } n∈Z belongs to l q (Z) for q > 1 iff n∈Z |a n | q < +∞, we are in position to state the following announced characterization of Lorentz spaces.
Proposition 3.1 Let be 1 < p < ∞ and 1 ≤ q ≤ ∞.For any v extended by 0 outside Ω satisfying (3.3), one has and in particular with C 1 , C 2 are positive constants independent on v and a v n is defined in (3.4).
Proof.Equivalence (3.7) is contained in [30], Proposition 3. We prove (3.8).Since the rearrangement is non increasing, we have Now following the idea of Tartar [30], m since the measure of the level sets is finite.We have that k and by the convolution Young inequality, we deduce the following inequality Combining (3.10) and (3.9), inequality (3.8) follows with C 2 = (log k) where the truncation at level M > 0 is defined as At the end of this section we recall the following useful lemma.
Lemma 3.2 (see [26, page 43] ) Let X be a rearrangement invariant space and let 0 ≤

Anisotropic inequalities
Let p = (p 1 , p 2 , ..., p N ) with p i > 1 for i = 1, ..., N and Ω be a bounded open set.As usual the anisotropic Sobolev space is the Banach space defined as It is well-known that in the anisotropic setting a Poincarè type inequality holds true (see [18]).Indeed for every u ∈ C ∞ 0 (Ω) with Ω a bounded open set with Lipschitz boundary we have for a constant C P depending on the diameter of Ω.Moreover, for u ∈ C ∞ 0 (R N ) the following anisotropic Sobolev inequality holds true (see [30]) where S N is an universal constant and p = p * and q = p whenever p < N , where p is defined in (1.2).Using the inequality between geometric and arithmetic mean we can replace the right-hand-side of (4.2) with N i=1 ∂ x i u L p i .When p < N and Ω is a bounded open set with Lipschitz boundary, the space , with p ∞ := max{p * , p max } as a consequence of (4.2) and (4.1).
In the following proposition we generalize the anisotropic Sobolev inequality (4.2) to the product of functions.Proposition 4.1 Let α i , β i ≥ 0 (but not both identically zero) and p 1 , • • • , p N ≥ 1 be such that 1 ≤ p < N .Then for every nonnegative functions φ, u ∈ C ∞ 0 (R N ) we have where p * is defined in (1.7) and α and β are defined by and C is a positive constant independent on u and φ.
It is clear that for example for β i = 1 the sign assumption on u can be dropped.
We point out that Lemma 3.2 with X = L p (Ω) and θ i = p p i N yields that with a positive constant C independent on u and φ.We emphasize that (4.3) is sharper than (4.4) as the following example shows. and ) We note that (4.6) implies (4.5) and under latest assumption we get u ∈ L p * (Ω), but u ∈ L p * +ε (Ω) for every ε > 0 (using again Lemma 7.1).
In order to prove Proposition 4.1 as first step we need the following result.
Proof.The proof is a generalization to product of functions of the Troisi's one ( see [31] Theorem 1.2).We have, for q > 0 that will be chosen later, the following inequality Now choosing q such that q(N −1) N − 1 = 0, we obtain inequality (4.7).
Proof of Proposition 4.1.Applying inequality (4.7) to the function we obtain where ω n and a n are defined as in (3.5) and (3.4), respectively.We stress that Moreover using the Hardy-Lyttlewood inequality, we have Let us consider the left-hand side of (4.8).By (3.6) and (4.9) we get Putting all together, inequality (4.8) becomes , where C denotes a positive constant that could change from line to line.Multiplying for k n p ′ , elevating to the power p and summing, we obtain On the other hand, by Proposition 3.1 with p = p * and q = p, it follows that We have, by the previous inequality and (4.10), that We observe that by (3.2), since k > 1 it follows that We stress that using inequality (4.7) we can prove the following well-known anisotropic Sobolev inequalities in Lebesgue spaces.For every u ∈ C ∞ 0 (R N ) there exists two positive constant C 1 , C 2 independent of u such that when p = N for every q ∈ [p 0 , ∞) and p 0 ≥ 1. Indeed the starting point is to take g i = u σ i in (4.7) with N i=1 σ i = N t for suitable t > 0.
5 Proof of Theorem 2.1 The proof consists in two steps.First we prove our regularity result assuming that b i L are small enough for i = 1, • • • , N .Then, last assumption is removed thanks (2.2).
Step 1. Proof assuming that b i L , where T k is defined as in (3.13).for all s ∈ R. We use w = ϕ q v with q > 0 (that we will choose later) as test function in the weak formulation (1.6) and we denote by Ω k = {|u| > k}.Using assumptions (1.3), (1.4) and (1.5), we get where C is a positive constant independent of u.By Young inequality, we have where here and in what follows ε denotes a positive constant that will be chosen later.
Using Young inequality, Hölder inequality and (4.4), we get and Finally Young and Hölder inequality yields Now putting together inequalities (5.2), (5.3), (5.4) and (5.5), rearranging and choosing ε small enough then inequality (5.1) becomes for suitable constant C = C( p, α, β, N ) that from now on could change from line to line.Now we note that and so . Making the product on the left and right sides of (5.7), we get (5.8) The assumption that the norms b i are small than a suitable constant depending on p, α, β, N and (5.8) allow us to obtain (5.9) Substituting (5.9) in (5.6), by easy calculations it follows At this point we multiply both sides of previous inequality by k γ for fixed γ > 0 that will be chosen later and we integrate with respect to k over [0, K] for fixed K > 0. A repeated use of Fubini's theorem gives where C = C(γ, α, N, p, β).Now by Hölder inequality and Young inequality we get where ǫ is a positive constant small enough.Substituting the previous inequality in (5.10) and by (4.4), we obtain Previous inequality give us an estimate of the j − th addendum of the sum at the left-hand side of (5.12) as well.Then elevating to the power 1 p j , making the product on the left and right sides of (5.12), we get (5.13) Using again that the norms b i are small than a suitable constant depending on γ, α, N, p, β from (5.13) we obtain Substituting (5.14) in (5.11) it follows that At this point, let us assume that u ∈ L µpmax loc (Ω) , where µ is defined in (1.8).Recalling that F i ∈ L r i loc (Ω) for every i = 1, ..., N , then F i ∈ L p i µ loc (Ω).Hence by Hölder inequality with exponents µ and µ ′ , from (5.15) we have with qµ.Now, by (2.1) we have α 2 > 0 and we can now choose q large enough to have α 1 > 0. Finally, we choose Thus, using Proposition 4.1, (5.16) becomes Rearranging the previous inequality it follows that (5.17) where that is positive by (2.1).Finally letting K → +∞ in (5.17), we conclude We observe that previous argument works directly when µp max ≤ p * , since in this case u ∈ W 1, p loc (Ω) implies that u ∈ L µp i loc (Ω) for every i = 1, ...., N .If otherwise there exist i ∈ {1, ..., N } such that µp i > p * we use a bootstrap procedure.
for a.e.x ∈ Ω and for every s ∈ R and b i ∈ L ∞ (Ω) for all i.In this case we obtain the same regularity result as in Remark 5.3.Indeed one can obtain the analogous inequality of (5.15) using Poincaré inequality (4.1) instead of Sobolev inequalities.Starting from the obtained estimate, one can conclude the proof as before under assumption (5.26).

Proof of Theorem 2.4
The first step of our proof is the boundedness of a local weak solution without the lower order terms in order to adapt Stampacchia's arguments [28] to the anisotropic case in the same spirit of Lemma 5.4 in [23] when one deals with local solutions.Finally, our assumptions on the summability of the coefficients b i allow us to apply Corollary 1.3 concluding also when B i ≡ 0.
Step 1. Proof assuming that B i vanishes.Using the same test function w as in Theorem 2.1 (Step 1) and assumptions (1.3) and (1.4), we get Since L 1 = I 1 and L 2 + L 3 = I 5 + I 6 , where I 1 , I 5 , I 6 are defined in Theorem 2.1 (Step 1), putting together inequalities (5.2), (5.5) and choosing ε small enough, inequality (6.1) became for suitable positive constant C = C(α, N, p, β) that will change from line to line.Now we choose the cut-off function ϕ.Let σ > τ > 0, we fix two concentric balls B τ ⊂ B σ ⊂⊂ Ω of radii τ and σ respectively.We consider ϕ such that 0 ≤ ϕ ≤ 1, ϕ ≡ 1 in B τ , ϕ ≡ 0 in Ω \ B σ and |∇ϕ| ≤ 2 σ−τ .We put Ω k,σ = Ω k ∩ B σ .By (6.2), taking q > max i p i and using Hölder inequality we get We stress that since F i ∈ L r i loc (Ω), then F i ∈ L p i µ loc (Ω) for every i = 1, ..., N .Moreover there exists k > 0 such that |Ω k,σ | < 1 for k ≥ k, and then 3) where δ is a positive constant that we choose later.Inequality (6.3) is the key ingredient to conclude that we can choose k 0 > 0 such that for a fixed 0 < σ 0 < σ, i.e. u is locally bounded when the norms of b i are small enough.The claim (6.4) will be proved in the next step following the idea of Lemma 5.4 of [23].
where ζ(s) is a continuously differentiable nonincreasing function on R that is equal to 1 for s ≤ σ 0 and equal to 0 for s ≥ 3 2 σ 0 .We stress that ζ n ≡ 1 inside the ball B ρ h+1 and ζ n ≡ 0 outside the ball B ρ h , where ρ h = 1 2 (ρ h+1 + ρ h ).We have Moreover by Poincaré inequality obtained by Sobolev and Hölder inequalities we get dx + ν j 2 p j h J j where C = C(α, F i L µp i , N, p, β).Let us find a bound for the measure of the set Ω k h+1 ,ρ h .We have Putting together (6.7), (6.6), inequality (6.5) can be rewrite as Now we are in position to apply Lemma 4.7 of [23] in order to obtain (6.4).

1 ,N −δ ′ + 1 . 2 maxp * and b = 2 (pmax+ p 2 maxp
where the last inequality follows taking δ < p N and observing that J m h are decreasing.Putting δ ′ = δp i p and summarizing left and right side of (6.8), we getY h+1 ≤ N C2 Denoting C 1 = N C max i (Y 0 ) p i N −δ ′ + 1 , ω = p min − p ) the previous inequality became Y h+1 ≤ C 1 b h k −ω 0 (Y h ) 1+δ ′ .Choosing k 0 such that k 0 = max{ k, 1, C 1/ω 1 b 1/[ωδ ′ (1+δ ′ )] a δ ′ /ω }, we obtain and so on.Since, if necessary, at the h-th step one has µ h = (p * − p max ) , it is now clear that, in a finite number of times, we can conclude the proof of Step 1.In the previous proof the definition of µ in (1.8) as a minimum obviously appears.Indeed, using Hölder inequality in(5.14)withexponent ζ and ζ ′ , it is enough to require p i ζ ≤ r i , i = 1, ..., N. Hence, the best choice of such ζ is µ.We stress that assumption p * > p max is essential for our technique.Indeed, at the end of the Step 1 of previous proof, in order to start with the bootstrap argument we need u ∈ L r i loc (Ω) with r i ≤ p * for every i = 1, .., N that means p i < p * , i = 1, ..., N. Remark 5.3 If Ω is bounded open set with Lipschitz boundary and we consider homogeneous Dirichlet problems we can argue as in Theorem 2.1 to obtain a regularity of solutions without restriction p max < p * .Precisely we have the following result.Assume that (1.3)-(1.5)are fulfilled, let 1 < p < N and let r 1 , • • • , r N be such that (2.1) holds.There exists a positive constant d = d( r, N, α, p) such that if is a weak solution to (1.1) with F i ∈ L r i (Ω), then u ∈ L s (Ω) with s = max{(µp) * , µp max }.Indeed when the max{(µp) * , µp max } = (µp) * the proof of (5.22) runs as Theorem 2.1 taking into account that we are managing not local solutions.Otherwise if max{(µp) * , µp max } = µp max we can reason as in Step 1 of Theorem 2.1 but instead of (5.15) we obtain * − p max ) , where p min = min i {p 1 , ..., p N }.Since p * > p max the exponent 1 + p i N − p i p > 0 and setting Y h = N m=1 J m h , one can rewrite the previous inequality as follows p (Y h )