L p regularity theory for linear elliptic systems

We consider the conormal derivative problem for an elliptic system 
in divergence form with discontinuous 
 coefficients in a more general geometric setting. 
 We obtain the $L^{p}$, $1 < p <\infty$, regularity of the maximum order derivatives 
of the weak solutions for such a problem.

1. Introduction. We consider the following conormal derivative problem for a linear elliptic system in divergence form: for i = 1, ..., m, where Ω is an open, bounded subset of R n , and ν = (ν 1 , ..., ν n ) is the outward pointing unit normal to the boundary ∂Ω of Ω. f = {f i α } ∈ L p (Ω; M m×n ) is a given m × n matrix for some 1 < p < ∞, and, as usual, repeated indices mean summation; α, β are summed from 1 to n and j from 1 to m; D α = ∂ ∂xα (α = 1, ..., n). We remark that the conormal derivative boundary condition is not well-defined in the classical sense on the boundary considered in this paper. Instead we define this conormal derivative boundary condition with its weak formulation (see Definition 1 and 5).
We henceforth assume that the coefficients A αβ ij (x) satisfy the uniform ellipticity condition; i.e., there exists constants λ, Λ > 0 such that for a.e. x ∈ R n and all ξ ∈ M m×n . In this work we want to ask what is a minimal condition on the coefficients A αβ ij and what is a more general geometric condition on the boundary ∂Ω under which we have the global W 1,p regularity theory. If one wants to use the classical approach, which uses the representation formulas in terms of singular operators and commutators, one needs to overcome the main obstacle coming from a reflection in the flattening argument. To the scalar-valued function the flattening argument works well provided the boundary of the domain is locally Lipschitz continuous (see [2,6]), while it does not when the domains are beyond Lipschitz domains. In recent paper [3] the authors were able to establish the global W 1,p regularity theory for linear elliptic equations in divergence form with the conormal boundary condition in the more general setting. Here we generalize the techniques used there to the systems of such equations. The main difficulty in our work comes from the systems of the equation (1). There have been many research activities concerning regularity theory for the solutions of elliptic systems (see [1,4,5,7,8,10,14,15,17]). But to the best of our knowledge, no rigorous results concerning boundary L p estimates for every 1 < p < ∞ beyond graph domains are available in this literature, and so our approach could be a new approach to the case of systems in this direction. Our approach is more flexible in dealing with the irregular domains than the classical approach since the latter always requires some differentiability conditions on the boundaries of the domains. Although we use similar functional framework and analytic procedure as in [3], more complicated analysis has to be carefully carried out with patience. We will use weak compactness method, the classical Hardy-Littlewood maximal function, the Vitali covering lemma, and standard arguments of measure theory (see [3]).
Solutions of (1) are taken in the weak sense. We use the following classical definition of a weak solution.
By the direct method (Lax-Milgram Lemma or the variational method), it is easy to prove the existence and uniqueness (up to a constant) of the 2-weak solution for the problem (1) in Sobolev space H 1 (Ω; R m ) (see [5,8]). In this paper, we will show that the gradient of the weak solutions for (1) preserves the same higher integrability as the matrix field f = {f i α } ∈ L p (Ω; M m×n ) for each 1 < p < ∞ under an appropriate discontinuity condition on the coefficients and a certain geometric condition on the domain which is beyond graph domains.
The assumption on the coefficients A αβ ij is formulated as follows.
Definition 2. We say that the coefficients A αβ ij are (δ, R)-vanishing if we have is the average of A αβ ij over B r (x). Our geometric setting for the domain is stated as the following.
Definition 3. We say that Ω is (δ, R)-Reifenberg flat if for every x ∈ ∂Ω and every r ∈ (0, R], there exists a coordinate system {y 1 , . . . , y n }, which can depend on r and x such that x = 0 in this coordinate system and that In the definitions above we mean δ to be a small positive constant, while one can assume R = 1 by a scaling. For the notion of δ-flatness we refer to the papers [9,11,12,13,18,16]. In this work we only consider the case p > 2. The case p = 2 is classical (see [5,8]), and a duality argument recovers the case 1 < p < 2 (see Theorem 2 below). We will hereafter focus our attention exclusively on the case that p > 2.
Let us state the main results of this paper. where C is a constant independent from u and f .
As a consequence of Theorem 1 we have the following.
Theorem 2. Suppose 1 < p < 2. Then there exists a small δ = δ(λ, Λ, p, n) > 0 such that for each (δ, R)-Reifenberg flat domain Ω, for each (δ, R)-vanishing A αβ ij and for each f ∈ L p (Ω; M m×n ), there is a unique (up to a constant) p-weak solution of (1) with estimate where C is a constant independent from u and f .
Proof. Fix 1 < p < 2. By a duality argument with use of Theorem 2, one can prove the existence of at least one p-weak solution u of (1) satisfying estimate (3). These W 1,p -weak solutions also have the regularity estimates as of H 1 -weak solutions: For each p < r < ∞, if u is a W 1,p -weak solution and f ∈ L r (Ω), then through the same argument as for H 1 -weak solutions, we have that u is a W 1,r -weak solution as well, and with estimate ∇u L r (Ω;M m×n ) ≤ C f L r (Ω;M m×n ) .

Now let us prove uniqueness.
Let v be another p-weak solution; then u−v is a W 1,pweak solution with f = 0. However, with the regularity theory we just outlined above, u − v is a W 1,r -weak solution and, in particular, a H 1 -weak solution which has to be constant. This completes the proof.
The organization of this paper is as follows: In the next section, we will study interior estimates. Section 3 will be devoted to deriving the global W 1,p regularity theory.

2.
Interior estimates in L p spaces. In this section we will obtain interior W 1,p , 2 ≤ p < ∞, estimates for As usual, a 2-weak solution of (4) will be a function u ∈ H 1 (Ω; R m ) such that . As announced in the introduction, the coefficients A αβ ij are assumed to be (δ, R)vanishing. We will use the following version of the Vitali covering lemma in our proof of Theorem 3.

Lemma 1. [19]
Assume that C and D are measurable sets with C ⊂ D ⊂ B 1 and that there exists an ǫ > 0 such that 1. |C| < ǫ|B 1 | and 2. for each x ∈ B 1 and for each r In this section we suppose that R ≥ 1 with B 6R ⊂ Ω. We want to find local estimates of 2-weak solutions of (1) from known regularity results for the following reference equation: Let us start with the following definition of a 2-weak solution.
Definition 4. We say that v ∈ H 1 (B R ; R m ) is a 2-weak solution of (5) if we have the following integral identity . We will use the following regularity for the PDE (5) in the proof of Corollary 1.
We need the following well known standard energy estimates for linear elliptic systems.

Lemma 3. [8]
Assume that u is a 2-weak solution of (4). Then we have Our approach is based on the following approximation lemma.
Lemma 4. For any ǫ > 0, there is a small δ = δ(ǫ) > 0 such that for any 2-weak solution u of (4) with and there exists a 2-weak solution v of (5) in B 4 such that Proof. We first fix any small positive number η. With the same technique used in the proof of Lemma 7, we see that there exists a small δ = δ(η) > 0 such that Now choose a standard cut-off function We can claim that w := u − v is a 2-weak solution of Then, combining (8)- (11) and (2), we estimate Hence we are done by selecting η and δ satisfying C δ + η 2 = ǫ 2 .
Theorem 3. Let u ∈ H 1 (Ω) be a 2-weak solution of (4). Denote by B the ball with 6B ⊂ Ω. Then there is a constant N 1 > 0 such that for any 0 < ǫ, r ≤ 1, there exists a small δ = δ(ǫ) > 0 such that if A αβ ij is (δ, 6)-vanishing and Proof. One can find the proof as an immediate consequence of Lemma 5 from a scaling argument.
Once we have a version of the Vitali covering lemma above, with the same technique used in [3,19] we have the following interior W 1,p estimates.  (4) with Ω ⊃ B 6 is locally a p-weak solution with estimate where the constant C is independent from u and f .

3.
Global estimates in L p spaces. In this section we will study the global W 1,p regularity for the p-weak solutions of a linear elliptic system with the conormal boundary condition.
For each ρ > 0 we write . Based on our definition of δ-Reifenberg flat domains (see Definition 3), and for some technical reasons, we normalize our situation as follows: . Remark 1. We remark that the constant 48 can be any number bigger than or equal to 1 by a scaling argument.
Hereafter let us assume 1 ≤ R ≤ 48 and first state a closely related problem, namely the following conormal problem: We will use the following definition.
We need the following regularity for 2-weak solutions of the conormal problem (12) on the flat boundaries.
is a 2-weak solution of (12), then v is locally Lipschitz continuous; that is, we have for each 0 < r < R ∇v 2 We will use the following compactness method.
Lemma 7. For any ǫ > 0 there exists a sufficiently small δ = δ(ǫ) > 0 such that if u is a 2-weak solution of (1) in a (δ, 5)-Reifenberg flat domain Ω with the following normalization conditions and then there exists a 2-weak solution v of (12) in B + 5 such that Proof. If not, there would exist with (20) But we have In view of Poincaré's inequality, (18) and (20), u k − u k B + 5 is bounded in . Thus there exist a subsequence, which we still denote by {u k − u k }, Noting that A k is uniformly bounded, we see that there exist a subsequence, which we still denote by A k αβ ij , and the constant A 0 αβ ij such that from (18) and (19). We recall Definition 4 and use (17) to (19), (23) and (24) to find upon passing to weak limits that u 0 is a 2-weak solution of the following elliptic system: Then we reach a contradiction to (21) and (22).

Corollary 1.
For any ǫ > 0 there exists a sufficiently small δ = δ(ǫ) > 0 such that if u is a 2-weak solution of (1) in a (δ, 5)-Reifenberg flat domain Ω with the following normalization conditions then there exists a 2-weak solution v of (12) in B + 5 such that where Proof. We first fix any small number η > 0. Applying the lemma above to u and v, with η replacing ǫ, we deduce that there exists a small δ = δ(η) > 0 such that if (26) and (27) hold true, then Now we define v δ (x) = v(x + δe n ) for x ∈ Ω 4 and choose a standard cut-off function We set ϕ := φ 2 (u − v δ ), and so ϕ ∈ H 1 (Ω). Now, since u is a 2-weak solution of (1), it follows from (30) that The term on the left of (31) is Here we used the fact that v is a 2-weak solution of (12). The uniform ellipticity condition implies Since A αβ ij ∈ L ∞ (R n ), we see from Cauchy's inequality with τ that In view of Lemma 6 and 30, we estimate But from Hölder's inequality and (26), we see that Since A αβ ij ∈ L ∞ (R n ), we see from Hölder's inequality that . But since translation is continuous in the L 2 -norm, we have Since A αβ ij ∈ L ∞ (R n ), we observe from Lemma 6 and Hölder's inequality that Using Hölder's inequality and (27), we estimate the term on the right of (31) as Now we combine (31)-(37) to discover Select τ = λ 2 and use (30) to obtain Here we used (29), (26) and Sobolev inequality. Thus this estimate and (38) imply Noting that δ depends only on η, we can first select η satisfying thereby there exists δ = δ(ǫ) > 0 such that the conclusion (28) is true. Hence we finish the proof.
We now come to state the scaling invariant form of Lemma 8.
Let us take N 1 , ǫ, and the corresponding δ > 0 given by the theorem above. We turn now to our primary task of showing the decay estimates on the size of distribution functions of maximal function M(|∇u| 2 ), which is based on the following version of the Vitali covering lemma.
As a consequence we have the following result: