Local boundedness and Harnack inequality for solutions of linear non-uniformly elliptic equations

We study local regularity properties for solutions of linear, non-uniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results by Trudinger [ARMA 1971]. We then apply the deterministic regularity results to the corrector equation in stochastic homogenization and establish sublinearity of the corrector.


Introduction and Main Results
We consider linear, second order, scalar elliptic equations in divergence form, (1.1) r ¡ aru h 0; where a 3 R d¢d is a measurable matrix field on a domain & R d , d ! 2.
In order to measure ellipticity of a, we introduce (1.2) . and suppose that and are measurable nonnegative functions. If 1 and are essentially bounded (i.e., a is uniformly elliptic), the seminal contributions of De Giorgi [14] and Nash [28] ensure that weak solutions of (1.1) are Hölder continuous. Moreover, Moser [25,26] showed that weak solutions of (1.1) satisfy the Harnack inequality, which then implies Hölder continuity. Here, we are interested in situations beyond uniform ellipticity. In [32] Trudinger considered nonuniformly elliptic equations of the type (1.1).
Instead of essential boundedness, he assumed that 1 P L q ./ and P L p ./ with 1 p g 1 q < 2 d and proved that weak solutions to (1.1) are locally bounded and satisfy the Harnack inequality. In this paper, we prove both results under the less restrictive and essentially optimal assumption 1 p g 1 q < 2 d 1 . More precisely, we establish the following: where c h c.d; p; q; .B R // P 1; I/.
Remark 1.2. As mentioned above, the conclusions of Theorem 1.1 are proven in the classical paper of Trudinger [32] under the more restrictive integrability condition 1 P L q ./, P L p ./ with (1.7) p; q P .1; I; 1 p g 1 q < 2 d s see also the paper by Murthy and Stampacchia [27] for related results. To the best of our knowledge, Theorem 1.1 contains the first improvements with respect to global integrability of 1 and compared to the corresponding results in [27,32] (see [13] for a recent generalization of the findings in [27,32] to nonlinear, nonuniformly elliptic equations under assumptions that match (1.7) in the linear case). Assumption (1.3) is essentially sharp in order to establish local boundedness (and thus also the validity of the Harnack inequality) for weak solutions of (1.1). Indeed, in view of a counterexample by Franchi, Serapioni, and Serra Cassano [20], the conclusion of Theorem 1.1 is false if condition (1.3) is replaced by 1 p g 1 q < 2 d 1 g " for any " > 0; see Remark 3.5 below. However, we emphasize that under an additional local assumption (e.g., that ; are in the Muckenhoupt class A 2 ) stronger results are available under weaker global integrability assumptions; see, e.g., [11,17]. As an application of Theorem 1.1 we consider the corrector equation in stochastic homogenization. Currently, homogenization and large-scale regularity for equations with random and degenerate coefficients is an active field of research; see e.g., [1-4, 6, 12, 15, 18, 19, 29]. Recently sublinearity (in L I ) of the corrector in stochastic homogenization was proven in [12] (see also [18]) under certain moment conditions that are comparable to (1.7) (see also [1,15] for related results in the discrete setting). In [1,12,15,18], the L I -sublinearity of the corrector is the key ingredient to prove quenched invariance principles for random walks [1,15] or diffusion [12,18] in a random environment with degenerate and/or unbounded coefficients. In this paper, we establish L I -sublinearity of the corrector under relaxed moment conditions; see Proposition 5.3.
The paper is organized as follows: In Section 2, we present a technical lemma that implies an improved version of the Caccioppoli inequality. This lemma plays a prominent role in the proof of Theorem 1.1 and is the main source for the improvement compared to the previous results in [27,32,33]. In Section 3, we make precise the notion of weak solution and prove part (i) of Theorem 1.1 and local boundedness for weak subsolution of (1.1). Section 3 contains an improvement of part (i) of Theorem 1.1 valid only in two dimensions; see Proposition 3.4. In Section 4, we establish part (ii) of Theorem 1.1 as a consequence of a weak Harnack inequality for nonnegative weak supersolutions of (1.1) and the local boundedness. Moreover, we list in Section 4 several direct consequences of the Harnack inequality. In the final section, Section 5, we apply Theorem 1.1 to the corrector equation of stochastic homogenization and prove L I -sublinearity of the corrector.

An Auxiliary Lemma
In this section, we provide a key estimate, formulated in Lemma 2.1 below, that is central in our proof of Theorem 1. Sending " to 0, we obtain (2.2) with h s 1 > 0.
To estimate the right-hand side of (2.2) we use the Sobolev inequality on spheres, ; which is the desired estimate.
Finally, we suppose d h 2. In this case we have p £ h 1. Instead of (2.4), we use the one-dimensional Sobolev inequality k'k L I .S 1 / ck'k W 1;1 .S 1 / to obtain the estimate (2.1) as above (but now also in the borderline case p h 1).

Local Boundedness Proof of Part (i) of Theorem 1.1
In this section we prove part (i) of Theorem 1.1 as a consequence of a local boundedness result for weak subsolutions of (1.1). Before we state the result, we first define the notion of weak solution to (1.1) that we consider here.   [32,33]. We only recall here the chain rule.
Remark 3.2. Let g R 3 R be uniformly Lipschitz-continuous with g.0/ h 0, and consider the composition F h g.u/. Then, u P H 1 0 .; a/ (or P H 1 .; a/) implies F P H 1 0 .; a/ (or P H 1 .; a/), and it holds that rF h g H .u/ru a.e. (see, e.g., [33, lemma 1.3]). In particular, if u satisfies u P H 1 .; a/ (or P H 1 .; a/), then also the truncations (3.3) u g h maxfu; 0g; u h minfu; 0g; satisfy u g ; u P H 1 .; a/ (or P H 1 .; a/). Now we come to the local boundedness from above for weak subsolutions of (1.1). THEOREM 3.3. Fix d ! 2, a domain & R d , and p; q P .1; I satisfying (1.3). Let a 3 R d¢d be such that and given in (1.2) are nonnegative and satisfy 1 P L q ./, P L p ./. Then every weak subsolution u of (1.1) in is locally bounded from above, and for every > 0 there exists c h c.; d; p; q/ P 1; I/ such that for any ball B R & and P .0; 1/  ; P L 1 ./. Then there exists c P 1; I/ such that for every weak solution u of (1.1) and for any ball Our proof of Theorem 3.3 is similar to that of [32, theorem 3.1] and relies on a modification of the Moser iteration method [25,26]. Let us now briefly highlight the main difference of our approach compared to the arguments given in [32] and discuss where our improvement comes from. A simple consequence of the Hölder and Sobolev inequalities combined with the relation 1 p g 1 q < 2 d is the following weighted Poincaré inequality: there exists h .p; q; d / > 1 (in fact, 1 h p p 1 .1 g 1 q 2 d /) such that for any ball B R and u with compact support in B R where c h c.d / P 1; I/. Inequality (3.6) and the Caccioppoli inequality are enough to use Moser's iteration argument to prove local boundedness. In the situation of Theorem 3.3, i.e., with the relaxed assumption 1 p g 1 q < 2 d 1 , we do not have a weighted Poincaré inequality in the form of (3.6) at hand. However, a version of (3.6) is valid if we replace the d -dimensional balls by .d 1/-dimensional spheres. In order to exploit this observation, we need an additional optimization step compared to the usual Caccioppoli inequality, which is gathered in Lemma 2.1. A similar idea, namely the use of the Sobolev inequality on spheres and a smart choice of certain cutoff functions, was previously used in [10] to obtain a new div-curl lemma and in [31] to show nonexistence of solutions for the Lane-Emden equation. The argument for Proposition 3.4 is different and in fact much simpler. It is mainly based on the maximum principle and Sobolev inequality in one dimension; see [20, prop  Step 2. The general case. It is well-known how to lift the result of Step 1 to prove the claim. For the convenience of the reader, we provide the arguments following the presentation in [22]. First, by scaling we deduce from (3.7) that for ! 2p H and R > 0 where c h c.; d; p; q/ P 1; I/ is the same as in (3.7 Before proving Theorem 4.1, we list some consequences of Theorem 1.1, which are by now standard and thus we only give the statements without proofs. In the uniformly elliptic setting, the Harnack inequality implies Hölder continuity of weak solutions to (1.1). As observed in [32], due to the explicit dependence of the constant c in (1.6) on .B R /, this is in general not true anymore in the nonuniformly elliptic setting. However, Theorem 1.1 yields the following largescale Hölder continuity:   Appealing to (1.2) and Young's inequality, we estimate the right-hand side of (4.9): Note that the first term on the right-hand side in (4.10) can be absorbed into the left-hand side of (4.9) and we obtain, using ! 1 and the definition of w, (4.11) 2 w 1 arw ¡ rw 16 .w g1 g .2/ /jrj 2 : Minimizing the right-hand side of (4.11) among such cutoff functions, we obtain with the help of Lemma 2. Recalling the fact w h v , estimates (4.5) and (4.6) and the fact that p £ 2q qg1 (by (1.3)), we obtain for s ! 1 which proves the claim.
A central object in the homogenization of linear elliptic equations is the so- Here, we assume that the coefficient fields fa ! .x/g xPR d & R d¢d are statistically homogeneous and ergodic, and nonuniformly elliptic (see below for the precise assumptions). In [1,12,15,18], the corrector is used prominently to prove quenched invariance principles for the random walk [1,15] or diffusion [12,18] in a random environment with degenerate and/or unbounded coefficients. The key ingredient in [1,12,15,18] is to upgrade the L 1 -sublinearity into L I -sublinearity, i.e., to show 1 R k k L I .B R / 3 0 as R 3 I. In this section, we show that the results of Section 3 can be used to weaken the assumption of [12,18] in order to establish L I -sublinearity of the corrector. The application to the quenched invariance principle for the random walk in a random degenerate environment can be found in [7]. Let us now be more precise and phrase the assumptions on the coefficient fields by appealing to the language of ergodic, measure-preserving dynamical systems (which is a standard in the theory of stochastic homogenization; see, e.g., the seminal paper [30]): Let .; F; P/ denote a probability space and h . x / xPR d a family of measurable mappings x 3 satisfying (group property) 0 ! h ! for all ! P and xgy h x y for all x; y P R d .
(stationarity) For every x P R d and B P F it holds that P. x B/ h P.B/.
(ergodicity) All B P F with x B h B for all x P R d satisfy P.B/ P f0; 1g. For a random field a 3 R d¢d and ! P , we denote by a ! R d 3 R d¢d its stationary extension given by a ! .x/ h a. x !/. Appealing to an additional truncation argument as e.g., in [5,18]  d (see also [1] for a similar result in the discrete setting and [18] for a corresponding statements with strictly elliptic, unbounded coefficients). In two dimensions, Proposition 5.3 might be not surprising since an analogous statement in a discrete setting was already proven by Biskup in [8]. For completeness and since the argument in [8] uses the discrete structure, we include this case here. The counterexample to local boundedness in [20] suggests that Assumption 5.1 should be almost optimal for the conclusion of Proposition 5.3. In fact, it was recently shown by Biskup and Kumagai [9] in a discrete setting that the corresponding statement of to a positive constant that depends only on d , p, and q. Before we give the details of the proof, we briefly explain the idea. There are two obstructions to deducing the statement directly from Theorem 1.1: First, we are not able to prove local boundedness of the corrector by considering (5.4) as an equation for i with the right-hand side r¡ae i as it is, e.g., done in [12]. Second, and more severe, the righthand side r ¡ae i is not small in general. We overcome these issues by appealing to a two-scale argument: We introduce an additional length scale R with 0 < ( 1 and compare i on balls with radius $ R with a-harmonic functions i ge i ¡x c with a suitable chosen c P R. Using the L 1 -sublinearity of i and the fact that the linear part coming from e i ¡ x can be controlled by > 0 on each ball of radius $ R, we obtain the desired claim. Step 1. As a preliminary step, we recall the needed input from ergodic theory. In view of the spatial ergodic theorem, we obtain from the moment condition  The arbitrariness of > 0 implies (5.7) and finishes the proof.
Step 3. The remaining case: d h 2 and p h q h 1. Let 2 be as in Step 2. From now on we fix ! P 2 and use the same notation as in Step 2.