Fundamental solutions for anisotropic elliptic equations: existence and a priori estimates

Transcription

1 Fundamental solutions for anisotropic elliptic equations: existence and a priori estimates Florica C. Cîrstea Jérôme Vétois April 30, 2014 Abstract Let be a domain in R n with n 2 and 0. We study anisotropic elliptic equations such as n x i xi u pi 2 xi u = δ 0 in with Dirac mass δ 0 at zero, subject to u = 0 on. We assume that all p i are in 1, with their harmonic mean p satisfying either Case 1: p < n and max 1 i n p i } < pn 1 n p or Case 2: p = n and is bounded. We introduce a suitable notion of fundamental solution or Green s function and establish its existence, together with sharp pointwise upper bound estimates near the origin for the solution and its derivatives. The latter is based on a Moser-type iteration scheme specific to each case, which is intricate due to our anisotropic analogue of the reverse Hölder inequality. We also derive some generalized anisotropic Sobolev inequalities and estimates in weak Lebesgue spaces as critical tools in our proof. Keywords. Anisotropic equations, Green s function, Moser-type iteration scheme 1 Introduction and main result Anisotropic elliptic equations have received much attention in recent years see, for example, [2, 5, 10, 11, 15, 17, 21, 22, 24, 25, 33, 37, 42, 48] and their references. Timedependent versions of these equations have been used as mathematical models to describe the spread of an epidemic disease, see [4, 6]. Such evolution models also arise in fluid dynamics when the media has different conductivities in different directions, see [1, 2]. We study anisotropic elliptic equations with right-hand side the Dirac mass δ 0 at zero in a domain of R n with n 2 and 0, namely p u = δ 0 in, 1.1 u = 0 on. F. C. Cîrstea: School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia; florica.cirstea@sydney.edu.au J. Vétois: Université de Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, Nice, France; vetois@unice.fr 1

2 2 Florica C. Cîrstea, Jérôme Vétois We use p u to denote div A u, where A u is the vector field whose ith component is given by i u p i 2 i u with p i > 1 and 1 i n. As usual, u = 1 u,..., n u stands for the gradient of u. Let meas E denote the measure of a measurable set E R n. Let p := p 1,..., p n. Without loss of generality, we assume throughout that 1 < p 1 p 2... p n <. 1.2 In what follows, p denotes the harmonic mean of p 1,..., p n, that is 1 p := 1 n 1 p i. 1.3 By a fundamental solution or Green s function of 1.1, we mean any non-negative solution of 1.1. Our main result gives the existence of a fundamental solution of 1.1 in appropriate spaces and sharp pointwise upper bound estimates near zero for the solution and its derivatives in two cases: CASE 1: Let be an arbitrary domain of R n. Assume that p < n and p n < p, where p := p n 1 /n p. 1.4 CASE 2: Let p = n and be a bounded domain. The difficulty of this problem arises from the anisotropic character of 1.1 since there is no explicit formula known for a fundamental solution of 1.1 when p i are not all equal. Our fundamental solution lies in a suitable anisotropic space, denoted by T 1, p 0, which is a natural generalization of the spaces introduced by Bénilan et al. [7] for this type of problem when p i are all equal. For n 2 and R > 0, we define E p R as follows E p R := x R n : 0 < } x i s i < R p i p in Case 1, and s i := p p i p i in Case In Section 2, we define the space T 1, p 0 and introduce a notion of weak solution of 1.1, whose existence is proved by Theorem 1.1 below. Theorem 1.1. Let Case 1 or Case 2 hold and R > 0 be such that E p 2R. Then 1.1 admits a non-negative weak solution Φ T 1, p 0 W 1, loc \ 0} and, hence, Φ is continuous in \ 0} with the following properties: 1. There exists a positive constant C 0 = C 0 n, p such that Φ x C 0 p i 1 p p x i i for all x E p R in Case

3 Fundamental solutions for anisotropic elliptic equations 3 2. There exist positive constants a = an, p, meas and b = bn, p so that Φ x a + b ln x i i p for all x E p R in Case There exists a positive constant C 1 = C 1 n, p such that i Φx p i C 1 where s i is defined by 1.5. x i s i 1 for a.e. x E p R in both cases, 1.8 The main contribution of this paper is to establish the upper bound estimates in Theorem 1.1. The asymptotic behaviour near zero of a fundamental solution of 1.1 is an open question when p i are not all equal. In Case 1, Namlyeyeva Shishkov Skrypnik conjectured in [37] that the asymptotic behaviour of a fundamental or source-type solution of 1.1 near zero is determined by the function U 0 defined by 1 p i p p U 0 x := x i i for x = x 1,..., x n By [37, Theorem 2.2], we only know that for our fundamental solution Φ of 1.1, the limit inferior of Φx/U 0 x is finite, whereas its limit superior is greater than zero as x tends to zero. Note that U 0 is given by 1.9 in Case 1, whereas in Case 2 we define U 0 by U 0 x := ln x i p i for x R n \ 0} In Theorem 1.1 we prove that the limit superior of Φx/U 0 x is finite as x approaches zero. Section 5 contains the major technical work which goes into proving 1.6 and 1.7. We outline the strategy in Section 1.1. The proof of the gradient estimates in 1.8 follows essentially from 1.6 and 1.7 using rescaling arguments and Lieberman s results [33], see Corollary 5.3. We prove the existence of a fundamental solution of 1.1 in Proposition 4.1 see Section 4 using an approach inspired by Bénilan et al. [7] see also Boccardo Gallöuet [8, 9]. The idea is to consider approximate problems for which distributional solutions exist see Lemma 3.1 and show that we can pass to the limit in the weak formulation to obtain a non-negative weak solution of 1.1. This plan is validated by Lemma 4.2, whose proof relies essentially on local estimates in weak Lebesgue spaces in Lemma 3.3. These estimates in Case 2 of Theorem 1.1 are new and more delicate since they require a refined anisotropic Sobolev inequality, also established in this paper: See Lemma B.1 in Appendix B, whose critical ingredient is an inequality of Moser Trudinger type.

4 4 Florica C. Cîrstea, Jérôme Vétois In Case 1, it is usually assumed that is bounded and p 1 > pn 1, so that a fundamental solution Φ of 1.1 can be obtained in W 1,1 0 see Bendahmane Karlsen [5] np 1 and Boccardo Gallouët Marcellini [10]. In this case, using an anisotropic Sobolev embedding see [5, Theorem 2.1], jointly with i Φ L q i for all q i [1, p ip 1 p and i = 1,..., n see [10], we find that Φ L q for all q [1, p 1. This result was improved in [5], where the fundamental solution Φ was shown to belong to the Marcinkiewicz space M p 1 or, equivalently, the weak Lebesgue space L p 1,, see Section 3.2 and i Φ M p i p 1 p for i = 1,..., n. However, if p n pn 1, which np 1 implies that p 1, 2 1 ], the fundamental solution is not expected to belong to W 1,1 n and thus the notions of weak derivatives and distributional solutions do not apply anymore see [5]. In Theorem 1.1, we remove the restriction p 1 > pn 1 in Case 1 and extend the np 1 existence of a fundamental solution of 1.1 on any domain. Hence, our fundamental solution must be understood in an appropriate weak sense see Definition 2.3 rather than in a distribution sense considered in [5, 10] see also Corollary 4.4. We do not know whether the fundamental solution of 1.1 is unique in our general framework. However, in a more restrictive setting, it is known that the fundamental solution of the p-laplacian operator, div u p 2 u with 1 < p n, is unique due to work of Kichenassamy Véron [30] based on rescaling techniques and a strong maximum principle. For a different approach, we refer to Trudinger Wang [45, 46]. The properties of the fundamental solutions of elliptic equations and their constructions have been investigated in many contexts. In the case of a linear operator, see the pioneer work of Krasovskiĭ [31]. For a construction in the context of a Riemannian manifold, see Druet Hebey Robert [20, Appendix A]. Recent progress in the study of Green s functions for the biharmonic or polyharmonic operators has been made, for instance, by Dall Acqua Sweers [18] and Grunau Robert [28]. The study of the local behaviour of singular solutions to nonlinear elliptic equations relies heavily on the properties of fundamental solutions. In the case of quasilinear equations, see the seminal papers by Serrin [40, 41]. For background and other important contributions on the topic of singularities of solutions, we refer to Véron [47]. More recent progress on the classification of the isolated singularities of solutions has been made for nonlinear equations involving, for example, divergence-form operators Brandolini et al. [12] or Hardy Sobolev operators Cîrstea [16]. The fundamental solutions also play a critical role for singular solutions of fully nonlinear uniformly elliptic equations see e.g., Labutin [32], Felmer Quaas [23], Armstrong Sirakov Smart [3] and their references. It is worth mentioning that for anisotropic elliptic equations there is little known about the local behaviour of singular solutions. By analogy with the p-laplacian, it is expected that the properties of fundamental solutions will have many consequences for the singular solutions of anisotropic elliptic equations.

5 Fundamental solutions for anisotropic elliptic equations Sketch of the proof of 1.6 and 1.7 We assume that p i are not all equal, otherwise the estimates are trivial since an explicit fundamental solution can be found see Section 2.1. Let E p R be given by 1.5. For λ 0, 1 and r 0, R, we define A r λ as the set E p 1 + λr\e p 1 λr. Since E p R is included in 0<r<R A r 1/2, the estimates in 1.6 and 1.7 would follow from Φ L A r1/2 Cr 1/p in Case 1, a + b ln r in Case Using the weak solution Φ ε of the approximate problem 3.2 in Lemma 3.1, it is enough to show that 1.11 holds for Φ ε instead of Φ. This requires a delicate analysis, which is carried out separately for Case 1 see Proposition 5.5 and Case 2 see Proposition 5.6. Our approach is based on an anisotropic version of the Moser-type iteration scheme. One difficulty arises in the running step of the iteration process, which unlike the isotropic case, does not give a pure reverse Hölder inequality between two precisely determined Lebesgue norms. Our anisotropic analogue renders one Lebesgue norm being dominated by one out of n possible different Lebesgue norms raised to different powers as follows. Proposition 1.2 An anisotropic reverse Hölder inequality. Let m = n/n p in Case 1 and m > 1/n 1 in Case 2. For every > mp 1 and 0 < λ < λ 3/4, we have Φ ε m Cr 1 L m n p n A rλ max,...,n r pin p pn 1 1 λ λ p i Φ ε m +p i p L m +p i p A rλ, 1.12 where C is a positive constant of the form cn, p max1, /m p + 1 pn } p. The proof of Proposition 1.2 see Section 5.1 relies essentially on: 1 a weighted anisotropic Sobolev inequality see Lemma A.1 in Appendix A, which is applied to ηφ ε for some suitable function η in Cc 1 ε \ B ε 0 and 2 key estimates derived by using η m p+p i Φ m p+1 ε as a test function in the weak formulation of the approximate problem 3.2. Since the definition of m in Proposition 1.2 is different in Case 1 compared with Case 2, the iteration scheme needs to be devised carefully for each case as follows: I In Case 1, the condition on reads as > p 1. Since C in 1.12 blows-up as decreases to p 1, in the running step we shall require p 1 + δ for some fixed positive δ = δn, p. The choice of such a δ is possible because p n < p. II In Case 2, we apply Proposition 1.2 for m = 2 and q 1, where q is arbitrarily larger than some value C 0 > 0, say C 0 = max2n + 1, 2p n n + 1}. Unlike Case 1, the exact value of the threshold is not essential as long as it is big enough.

6 6 Florica C. Cîrstea, Jérôme Vétois With each subsequent iteration, we would need to ensure that Proposition 1.2 can be applied to all norms in the right-hand side of 1.12, that is for all i = 1,..., n we require /m + p i p p 1 + δ in Case 1, q 1 in Case Let k k 1 satisfy lim k k = and λ k = 1/2 + 1/2 k+1 for k 1. For k k 0 large and N a fixed large positive integer independent of k, we apply Proposition 1.2 with = k, λ = λ k+n+1 and λ = λ k+n. Let l denote the maximum number of iterations permitted under the restrictions in Due to the anisotropy in 1.12, we cannot determine the precise number of iterations l. However, with a careful choice of k and N, we can guarantee that k < lk k + N. After running our iteration scheme, jointly with various estimates from above, we find that Φ ε L k Arλk+N+1 is dominated by Cr lj=1 m j pτ j p p k C 2l n l j=1 2 j k q k m l k,τ1...τ l Φ ε k L k,τ 1...τ l Arλk+N l+1 lj=1 r 2 j 1 k 2 l k,τ1...τ l Φ ε k L k,τ 1...τ l Arλk+N l+1 in Case 1, see 5.32, in Case 2, see Here, C denotes a positive constant depending only on n and p, while τ j 1, 2,..., n} is in some sense a maximizer for each j = 1,..., l. For example, τ 1 is the index for which the right-hand side of 1.12 reaches its maximum and k,τ1 is given by k /m + p τ1 p. From the definition of l, we have k,τ1...τ l < p 1 + δ in Case 1 and k,τ1...τ l < q 1 in Case 2. To further bound Φ ε L k,τ1...τ l Arλ from above, we invoke interpolation inequalities, together with the local estimates in weak Lebesgue spaces found in k+n l+1 Lemma 3.3. To conclude 1.11 in Case 1, we let k. In case 2, after some further manipulations we can also pass to the limit, leading to 5.59 in which we choose q = C 0 max ln r, 1} to reach This concludes the sketch of the proof of the upper bound estimates in Theorem 1.1. The Moser iteration scheme represents a milestone in the development of the regularity theory of elliptic equations see Gilbarg Trudinger [26] or Han Lin [29] for more details. We mention that for anisotropic equations, other Moser-type iteration schemes, which are different from ours, have been used to derive local boundedness or gradient estimates of solutions. See, for instance, works by Fusco Sbordone [25], Lieberman [33], or the more recent paper by Cupini Marcellini Mascolo [17] and the references therein. 1.2 Plan of the paper In Section 2, we define the appropriate anisotropic Sobolev spaces for our problem. In Section 3, we construct the above-mentioned family of solutions Φ ε to approximate

7 Fundamental solutions for anisotropic elliptic equations 7 problems see Lemma 3.1 for which we also establish key estimates in weak Lebesgue spaces see Lemma 3.3. Section 4 is dedicated to the proof of the existence of a fundamental solution of 1.1 see Proposition 4.1. In Section 5, we prove our upper bound estimates by performing an iteration scheme as explained above. In Appendix A, we prove a weighted anisotropic Sobolev inequality, see Lemma A.1, which is invoked in the proof of the anisotropic reverse Hölder inequality of Proposition 1.2. Finally, in Appendix B, we establish another anisotropic inequality based on an inequality of Moser Trudinger type, see Lemma B.1, which is essential in the derivation of our estimates in weak Lebesgue spaces pertaining to Case 2 of Theorem Functional spaces In this section, we give a suitable notion of weak solution for studying 1.1. To this end, we need to introduce appropriate anisotropic spaces since the solutions of 1.1 cannot be defined in general in the usual distribution sense. 2.1 Motivation If p i = 2 n for all i = 1,..., n, then p u gives the usual Laplacian, whose fundamental solution is well-known. When all p i are equal to p 1, n], a non-negative solution of 1.1 can still be found explicitly. For simplicity, let = n x i p 1 } p 1 < R n 1. By direct inspection, a fundamental solution of 1.1 takes the form p n C x i p p p 1 R p n pn 1 if 1 < p < n, Φx = 2.1 C ln R 1 n 1 ln x i n n 1 if p = n for all x = x 1,..., x n \ 0} and some normalizing constant C = Cn, p > 0. If all p i are equal to p 1, 2 1/n], then the fundamental solution Φ in 2.1 does not belong to W 1,1 loc. Indeed, a simple calculation gives that Φ = C n n p x i p p 1 p 1 p 1 x i 2 2 p 1, which shows that Φ L 1 loc if and only if p 2 1/n, n. It follows that for p 1, 2 1/n], we cannot take the gradient of Φ in the anisotropic p -Laplacian in the usual distribution sense. We shall tackle this difficulty as in Bénilan et al. [7] by introducing the space T 1,1 loc, which allows us to give a sense to the gradient of u even if it is not locally integrable in general. This will be done in Section 2.2, see Definition 2.1.

8 8 Florica C. Cîrstea, Jérôme Vétois 2.2 Anisotropic Sobolev spaces We assume that 1.2 holds and is an open subset of R n with n 2. For any h > 0 and u : R, let T h u denote the truncation of u at height h, namely T h u : R, T h u x := ux if u x h, h sgnux if u x > h. 2.2 Let 1 E denote the characteristic function of a measurable set E in R n. Definition 2.1. i The functional space T 1,1 loc is defined as the set of all measurable functions u : R whose truncated function T h u belongs to W 1,1 loc for all h > 0. ii For any u T 1,1 loc, the gradient u is defined as the unique measurable function v : R n such that for all h > 0, we have T h u = v1 u <h} a.e. in. 2.3 For the existence and uniqueness up to a set of measure zero of v, see [7, Lemma 2.1]. Remark 2.2. i The set T 1,1 loc is not even a vector space, although if u 1 T 1,1 loc and u 2 W 1,1 loc L loc, then u 1 + u 2 T 1,1 loc see [7, p. 245]. ii The above definition of derivative for u T 1,1 loc is not a definition in the sense of distributions since, in general, if u T 1,1 loc L1 loc, then u need not belong to L 1 1,1 loc. However, for u Tloc we have u L1 1,1 loc if and only if u Wloc, and in this case, v in 2.3 coincides with u in the usual weak sense, that is i T h u = 1 u <h} i u a.e. in for every h > 0 and all i = 1,..., n. Given the definition of T 1,1 loc, we now introduce a concept of weak solution of 1.1. Definition 2.3. We say a function u : R is a weak solution of 1.1 if u T 1,1 loc and i u pi 1 L 1 loc for all i = 1,..., n such that i u i u pi 2 i ϕ dx = ϕ 0 for all ϕ Cc. 2.4 We next introduce other functional spaces needed in our paper: i Let T 1, p loc be the set of all u T 1,1 loc such that it h u L p i loc for every h > 0 and all i = 1,..., n. Notice that W 1, p loc T 1, p loc and T 1, p loc L loc = W 1, p loc L loc. ii The set T 1, p consists of all functions u T 1,1 loc with the property that i T h u L p i for every h > 0 and all i = 1,..., n.

9 Fundamental solutions for anisotropic elliptic equations 9 iii We define T 1, p 0 as the set of functions u T 1, p such that for any h > 0, there exists a sequence ϕ k k N in Cc such that as k, it holds ϕk T h u in L 1 loc, i ϕ k i T h u in L p i for all i = 1,..., n. 2.5 Note that, similar to Bénilan et al. [7], in the definition of T 1, p and T 1, p 0 we do not require T h u to belong to any L q with q 1. This condition is obviously satisfied when is bounded, but it makes a real difference when is unbounded. As in Appendix II of [7], it can be proved that the above definition of u T 1, p 0 is equivalent to the following: The function u T 1, p and there exists a sequence ζ k k in Cc such that as k, we have ζk u a.e. in ; i T h ζ k i T h u in L p i for every h > 0 and all i = 1,..., n. 2.6 Note that the main difference between 2.5 and 2.6 is that in the former case the sequence ϕ k k depends on h. iv Let W 1, p denote the set of all functions u L p whose weak partial derivative i u exists and belongs to L p i for each i = 1,..., n. The anisotropic Sobolev space W 1, p is a reflexive and separable Banach space equipped with the norm u W 1, p := u L p + i u L p i. We write u n u in W 1, p loc to mean u n u in W 1, p ω for each ω. By ω, we mean that ω is an open set such that ω is compact and ω. v The set W 1, p 0 is defined as the closure of Cc with respect to W 1, p. Hence, we have the inclusion W 1, p 0 T 1, p 0. If T h u W 1, p 0 for every h > 0, then u T 1, p 0 ; The converse is also true provided that is bounded. It can be proved that the definition of u W 1, p 0 is equivalent to u T 1, p 0 and i u L p i for all i = 1,..., n when is bounded. We recall the following Sobolev inequality due to Troisi see Theorem 1.2 in [43]: Lemma 2.4. Let p be given by 1.3. Then there exists a constant c > 0 such that u L q c n i u 1/n L p i for all u C c, 2.7 where c = cn, p and q = np/n p if p < n, while c = cn, p, q, meas supp u and q is any positive number if p n.

10 10 Florica C. Cîrstea, Jérôme Vétois Remark 2.5. If p < n, then 2.7 holds for any u W 1, p 0 by a density argument. Remark 2.6. If p < n, then T 1, p 0 L 0, where L 0 denotes the set of measurable functions u : R such that the set u > ε} has finite measure for every ε > 0. Indeed, i T h u L p i for every h > 0 so that T h u L np/n p by a density argument using 2.7. Thus, T h u L 0 for every h > 0. From our definition of W 1, p and W 1, p 0, we recover the usual Sobolev space W 1,p and W 1,p 0, respectively when p i = p for all i = 1,..., n. On the other hand, if is bounded and 1 < p <, we infer that 2.7 is true with q = p and any u W 1, p 0 so that an equivalent norm on W 1, p 0 can be taken as u = n iu L p i. We mention in passing that there are other versions of anisotropic spaces, which may not coincide with the ones introduced here see, for example, Nikol skiĭ [38], Rákosník [39] and Troisi [43]. 3 Auxiliary tools In Lemma 3.1 we establish the existence of a family of approximate solutions for which later in Lemma 3.3 we give crucial uniform estimates in weak Lebesgue spaces. These estimates in Case 2 p = n and is bounded are new and different from those in Case 1 p < n. In the former case, they are obtained via anisotropic Sobolev inequalities of Moser Trudinger type see Lemma B.1 in Appendix B. For the reader s convenience, in Section 3.2 we introduce the weak Lebesgue spaces and their properties. 3.1 Approximate solutions Let be a domain of R n such that 0. In this section, we are in either Case 1 see 1.4 or Case 2 p = n and is bounded. To prove the existence of weak solutions of 1.1, we consider suitable approximate problems for which existence results can be obtained easily. We use B 1/ε 0 to denote the ball of center 0 and radius 1/ε. Since may be unbounded when p < n, we approximate using a sequence of bounded domains ε with ε as ε 0. We fix ε 0 0, 1. For ε 0, ε 0 ], we define B1/ε 0 if p < n, ε := if p = n. Moreover, for every ε 0, ε 0 ], we construct a function f ε with the following properties: fε Cc B ε 0, f ε 0, and f ε L 1 ε 1, f 3.1 ε δ 0 in the sense of measures as ε 0.

11 Fundamental solutions for anisotropic elliptic equations 11 Let f 1 C c B 1 0 be such that f 1 0 and B 1 0 f 1 x dx = 1. For any ε 0, ε 0 ], we set f ε x := ε n f 1 ε 1 x for all x B ε 0 and f ε = 0 on R n \ B ε 0. Hence, f ε satisfies 3.1. Lemma 3.1. Let ε 0, ε 0 ] be arbitrary. Then the problem p Φ ε = f ε in ε, Φ ε = 0 on ε 3.2 admits a non-negative weak solution Φ ε W 1, p 0 ε L loc ε in the sense that i Φ ε i Φ ε pi 2 i ϕ dx = f ε ϕ dx for all ϕ W 1, p 0 ε. 3.3 ε ε Furthermore, for every h > 0, the solution Φ ε satisfies i Φ ε p i dx h. 3.4 Φ ε h} Proof. For u W 1, p 0 ε fixed, we define A ε u : W 1, p 0 ε R by A ε u, v := i u i u pi 2 i v dx for every v W 1, p 0 ε. ε Let p i denote the Hölder conjugate of p i, that is p i = p i /p i 1 for i = 1,..., n. Clearly, A ε u belongs to the dual of W 1, p 0 ε, denoted by W 1, p ε with p := p 1,..., p n. One can easily check that the operator A ε : W 1, p 0 ε W 1, p ε is bounded, monotone, coercive and hemicontinuous see Bendahmane Karlsen [5] for more details. Then, A ε is a surjective operator see Lions [34, Chapter 2, Theorem 2.1]. Let B ε ϕ denote the right-hand side of 3.3. Since B ε W 1, p ε, the surjectivity of A ε proves the existence of Φ ε W 1, p 0 ε such that A ε Φ ε = B ε, that is 3.3 holds. Moreover, by Fusco Sbordone [25, Remark 3.5], we obtain that Φ ε L loc ε. To prove that Φ ε 0 a.e. in ε, we denote N ε := x ε : Φ ε x < 0}. If meas N ε 0, then by using Φ ε 1 Nε as a test function in 3.3, we find that i Φ ε p i dx = N ε f ε Φ ε dx 0, N ε 3.5 which implies that Φ ε = 0 a.e. in N ε. This contradiction shows that Φ ε 0 a.e. in ε. For any h > 0, we have T h Φ ε = minφ ε, h} W 1, p 0 ε. By using the truncated function T h Φ ε as a test function in 3.3, we obtain that i Φ ε p i dx = i Φ ε i Φ ε pi 2 i T h Φ ε dx Φ ε h} ε 3.6 = f ε T h Φ ε dx h f ε dx h, ε ε which proves 3.4. This ends the proof of Lemma 3.1.

12 12 Florica C. Cîrstea, Jérôme Vétois 3.2 Weak Lebesgue spaces The weak L space is by definition the usual L space. Therefore, unless otherwise stated, we assume throughout this subsection that 0 < q <. Recall that L q denotes the set of all real-valued measurable functions u on such that u q is integrable. The quasi-norm of such a function u L q is defined by u L q := 1 ux q q dx. Whenever 1 q <, the Minkowski inequality holds: f + g L q f L q + g L q for all f, g L q, 3.7 whereas for 0 < q < 1, the inequality 3.7 is reversed when f, g 0. However, for the case 0 < q < 1, the inequality 3.7 is replaced by the following f + g L q 2 1 q q f L q + g L q for all f, g L q. Note that L q are Banach spaces for q 1 and quasi-banach spaces for q 0, 1. For a measurable function u : R, the distribution function of u is the function d u defined on [0, as follows d u h := measx : u x > h}. 3.8 The distribution function h d u h is a decreasing function. Definition 3.2. For 0 < q <, we define the weak L q space, denoted by L q,, as the set of all measurable functions u : R such that } u L q, = sup h>0 h d u h 1 q <. 3.9 As an analogue of 3.7, for any 0 < q < we have f + g L q, max2, 2 1/q } f L q, + g L q, for all f, g L q,. Notice that L q, does not define a norm for q 0,, but a quasi-norm in L q,. It can be shown that L q, is a complete quasi-normed space for q 0,. The weak L q spaces are larger than the usual L q spaces, that is L q L q, for 0 < q < Indeed, we have u L q, u L q for any u L q by Chebyshev s inequality: h q d u h ux q dx. x : ux >h}

13 Fundamental solutions for anisotropic elliptic equations 13 Furthermore, the inclusion in 3.10 is strict. Indeed, for any q 0,, the function ux = x n/q is not in L q R n, whereas u belongs to L q, R n with u L q, R n being the measure of the unit ball in R n to the 1/qth power. If is bounded, then for 0 < q 0 < q <, we have L q, L q 0 and 1 q q 0 1 u L q 0 meas 1 q 0 q u L q q q, for all u L q, The inequality in 3.11 will be applied several times in the paper. For more details on weak Lebesgue spaces, we refer to Grafakos [27]. 3.3 Key estimates in weak Lebesgue spaces When p < n, Bendahmane Karlsen [5] proved 3.12 under more general structure conditions on the anisotropic operator, but with a constant depending on the domain. In our framework, we prove that the constant in 3.12 depends only on n and p. Furthermore, the case p = n included in Lemma 3.3 is new compared with [5]. Lemma 3.3. For any ε 0, ε 0 ], let Φ ε be the weak solution of If 1.4 holds, then there exists a positive constant C = C n, p such that i Φ ε p i p 1 + Φ, ε L L p p 1, ε C ε 2 If p = n and is bounded, then there exists a positive constant C = C n, p such that for all q 1, we have } Φ ε L q, max Cq, meas 1+ 1 q, p i q 3.13 q+1 i Φ ε p i q Cq q + n meas q+1. L q+1, Proof. 1 We proceed using essentially the same ideas as in [5]. Let p := np/ n p. Since T h Φ ε W 1, p 0 ε and p < n, by Remark 2.5 and 3.6, there exists a constant C = C n, p > 0 such that T h Φ ε L p ε C n We conclude 3.12 by showing that i T h Φ ε 1/n L p i ε Ch1/p for every h > Φ ε L p 1, ε C p p 1 and i Φ ε L p i p 1 p, ε C p + 1 p p i p

14 14 Florica C. Cîrstea, Jérôme Vétois for each i = 1,..., n. From 2.2, 3.8, and 3.14, we obtain that 1 h d Φε h 1 p p = T h Φ ε p dx T h Φ ε L p Ch 1 p ε for all h > Φ ε>h} The first inequality in 3.15 follows from 3.16 since d Φε h C p h 1 p for every h > Moreover, using 3.6 and 3.17, for each i = 1,..., n and h > 0, we find that d i Φ ε h d Φε h p i /p + meas x ε : Φ ε x h p i/p and i Φ ε x > h } C p + 1 h p ip 1/p. This implies the second inequality in Hence, the assertion of a holds. 2 By Lemma B.1 in Appendix B, there exists a constant c = c n, p > 0 such that for all q 1 and h > 0, we have T h Φ ε n cq + 1 meas n 1 + L q+1n n 1 Since T h Φ ε = minφ ε, h}, by using 3.8 we see that Φ ε h} i Φ ε p i dx q+n q n 1 T h Φ ε n h nq+1 q+1 L q+1n n 1 dx = h n d Φε h n 1 q+1 for all h > n 1 Φ ε>h} From 3.4, 3.18, and 3.19, we infer that h n d Φε h n 1 q+1 cq + 1 n 1 meas + h q+n q+1 for every h > 0. Hence, for all h > 0, we have h d Φε h 1 q+1 q+1 q c qn 1 q + 1 q h 1 meas + 1 q+n qn 1 c n 1 6q h 1 meas + 1 n+1 n If 0 < h < meas, then h d Φε h 1 q meas 1 q +1, whereas for h meas we obtain from 3.20 that h d Φε h 1 q Φ ε L q, = sup h>0 Cq, where C = Cn, p > 1. It follows that } } h d Φε h 1 q max Cq, meas 1+ 1 q. This proves the first inequality in Using 3.4 and 3.8, for any h > 0, we find that d i Φ ε h = meas x : i Φ ε x > h} } meas x : Φ ε x h p i q+1 and i Φ ε x > h + d Φε h p i q+1 h p i p i } i Φ ε p i dx + d Φε h p i q+1 h p i q q Φ ε q L q,. Φ ε h q+1

15 Fundamental solutions for anisotropic elliptic equations 15 Hence, for every i = 1,..., n, we obtain that p i q } q+1 i Φ ε p i q = sup h p i q q+1 d i L q+1, Φ ε h 1 + max Cq q, meas q+1} h>0 The second inequality in 3.13 follows from This completes the proof. Remark 3.4. In both cases of Lemma 3.3, we find that lim d Φ ε h = 0 and lim d i Φ ε h = 0 for every i = 1,..., n, 3.22 h h where all limits hold uniformly with respect to ε 0, ε 0 ]. 4 Existence of a fundamental solution In this section, we adapt ideas of Bénilan et al. [7] to show that, under the assumptions of Theorem 1.1, problem 1.1 admits a fundamental solution Φ T 1, p 0. This is obtained using an approximation procedure as in Boccardo Gallöuet [8, 9] and Dall Aglio [19]. For more general existence and regularity results for nonlinear measure data problems, see Mingione [35]. Proposition 4.1. Suppose that either Case 1 or Case 2 in Theorem 1.1 holds. Then 1.1 has a non-negative weak solution Φ T 1, p 0. We later prove that Φ in Proposition 4.1 belongs to W 1, loc \ 0} see Remark 5.2. Proof. Let ε k k N be a sequence in 0, ε 0 ] such that ε k 0 as k. For any k N, let Φ εk be as in Lemma 3.1. We define Φ εk = 0 on R n \ εk. For every h > 0 and i = 1,..., n, we have the following facts about T h Φ εk k : a From 2.2 and Φ εk W 1, p 0 εk, we have T h Φ εk W 1, p 0 εk for all k N and i T h Φ εk = 1 Φεk <h} i Φ εk a.e. in. 4.1 b For any 1 q, the family T h Φ εk k is uniformly bounded in L q loc Rn. c The family i T h Φ εk k is uniformly bounded in L p i R n see 3.4. d Up to a subsequence, T h Φ εk converges in L q loc Rn for all q [1,. We need only prove d. Fix R > 0. From c, we see that T h Φ εk k is uniformly bounded in W 1,p 1 B R 0. So, up to a subsequence, T h Φ εk k converges in L q B R 0 for every q [1, p 1 ] due to the compactness of the embedding W 1,p 1 B R 0 L p 1 B R 0. Hence, using b and an interpolation between L 1 B R 0 and L B R 0, we get that, up to a subsequence, T h Φ εk k converges in L q B R 0 for all q [1,. Since R > 0 is arbitrary, by a diagonal argument, we conclude d. In the framework of Proposition 4.1, we establish the following auxiliary result.

16 16 Florica C. Cîrstea, Jérôme Vétois Lemma 4.2. Fix K an arbitrary compact subset of. Let ε 0 > 0 be small such that K εk for every k N. Then, for every i = 1,..., n, we have: i The family i Φ εk pi 2 i Φ εk }k is uniformly bounded in L1 K and lim sup h k N x εk : i Φ εk >h} iφ εk pi 1 dx = ii There exists a measurable function Φ : R such that, up to a subsequence, Φ εk Φ locally in measure and a.e. in as k. 4.3 iii Moreover, Φ T 1,1 loc and, up to a subsequence, iφ εk satisfies i Φ εk i Φ locally in measure and a.e. in as k. 4.4 Remark 4.3. Let h > 0 be arbitrary. From d and 4.3, we get that, up to a subsequence, T h Φ εk T h Φ in L q loc Rn and a.e. in for every q [1,. 4.5 We postpone the proof of Lemma 4.2 to complete the proof of Proposition 4.1. Proof of Proposition 4.1 concluded. We first show that Φ is a weak solution of 1.1 see Definition 2.3. Let ϕ C c be arbitrary. Then supp ϕ εk0 if k 0 N is large enough. Since Φ εk is a weak solution of 3.2 with ε = ε k and = εk, we have i Φ εk i Φ εk pi 2 i ϕ dx = εk f εk ϕ dx ε for every k k From Lemma 4.2i, the family i Φ εk p i 2 i Φ εk }k is uniformly integrable in L1 K. In view of Lemma 4.2iii, by Vitali s convergence theorem see Brezis [13, p. 122], we conclude that i Φ p i 1 L 1 loc and, up to a subsequence of ε k, relabelled ε k, we have i Φ εk p i 2 i Φ εk i Φ p i 2 i Φ in L 1 loc as k. 4.7 Using 4.7 and 3.1, we can pass to the limit in 4.6 to conclude that i Φ i Φ p i 2 i ϕ dx = ϕ0 for every ϕ C c. Hence, Φ is a weak solution of 1.1. We next show that Φ T 1, p 0. Since T h Φ εk W 1, p 0 εk for any h > 0 and k N, we find that there exists ϕ h,k C c εk such that ϕ h,k T h Φ εk W 1, p 0 εk 1/k. 4.8

17 Fundamental solutions for anisotropic elliptic equations 17 For h > 0 fixed, since i T h Φ εk k is uniformly bounded in L p i, we deduce that i ϕ h,k k is uniformly bounded in L p i. By Lemma 4.2iii, up to a subsequence, it holds i T h Φ εk i T h Φ a.e. in as k. Thus, using 4.5 and 4.8, we get that, up to a subsequence, ϕ h,k k satisfies ϕ h,k T h Φ in L 1 loc and i ϕ h,k i T h Φ in L p i as k for i = 1,..., n. By Mazur s lemma see Brezis [13, p. 61], there exists ϕ h,k k in C c such that ϕ h,k T h Φ in L 1 loc and i ϕ h,k i T h Φ in L p i as k for all i = 1,..., n. This proves that Φ T 1, p 0. This completes the proof. Proof of Lemma 4.2. i Let i = 1,..., n be fixed arbitrarily. Let q i = p i p 1/p in Case 1 and q i > maxp i 1, 1} in Case 2. If m i := q i /q i p i + 1, then m i > 0 in both cases. Hence, using 3.8 and 3.11, for every k N and h > 0, we have K i Φ εk p i 1 dx m i meas K 1/m i i Φ εk p i 1 L q i, K, x εk : i Φ εk >h} i Φ εk p i 1 dx m i d i Φ εk h 1/m i i Φ εk p i 1 L q i, εk. By Lemma 3.3 and Remark 3.4, we conclude the assertion of i. ii We prove that, up to a subsequence, Φ εk k is a Cauchy sequence with respect to convergence in measure in K: For every ν, τ 0,, there exists N ν,τ N such that meas x K : Φεk x Φ εk x > ν } < τ for all k, k N ν,τ. 4.9 Let ν, τ and h be fixed arbitrarily in 0,. For k, k N, we use the notation d Φεk h and d h as in 3.8. We define I Φεk h,ν,k,k as follows I h,ν,k,k := meas x K : Φ εk x h, Φ εk x h and Φεk Φ εk x > ν }. Then for every h > 0, we have the following estimates: meas x K : Φ εk Φ εk x > ν } d Φεk h + d Φ ε k h + I h,ν,k,k, I h,ν,k,k meas x K : T h Φ εk T h Φ εk x > ν} From d above, we infer that, up to a subsequence, T h Φ εk k is a Cauchy sequence with respect to convergence in measure in K. Using Remark 3.4 and 4.10, we conclude up to a subsequence, Φ εk k is a Cauchy sequence with respect to convergence in measure in K. Hence, up to a subsequence, Φ εk k converges in measure in K to a measurable function Φ : K R. By Riesz Theorem, we can further pass to a subsequence, relabelled ε k, such that Φ εk Φ a.e. in K. By a diagonal argument, we conclude the proof of ii.

18 18 Florica C. Cîrstea, Jérôme Vétois iii Let i = 1,..., n be arbitrary. We show that, up to a subsequence, i Φ εk k is a Cauchy sequence with respect to convergence in measure in K. For any ν, h 0, and k, k N, we introduce the following notation: U ν,k,k := x K : i Φ εk i Φ εk x } > ν, V h,k,k := x K : i Φ εk x h, i Φ εk x h }, W h,k,k := x K : Φ εk Φ εk x 1/h } 4.11, J h,ν,k,k := meas U ν,k,k V h,k,k W h,k,k. We want to get an upper bound estimate for meas U ν,k,k. Let d i Φ εk h and d i Φ εk h be given as in 3.8. Then for all h, ν > 0 and k, k N, we find that meas U ν,k,k d i Φ εk h + d i Φ h + meask \ W εk h,k,k + J h,ν,k,k We next show that J h,ν,k,k 0 as h uniformly with respect to k, k N. By testing problem 3.2 with the truncated function T 1/h Φεk Φ εk, we obtain that W h,k,k i Φ εk p i 2 i Φ εk i Φ εk p i 2 i Φ εk i Φ εk i Φ εk dx 2 h Indeed, the left-hand side LHS of 4.13 satisfies 1 LHS of 4.13 = fεk f εk T1/h Φεk Φ εk dx fεk f εk 2 dx R h n R h. n Moreover, there exists a positive constant C, independent of h > 0, such that s p i 2 s t p i 2 t Ch p i 2 s t if p i < 2 C s t p i 1 if p i for any s, t h, h. By 4.11, 4.13 and 4.14, we infer that 0 J h,ν,k,k ν max2,p i} max2,pi } i Φ εk i Φ εk dx V h,k,k W h,k,k 2 Cν max2,p i} h p i max2,p i }+1 0 as h. Hence, using Remark 3.4, 4.9 and 4.12, we find that, up to a subsequence, i Φ εk k is a Cauchy sequence with respect to convergence in measure in K. By Riesz Theorem, we can pass to a subsequence such that i Φ εk Ψ i a.e. in K for some measurable function Ψ i : K R and all i = 1,..., n. A standard diagonal argument gives that Ψ i : R is measurable and, up to a subsequence, i Φ εk satisfies for i = 1,..., n i Φ εk Ψ i locally in measure and a.e. in as k. 4.15

19 Fundamental solutions for anisotropic elliptic equations 19 We next prove that Φ T 1,1 loc and Ψ i = i Φ for all i = 1,..., n. Let h > 0 and R > 0 be arbitrarily fixed. From c, we have i T h Φ εk k is uniformly bounded in L p i R n so that, up to a subsequence, i T h Φ εk converges weakly to some function Ψ i,h in L p i R n and, hence, in L p 1 loc Rn. Using 4.5 for q = p 1 and the compactness of the embedding W 1,p 1 B R 0 L p 1 B R 0 for all R > 0, we get that T h Φ W 1,p 1 B R 0 and, up to a subsequence, T h Φ εk T h Φ in W 1,p 1 B R 0. We thus deduce that Ψ i,h = i T h Φ and, hence, up to a subsequence, i T h Φ εk i T h Φ in L p i R n In particular, we have Φ T 1,1 loc. From 4.1 and 4.15, we find that, up to a subsequence, i T h Φ εk 1 Φ<h} Ψ i a.e. in. This, jointly with 4.16, implies that i T h Φ = 1 Φ<h} Ψ i a.e. in B R 0 for every R > 0. Thus i Φ = Ψ i a.e. in for i = 1,..., n, which together with 4.15, proves iii. This finishes the proof of Lemma 4.2. We next discuss the situations when our fundamental solution Φ in Proposition 4.1 becomes a distributional solution. Corollary 4.4. If in Case 1 of Theorem 1.1 we let be bounded and p 1 > pn 1, then Φ np 1 in Proposition 4.1 is a W 1, q 0 -distributional solution of 1.1, where q = q 1,..., q n np 1 and 1 q i < p i for i = 1,..., n. The same conclusion holds in Case 2. pn 1 Proof. The same argument applies for both Case 1 and Case 2. Let i 1,..., n} be fixed arbitrarily. Since T 1, p 0 T 1, q 0, by v in Section 2.2, it is enough to show that i Φ L q i np 1 with 1 q i < p i. Let Φ pn 1 ε k k be such that 4.4 holds. By 3.11 and the weak Lebesgue estimates in Lemma 3.3, we infer that i Φ εk k is uniformly bounded in L q i and, hence, up to a subsequence, i Φ εk converges weakly in L q i. Since i Φ εk i Φ a.e. in, we conclude that i Φ L q i for i = 1,..., n. 5 Sharp upper bound estimates Let Φ be the non-negative fundamental solution of 1.1, which was constructed in Section 4. In Theorem 5.1, we prove that Φ satisfies 1.11 from which we get 1.6 in Case 1 and 1.7 in Case 2. We follow the sketch of the proof outlined in Section 1.1. For λ 0, 1 and r > 0, we define A r λ := E p 1 + λr \ E p 1 λr, where E p R is given by 1.5. Theorem 5.1. Under the assumptions of Theorem 1.1, the fundamental solution Φ given by Proposition 4.1 satisfies 1.11 for every r 0, R.

20 20 Florica C. Cîrstea, Jérôme Vétois Remark 5.2. For any compact subset K in \ 0}, we deduce from 1.11 that Φ = T h Φ on K for h > 0 large so that Φ W 1, p K L K. It follows from Lieberman see [33] that Φ W 1, loc \ 0} and, hence, Φ C \ 0}. Our proof uses an iteration scheme of Moser-type see Section 5.2 for Case 1 and Section 5.3 for Case 2, whose running step is given by Proposition 1.2 proved in Section 5.1. Using 1.11, we next prove our gradient estimates in 1.8. Corollary 5.3 Gradient estimates. For the fundamental solution Φ constructed in Proposition 4.1, there exists a positive constant C 1 = C 1 n, p such that 1.8 holds. Proof. We define Φ r x := r n p pn 1 Φ r 1 s 1 x 1,..., r 1 sn x n for all r 0, R and x E p 2, where s i is given by 1.5. Rescaling the equation satisfied by Φ, we get that div A Φ r = 0 in E p 2 \ E p 1/8. Using 1.11, we find that there exists a constant c = cn, p > 0 such that Φ r x c for all x E p 3/2 \E p 1/4. From Lieberman s gradient estimates in [33], there exists a constant C = C n, p > 0 such that Φ r x C for a.e. x E p 1 \E p 1/2. Hence, we have i Φ x p i = r 1 p i i Φ r r 1 s 1 x 1,..., r 1 p i sn x n r 1 C p i for a.e. x E p r \E p r/2. In view of E p R 0<r<R E p r \E p r/2, we get the desired estimate 1.8. Remark 5.4. Since in Case 1, the function U 0 in 1.9 belongs to L p 1,, we infer from 1.6 and 1.8 that Φ L p 1, and i Φ L p i p 1 p, for i = 1,..., n. In Case 2, we derive that Φ L q for every 1 q < and i Φ L pi, for i = 1,..., n by using 1.7 and Proof of Proposition 1.2 In this subsection, we are in either Case 1 or Case 2 of Theorem 1.1. Let ε 0, ε 0 ] and r > 0 be such that A r 3/4 ε \ B ε 0. We use m to denote n/n p in Case 1 and any number greater than 1/n 1 in Case 2. Let > mp 1 be arbitrary. We define β and Θ j with j = 1,..., n as follows β = m p and Θ j := r pj n p pn 1 1 λ λ p j Φ ε β+p j L β+pj A rλ. 5.1

21 Fundamental solutions for anisotropic elliptic equations 21 Our assumption on gives that β > 1. We show that 1.12 holds, meaning that there exists a positive constant c = cn, p such that for every 0 < λ < λ 3/4, we have Φ ε m c max1, β + L A rλ 1 pn } p r 1 m n p n max Θ j. 5.2 j=1,...,n We split the proof into four steps. We choose a suitable function η C 1 c ε \ B ε 0 as in Step 1 below. We can suppose that Φ ε ν > 0 a.e. on supp η for some ν > 0. If this is not true, we can consider Φ ε + ν in our argument and then let ν 0. Step 1. We construct a function η C 1 ε \ B ε 0 with the following properties: a 0 η 1 in ε, η = 0 in ε \ A r λ and η = 1 in A r λ. b η β+p 1 pn C 1 ε and there exists a positive constant c = cn, p such that η β+p 1 pn 1 j η c p 1 + β r n p pn 1 1 p j λ λ for every j = 1,..., n. To this aim, we choose a function η 1 C 1 [0, such that 0 η 1 1 in [0,, as well as η 1 = 0 on [0, 1 λ ] [1 + λ, and η 1 = 1 on [1 λ, 1 + λ]. We further assume that η 2 := η β+p 1/p n 1 C 1 [0, and η 2 2/λ λ. We now define ηx = η 1 r 1 x i s i for all x ε, 5.3 where s i is given by 1.5. It is easy to check that η in 5.3 satisfies a and b. Notation. For every i, j = 1,..., n, we define V ij and T ij by V ij := η β+p i p j j η p j Φ β+p j ε dx and T ij := ε j Φ ε p j η β+p i Φ β ε dx. ε 5.4 Step 2. There exists a constant C 0 = C 0 n, p > 0 such that for every i = 1,..., n, pj β + pi T ii C 0 V ij <. 5.5 β + 1 j=1 From Step 1 and the definition of Θ j in 5.1, we obtain that max V ij η pj β+p1 pn 1p j j η p j Φ β+p j c ε dx Θ j <. 5.6,...,n ε β + p 1 Since β + p i > β + p 1 /p n, using Step 1 and our assumption A r λ ε \ B ε 0, we have η β+p i Cc 1 ε \ B ε 0. Taking η β+p i Φ β+1 ε as a test function in 3.3, we get T ij + β + p i j Φ ε pj 2 j Φ ε j η η β+pi 1 Φ β+1 ε dx = 0, β + 1 ε j=1 j=1

22 22 Florica C. Cîrstea, Jérôme Vétois which implies that j=1 T ij β + p i β + 1 j=1 ε j Φ ε p j 1 j η η β+p i 1 Φ β+1 ε dx. 5.7 Applying Young s inequality ab a p j /p j + b p j /p j in the following situation p j = we arrive at p j p j 1, a = τ 1+1/p j j η η 1 Φ ε, b = τ 1 1/p j j Φ ε pj 1, and τ = β + 1, β + p i j Φ ε p j 1 j η η 1 Φ ε β + 1p j 1 β + p i p j j Φ ε p j + 1 p j Then, the right-hand side RHS of 5.7 satisfies β + 1 β + p i 1 pj j η p j η p j Φ p j ε. RHS of 5.7 j=1 p j 1 p j T ij + 1 p j=1 j pj β + pi V ij, 5.8 β + 1 where V ij and T ij are given by 5.4. From 5.7 and 5.8, we infer that j=1 1 p j T ij 1 p j=1 j pj β + pi V ij. 5.9 β + 1 The claim of Step 2 follows immediately from 5.9. Step 3. There exists a constant C 1 = C 1 n, p > 0 such that for every i = 1,..., n H i := ηφ ε β/p i i ηφ ε p i L p i ε C 1 max1, β + 1 pn } max j=1,...,n Θ j, 5.10 where Θ j is defined by 5.1. Indeed, for any i = 1,..., n, we find that i ηφ ε p i 2 p i 1 η p i i Φ ε p i + Φ p i ε i η p i. Hence, by 5.6 and Step 2, there exist positive constants C 0, C 0, and C 1, depending only on n and p, such that H i 2 p i 1 T ii + V ii 2 p i 1 C C 0 β + 1 p j j=1 pn + β p 1 + β j=1 pj β + pi V ij β + 1 pj Θ j C 1 max1, β + 1 pn } max j=1,...,n Θ j This proves the claim of Step 3.

23 Fundamental solutions for anisotropic elliptic equations 23 Step 4. Proof of 5.2 concluded. We apply Lemma A.1 in Appendix A with = ε and ϕ = ηφ ε. Clearly, we have ϕ L ε and supp ϕ is a compact subset of ε. We denote ξ = β + p/p. The assumption > mp 1 gives that 1 + pξ 1/p 1 > 0. If ξ 1 i.e., mp, then ηφ ε W 1, p ε since Φ ε W 1, p ε and η Cc 1 ε. 1+ pξ 1 pξ 1 1+ p If ξ < 1, then we find that ηφ ε 1 W 1, p p ε since Φ 1 ε W 1, p ε pξ 1 1+ p using that Φ ε ν > 0 a.e. on supp η and η 1 Cc 1 ε see Step 1. Then, by Lemma A.1 in Appendix A, there exists a constant c = cn, p > 0 so that ηφ ε m L ε c p n p H i n i in Case 1, c n meas A r λ 1 m p n 1 p H i i in Case 2, 5.12 where H i is defined by Since η = 1 in A r λ and n 1/p i = n/p, from 5.12 and Step 3, we reach 5.2. This concludes the proof of Proposition Proof of Theorem 5.1 in Case 1 Throughout this subsection, we assume that 1.4 holds. Recall that m := n/n p. Let ε 0, ε 0 ] and r > 0 be such that A r 3/4 ε \ B ε 0. In what follows, we denote L q A rλ := q;λ To keep the notation short, we shall not include r since it is fixed everywhere in the proof. The desired estimate 1.11 follows from 5.14 below applied to Φ εk in Lemma 4.2 and passing to the limit, up to a subsequence, as k. Proposition 5.5. There exists a positive constant C = C n, p such that Φ ε L A r1/2 Cr 1/p Proof. For any > p 1 and i = 1,..., n, we define,i as follows,i := σ i = /m + p i p In this proof, we fix δ = δn, p so that 0 < δ < p p n n/2n p. Hence, we have σ i p 1 + δ < p 1 δ for all i = 1,..., n By Proposition 1.2, we know that for any p 1 + δ, there exists a positive constant C = Cn, p such that [ ] Φ ε ; λ C p r p i p 1 m/ max,...,n λ λ Φ ε,i p i,i ; λ. 5.17

24 24 Florica C. Cîrstea, Jérôme Vétois The proof of Proposition 5.5 consists of iterating We provide the details below. The iterative scheme. Let λ k := 1/2 + 1/2 k+1 for every k 1. Hence, we have λ 1 = 3/4 and λ k+1 = λ k 2 k 2 for all k We see that λ k k 1 is decreasing and converges to 1/2 as k. Since p n < p and m > 1, we can find a large positive integer N so that We construct a sequence k k 1 such that n p p 1 /p + δ < m N n p p n /p n p p 1 /p + δ < k /m k < m N n p p n /p Since k as k, there exists a large integer k 0 1 such that k > p 1 + δ for every k > k 0. In what follows, we fix k > k 0. For each i 1, i 2 1,..., n}, we define k,i1 as in 5.15 and k,i1 i 2 = σ i2 k,i1, that is k,i1 := k /m + p i1 p, k,i1 i 2 := k,i1 /m + p i2 p We also need to introduce B i1 and B i1 i 2 as follows m mk+n+2p i1 mp i1 p k,i1 B i1 := 2 k p r k Φ ε k k,i1 ; λ k+n, m 2 2j=1 p m j k+n j+3p 2j=1 ij m j p ij p m 2 k,i1 i 2 B i1 i 2 := k k,i 1 2 k p r k Φ ε k k,i1 i 2 ; λ k+n Let τ 1, τ 2 1,..., n} be such that B τ1 = max i1 =1,...,n B i1 and B τ1 τ 2 = max i2 =1,...,n B τ1 i 2. We first apply 5.17 with = k, λ = λ k+n+1 and λ = λ k+n. We obtain that Φ ε k ; λ k+n+1 C m mp k k k B τ Our choice of k ensures that k,i1 > p 1 + δ for all i 1 = 1,..., n. Thus in 5.23, we can continue using 5.17 to estimate each Φ ε k,i1 ; λ k+n with i 1 = 1,..., n as follows m k,i1 Φ ε k k,i1 ; λ k+n C m 2 m 2 p m 2 k+n+1p i2 m 2 m p i2 p 2 k,i1 i 2 k k k,i 1 2 k p r k Φ ε k k,i1 i 2 ; λ k+n 1. This, jointly with 5.23, leads to max i 2 =1,...,n Φ ε k ; λ k+n+1 C m+m 2 k mp k k B τ1 τ If this iteration process works l times, it gives rise to k,i1...i l, A i1...i l and B i1...i l, namely k,i1...i l := k,i1...i l 1 /m + p il p, l m j lj=1 p m j k+n j+3p ij A i1...i l := k k,i 1...i j 1 2 k, 5.25 j=2 B i1...i l := A i1...i l r lj=1 m j p ij p p k m l k,i1...i l Φ ε k k,i1...i l ; λ k+n l+1.

25 Fundamental solutions for anisotropic elliptic equations 25 For the definition of k,i1 and k,i1,i 2, see 5.21, whereas for B i1 and B i1 i 2 see For each j 2,..., l}, let τ j 1,..., n} be such that After l uses of 5.17, we arrive at B τ1...τ j = max i j 1,...,n} B τ 1...τ j 1 i j. Φ ε k ; λ k+n+1 C lj=1 m j k mp k k B τ1...τ l Let l denote the maximum number for which 5.17 can be used iteratively, that is k,τ1...τ j p 1 + δ for all 1 j l 1 and k,τ1...τ l < p 1 + δ We next prove that k < l k + N. Using 5.25, we find that k,i1...i j = k /m j + j m ν j p iν p for all 1 j l Since p 1 p i p n for all i = 1,..., n, it follows from 5.28 that ν=1 k /m j n p p 1 /p k,i1...i j k /m j + n p n p /p This, jointly with 5.20, implies that for any i 1,..., i k+n 1,..., n}, we have k,i1...i j > p 1 + δ for all j = 1,..., k and k,i1...i k+n < p Using 5.30, we conclude that k < l k + N. Thus from 5.20 and 5.29, we see that m l k < m N p np p 1, k < m N+k p 1 and k,τ1...τ j 1 < m N+k j+1 p for all j = 2,..., l. Using 5.31 and a simple calculation, we obtain that 1 k [ l N + k j + 1m j m = N + k lm l N + k + m 1 k m l < Nm l + m j < C m 1 n, p, k j=1 where C n, p > 0 is a positive constant depending only on n and p. Let A i1...i l be defined by It follows that j=1 ] l m j j=1 mp k k p l j=1 m j p l j=1 N+k j+1m j pn l j=1 m j k+n j+3 A τ1...τ l p k m k 2 k,

26 26 Florica C. Cîrstea, Jérôme Vétois which is dominated from above by a positive constant depending on n and p. Therefore, using 5.25 and 5.26, we infer that there exists a constant C = C n, p > 0 such that Φ ε k ; λ k+n+1 Cr 1 p k m l k,τ1...τ l l j=1 mj p τj p Φε k k,τ1...τ l ; λ k+n l We now estimate the quasi-norms in the right hand side of CLAIM. There exists a constant c = cn, p > 0 such that p 1 k,τ1...τ l Φ ε k,τ1...τ l ; λ k+n l+1 c r p k,τ1...τ l Suppose for the moment that 5.33 has been proved. We show how to conclude the proof of Proposition 5.5. Plugging 5.33 into 5.32, then using 5.28, we find positive constants C, c, and c, depending only on n and p, such that Φ ε k ; λ k+n+1 Cc Letting k in 5.34, we obtain Proof of From 5.27, we have k,τ1...τ l m l k r p 1 k p k c r p 1 k p k p 1 + δ > k,τ1...τ l = k,τ1...τ l 1 /m + p τl p > p 1/m + p 1 p = p We further distinguish two cases: CASE A: Let k,τ1...τ l p 1 δ. In view of 5.35, we obtain that p 1 p 1 k,τ1...τ l 1 k,τ1...τ l c0, where c 0 := p 1 δ 1 p From 5.18 and k + N l 0, we infer that meas A r λ k+n l+1 meas A r 3/4. Thus, using 3.11 with q 0 = k,i1...i l, q = p 1, and = A r λ k+n l+1, we find that p 1 k,τ1...τ l p 1 Φ ε k,τ1...τ l ; λ k+n l+1 c 0 meas A r 3/4 k,τ1...τ l Φ ε L p 1, A rλ k+n l+1. This, together with 3.12, proves the claim of 5.33 in Case A. CASE B: Let p 1 δ < k,τ1...τ l < p 1 + δ. We choose θ 0, 1 with the following property: 1 = 1 θ k,τ1...τ l p 1 δ + θ p 1 + δ.