Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

We consider linear elliptic systems whose prototype is 0.1divΛexp(-|x|)-log|x|IDu=divF+ginB.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} div \, \Lambda \left[ \,\exp (-|x|) - \log |x|\,\right] I \, Du = div \, F + g \text { in}\, B. \end{aligned}$$\end{document}Here B denotes the unit ball of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document}, for n>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n > 2$$\end{document}, centered in the origin, I is the identity matrix, F is a matrix in W1,2(B,Rn×n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1, 2}(B, \mathbb {R}^{n \times n})$$\end{document}, g is a vector in L2(B,Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(B, \mathbb {R}^n)$$\end{document} and Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} is a positive constant. Our result reads that the gradient of the solution u∈W01,2(B,Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \in W_0^{1, 2}(B, \mathbb {R}^n)$$\end{document} to Dirichlet problem for system (0.1) is weakly differentiable provided the constant Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} is not large enough.


Introduction
We consider the linear elliptic system div A(x)Du(x) = div F (x) + g (1.1) in a bounded domain Ω ⊂ R n , n > 2, with A(x) = (A ij (x)) symmetric, positive definite matrix with measurable coefficients, F given matrix field in W 1,2 (Ω, R n×n ) and g given vector field in L 2 (Ω, R n ). A vector field u in the Sobolev space W 1,2 0 (Ω, R n ) is a weak solution of the Dirichlet problem: with K(x) ∈ L n,∞ (Ω), and h ∈ R such that x+he i ∈ Ω. Thanks to a characterization of the Sobolev functions due to Hajlasz [20], the function K above plays the role of the derivative D x A. In fact this condition describes a weak form of continuity since the function K may blow up at some points.
In the account of the typical functions of BM O and L n,∞ respectively, it's obvious that the matrix A(x) = Λ (e −|x| − log |x|) I, (1.4) with x ∈ B(0, 1) = {x ∈ R n : 0 < |x| < 1}, satisfies the assumptions above. Here Λ is a positive constant and I denotes the identity matrix. Note that the hypothesis A(x) in BM O guarantees that the problem (1.1) admits a unique solution (see Theorem 1.1 of [28]).
The regularity results for linear systems with continuous coefficients can be considered classical. The first remarkable contribution is due to Agmon, Douglis and Nirenberg (see [2] and [3]). Later regularity results of Schauder type in the class of Hölderian functions are proved by Campanato [8] and Morrey [26]. See also [9]. A full discussion can be found in [15] and [16].
The aim of this paper is to study the second order regularity of the solution of (1.1). More precisely, we prove the following: Theorem 1.1. Let Ω be a regular Lipschitz domain. There exists ε 0 > 0, depending on n, such that, if (1.5) then u ∈ W 2,2 loc (Ω, R n ) and For the definition of regular domain, see the Sect. 2 below. Anyway, balls and cubes of R n are regular domains.
The condition (1.5) on the distance D K of K(x) to L ∞ is clearly satisfied if the derivatives of A(x) belong to any subspace of L n,∞ in which L ∞ is dense, and then, in particular, if they belong to L n,q with 1 < q < ∞, since their distances to L ∞ are null. On the contrary, L ∞ is not dense in L p,∞ for any p > 1. We point out that the condition (1.5) does not imply the smallness of the norm of K(x) in L n,∞ . In fact, if A(x) is the matrix in (1.4), an elementary calculation shows that Vol. 89 (2021) Second Order Regularity for a Linear Elliptic 415 it reduces to consider the constant Λ < ε 0 ω −1/n n , where ω n denotes the measure of unit ball in R n . A value of ε 0 is given in (3.13). It follows that assuming (1.5) is more general than considering a condition on the norm and allows us to present different settings of our result in a unified way. We explicitly remark that, thanks to the embedding theorem 2.2, our result applies if the entries of A(x) lie in W 1,n . The boundedness of the coefficients in a system of the type (1.1) is sometimes too restrictive in applications, as for example in phisical process of diffusion or in mathematical finance.
The novelty of Theorem 1.1 is to consider systems with BM O coefficients, which feature is that they are allowed to be very irregular. In this case the energy functional Ω A(x) Du, Du dx could not be bounded, then a priori we cannot use test functions in (1.3) proportional to the solution u. The Hodge decomposition and a generalization of Coifman, Rochberg and Weiss commutator result [12] allow us to establish an a priori estimate. Then the result follows by considering regularized approximating problems. If the boundary of Ω is more regular, a global version of Theorem 1.1 is also available (Proposition 4.1).
The study of the second order regularity of solutions to linear equations with discontinuous coefficients goes back to C. Miranda [25], who considered the case of coefficients in W 1,n . Then a significant improvement has been given in [5] and in [10,11]. Linear equations having coefficients in BM O with small norm have been addressed in [19]. More recently, a condition similar to (1.5) has been considered in [16] to study the L p -regularity of a linear Dirichlet problem. In connection with the regularity of minimizers of functionals of the Calculus of Variations [1], the study of the regularity theory for systems had a remarkable development in last years. Recently in [13] linear systems with coefficients having in some directions locally small mean oscillation have been studied. We refer to [23,24] and references therein for an almost complete recent treatment.

Preliminaries
This section is devoted to notation and preliminary results useful for our aims.

BMO Spaces
Definition 2.1 ([7,22]). Let Ω be a cube or the entire space R n . The BM O(Ω) space consists of all functions b which are integrable on every cube Q ⊂ Ω with sides parallel to those of Ω and satisfy: We also recall the following inclusion Theorem 2.2 ([7]). For any cube Q ⊂ R n the following inclusion holds with continuous embedding:

Hodge Decomposition
We shall now discuss briefly the Hodge decomposition of vector fields; for a more complete treatment see [21]. For a given vector field F = (f 1 , . . . , f n ) ∈ L p (R n , R n ), 1 < p < ∞, the Poisson equation Δu = divF can be solved by using the Riesz transforms in R n , R = (R 1 , . . . , R n ), Here the tensor product operator Notice that the range of the operator consists of the divergence free vector fields. We then arrive at the familiar Hodge decomposition of F Hence, L p -estimates for Riesz transform yield an uniform estimate Let Ω ⊂ R n be a domain and G = G(x, y) the Green's function. For h ∈ C ∞ 0 (Ω) the integral

then integration by parts yields
Hence the gradient of u is expressed by a singular integral The continuity of K Ω : L 2 (Ω, R n ) → L 2 (Ω, R n ) is easily established by interpreting K Ω F as the orthogonal projection of F into gradient fields. Let D p (Ω, R n ) denote the closure of the range of the gradient operator ∇ : If Ω is smooth, then K Ω extends continuously to all L p (Ω, R n ) spaces. Consequently the formula ∇u = K Ω F extends to all F ∈ L p (Ω, R n ) giving a solution with ∇u ∈ D p (Ω, R n ), 1 < p < ∞.

Definition 2.3 ([21]
). A domain Ω ⊂ R n will be called regular if the operator K Ω acts boundedly in all L p (Ω, R n )-spaces, for 1 < p < ∞.
For Ω a regular domain we introduce, as before, the operator Obviously, the range of H Ω consists of the divergence free vector fields on Ω. We have the Hodge decomposition of F ∈ L p (Ω, R n ), (2.1) We also have the uniform estimate We now turn to commutators. We need the following definition.

Definition 2.4 ([28]). Let k(x)
: R n → R n . We will call k a Calderon-Zygmund kernel (CZ kernel) if k satisfies the following properties Given such a kernel, one can define a bounded operator in L p , called Calderon-Zygmund singular operator, as follows Let ϕ ∈ BM O(R n ) and k a CZ kernel. Following [12], we define, for f ∈ L p (R n ) (1 < p < ∞), the commutator of ϕ and k as the principal value

Theorem 2.5 ([12]). Under the previous assumptions on ϕ and
We will state a generalization of Theorem 2.5 in finite -dimensional normed spaces. Let E be a finite-dimensional normed space of dimension m, a Calderon -Zygmund integral operator. Fixed a basis in E, we can associate to the operator T a m × m matrix of Calderon-Zygmund operators Given A ∈ BM O(R n , Aut(E)), where Aut(E) denotes the space of linear maps from E to E, and is the BM O norm, we consider the range of A: Let us state the theorem: for almost every z ∈ R n , then Since we would apply Theorem 2.6 to the projection T onto divergence free matrix fields, we need to identify the range of the compatibility condition (2.2). Take now A 0 a n × n-matrix, A 0 induces a linear transformation given by A 0 (X) = A 0 X, the row by column product of matrices. It is easy to verify that A 0 commutes with the operator T ; in fact, multiplying the decomposition of F by A 0 we get in the sense of distributions: where A 0t denotes the transpose of A 0 . It can be shown a local version of Theorem 2.5 in the case of Hodge decomposition of matrix fields. Therefore, given any

Lorentz Spaces
Let Ω be a bounded domain in R n . Given 1 < p, q < ∞, the Lorentz space L p,q (Ω) consists of all measurable functions g defined on Ω for which the quantity Note that || · || L p,q is equivalent to a norm and L p,q becomes a Banach space when endowed with it. For p = q, the Lorentz space L p,p reduces to the standard Lebesgue space L p . For q = ∞, the class L p,∞ consists of all measurable functions g defined on Ω such that and it coincides with the Marcinkiewicz class, weak-L p . For Lorentz spaces the following inclusions hold Fundamental to us will be the Sobolev embedding theorem in Lorentz spaces (see [4]).
Here p * = np n−p and S n, To find a formula for the distance, we consider the truncation operator. For k > 0 and y ∈ R, we set T k (y) = min{k, max{−k, y}}.
Let Ω be the unit ball of R n . The function and it does not depend on k. For more details, see [15]. We recall the following relevant properties.

Difference Quotients
Definition 2.11 ([18]). Let f (x) be a function defined in an open set Ω ⊂ R n , and let h be a real number. We call the difference quotient of f with respect to x s the function where e s denotes the direction of the x s axis and τ s,h is the finite difference operator.
When no confusion can arise, we shall omit the index s, and we shall write and hence in the set The following properties of the difference quotients are immediate: • If at least one of the functions f or g has support contained in Ω |h| , then In other words, we have Lip(Ω) = W 1,∞ (Ω).
For R > 0 and x 0 ∈ R n , we define but in the case no ambiguity arises, we shall use the short notation B R .
Finally we recall the following useful lemma: where A, B and α denote non-negative constants and ϑ ∈ (0, 1). Then we have Proof. If u ∈ W 1,2 0 (Ω) is the solution of (1.2), then for every ε > 0 we have that A(x)Du ∈ L 2−ε (Ω, R n×n ) since A belongs to L p for every 1 < p < ∞. We decompose as in (2.1)

A Priori Estimate
with DΨ ∈ L 2−ε (Ω, R n×n ) and H ∈ L 2−ε (Ω, R n×n ) divergence free vector field. Since u solves problem (1.2), we get then, by the classical theory, DΨ ∈ L 2 (Ω, R n×n ) and Let us examine the other term of the Hodge decomposition: H in (3.1) is a commutator with BM O-matrix of a gradient field in L 2 ; using Lemma 2.7 we conclude that H ∈ L 2 and where c = c(n). Finally from (3.1) we deduce that A(x)Du belongs to L 2 (Ω, R n×n ) and we get Now for a fixed ball B 2R ⊂⊂ Ω and radii R < s < t < 2R with R small enough, consider a function ξ ∈ C ∞ 0 (B t ), 0 ≤ ξ ≤ 1, ξ = 1 on B s , |∇ξ| ≤ 1 t−s and set ψ = ξ 2 τ h u for sufficiently small h. Since u is a weak solution of (1.1), we are able to use ϕ = τ −h ψ as test function in (1.3). Then and by virtue of the properties of difference quotients We remark that (3.3) Now let K 0 ∈ L ∞ (Ω). The use of Hölder's inequality in Lorentz spaces (Theorem 2.10), together with Young's inequality with a constant ν ∈ (0, 1) that will be chosen later, yields Finally by Theorem 2.8 Next we estimate I 2 .
Now we estimate I 3 . Again by Hölder's and Young's inequalities, we get Finally we estimate I 4 : and I 5 : (3.9) Next we divide by |h| 2 in (3.9) and, by Lemma 2.13, as h → 0 + , we get 4S 2,n . Let ε 0 be a number such that 0 < ε 0 < η. If then we can choose K 0 ∈ L ∞ (Ω) such that 1 − 5 ,n ε 2 0 > 0. Then, by reabsorbing the first term of the right hand side of (3.10) in the left hand side, since ξ = 1 on B s and 0 ≤ ξ ≤ 1, we get (3.12) Then by Lemma 2.14 |g(x)| 2 dx , (3.13) where c = c(n, D K ), and therefore we have the result.
Remark 3.2. The bound in (3.11) could be not optimal. Anyway it is comparable with analogous bounds in [14].

Regularity
Given a symmetric matrix-valued function A(x) ∈ BM O(Ω, R n×n ), Ω Lipschitz domain, we assume that for every matrix Y ∈ R n×n . We consider the following system with Dirichlet boundary conditions : where u : Ω → R n is a W 1,2 0 vector-valued function, F is a field in W 1,2 (Ω, R n×n ) and g is a field in L 2 (Ω, R n ). Notice that we do not require that A belongs to L ∞ , the system have to be understood in the weak sense we know that A ∈ L 2 but A doesn't need to be bounded; however we know, by Theorem 1.1 in [28], that the Dirichlet problem (4.1) admits a unique solution u in W 1,2 0 . Now we are in position to prove Theorem 1.1.
Proof of Theorem 1.1. We first extend the matrix A to R n , putting zero outside of Ω. Then we take ∈ C ∞ 0 (R n ) such that supp ⊂ B 1 (0), ≥ 0, ≡ 0 and ∞, and we consider the convolution A N = A N , with N = N n (Nx) and defined by: We notice that We find solutions u N ∈ W 1,2 0 (Ω, R n ) of the Dirichlet problems : div(A N Du N ) = div F + g in Ω u N = 0 on ∂Ω, (4.2) that converge weakly in W 1,2 0 , and strongly in L 2 , to u (see [28]) and it is well known that u N ∈ W 2,2 loc (Ω, R n ) (see [18]). Let ε 0 > 0 be the number fixed in (3.11) and let us assume We notice that, from Lemma 2.9, we have Since K 0 ∈ L p (Ω) for every p ≥ n, thanks to Theorem 2.10 the second term in the right hand side of the previous inequality goes to 0 as N → +∞. Then we can assume that for N sufficiently large. Now, arguing as in Theorem 3.1, we get the following relation where the constant C is the same of (3.12). Applying Lemma 2.14 we deduce with c = c(n, D K , ||A|| * ). From the previous relation, we deduce that |D u N | is a bounded sequence in W 1,2 (B R ). Then, by compactness, up to a sequence not relabeled, we deduce that |D u N | converges to |D u| in L 2 (B R ). Finally, by the semicontinuity of the norm, we get Now we prove a global version of Theorem 1.1.

Proposition 4.1.
Let Ω be a regular domain with C 2 boundary. There exists ε 1 > 0, depending on n and Ω, such that, if then u ∈ W 2,2 (Ω, R n ) and for a constant c, depending on n, D K , Ω and the BM O-norm of A.
We consider the solutions u N of the problems (4.2). CoveringΩ by a finite number of balls, we have that Now we focus on the boundary regularity of the solutions. Fixed l ∈ {1, . . . , m}, on every U l we can consider the diffeomorphism Φ(x) = (Φ 1 (x), . . . , Φ n (x)) which maps Ω l ≡ U l ∩ Ω to an open set of R n and defined by where ψ l : R n−1 → R is C 2 and whose graph coincides with ∂Ω in U l . Φ(x) =: y is such that Φ(U l ∩ Ω) ⊂ {y ∈ R n : y n > 0}, Φ(U l ∩ ∂Ω) ⊂ {y ∈ R n : y n = 0}.
It can be seen that Φ is invertible, both Φ and Φ −1 are C 2 functions. Letũ N be such that u N (x) = (ũ N •Φ)(x), x ∈ U l ∩Ω, and we check thatũ N solves inΩ l = Φ(U l ∩Ω) the system divÃ N (x)Dũ N = divF +g inΩ l u N = 0 on {y n = 0} ∩ ∂Ω l , These formulas can be derived starting from the weak formulation of the problem and applying a change of variables in order to express the different integrals in terms of the new coordinates. For instance, for ϕ ∈ C ∞ 0 (Ω l ), We have to prove that the conditions on A still hold true forÃ. About the uniform ellipticity, we have that Hence,Ã N satisfies the uniform ellipticity condition; moreover BM O and L n,∞ are preserved under C 2 transformations. Finally, we have |Ã N (y + h e i ) −Ã N (y)| ≤K N (y)|h|, y ∈Ω l , with DK N = D K N . Therefore, if D ũ N indicates any derivatives D sũN with s = n, we have that D ũ N ∈ W 1,2 loc (Ω l , R n ) and where B + R = {y = (y 1 , . . . , y n ) ∈ R n : ||y|| < R, y n > 0} and B + 2R ⊂⊂Ω l . In order to have the estimate for the derivatives D 2 nnũ N , the equation readily implies that D n (Ã i n N D nũ j N ) ∈ L 2 (Ω l ) for i, j ∈ {1, . . . , n}; by Lemma 2.13 the difference quotients Δ h (Ã i n D nũ j N ) have uniformly bounded L 2 norm inΩ l |h| and the same is true forÃ i n Δ h D nũ j N . The uniform ellipticity condition gives that D 2 nnũ N ∈ L 2 (Ω l ) and the analogous estimate for D 2 nnũ N . Taking the covering introduced above with U l = Φ −1 (B + 2R ) and V l = Φ −1 (B + R ) and coming back to the original variables, we get that Arguing as in the proof of the Theorem 1.1, we get the desired result.