On the existence of the Green function for elliptic systems in divergence form

We study the existence of the Green function for an elliptic system in divergence form $-\nabla\cdot a\nabla$ in $\mathbb{R}^d$, with $d>2$. The tensor field $a=a(x)$ is only assumed to be bounded and $\lambda$-coercive. For almost every point $y \in \mathbb{R}^d$, the existence of a Green's function $G(a; \cdot, y)$ centered in $y$ has been proven in [J. Conlon, A. Giunti and F.Otto,"Green's function for elliptic systems: Delmotte-Deuschel bounds", 2017]. In this paper, we show that the set of points $y \in \mathbb{R}^d$ for which $G(a; \cdot, y)$ does not exist has zero $p$-capacity, for an exponent $p>2$ depending only on the dimension $d$ and the ellipticity ratio of $a$.

This paper is an extension of [2] and further investigates the existence of a Green's function for the second-order elliptic operator −∇ · a∇ in R d , with d > 2. We focus on the case of systems, namely when a is a measurable tensor field a : R d → L(R m×d ; R m×d ), with m being any positive integer. We stress that in this paper we do only assume that a is bounded and λ-coercive, i.e. that there exists λ > 0 such that (0.1) In [2], J. Conlon and the authors show that a Green's function G(a; ·, y) centered in y, exists for every coefficient field a satisfying (0.1) and for (Lebesgue-)almost every point y ∈ R d . In this paper, we improve this result by showing that the exceptional set Σ of points y ∈ R d for which G(a; ·, y) may not exist has p-capacity zero, for an exponent p > 2 depending only on the dimension d and the ellipticity ratio λ. This, in particular, implies that for every coefficient field a that is λ-coercive and bounded, the Hausdorff dimension of Σ is strictly smaller than d − 2 [4][ Theorem 4.17].
The result of [2] crucially relies on the idea of studying the Green function as a map G(a; ·, ·) in both variables x, y ∈ R d . This yields optimal estimates for the L 2 -norm in y and x of G(a; ·, ·), ∇ x G(a; ·, ·) and ∇ x ∇ y G(a; ·, ·) both away from the diagonal {x = y} and close to it. By the standard properties of Lebesgue-integrable functions, these estimates allow to give a pointwise meaning in y to G(a; ·, y), up to a set of Lebesgue-measure zero. The main idea behind the result of this paper is to exploit the integrability of the mixed derivatives ∇ y ∇ x G(a; ·, ·) and extend the set of Lebesgue points y where G(a; ·, y) is well-defined up to the set Σ having zero p-capacity.
We remark that in the case of elliptic systems the set Σ is expected to be non-trivial. There are, indeed, coefficient fields a satisfying (0.1) for which one may construct unbounded a-harmonic vector fields. From this, and by means of representation formulas, it follows that the points where such vector fields are unbounded cannot be Lebesgue points for G(a; ·, y). A classical example of a discontinuous a-harmonic vector field is due to E. De Giorgi [3]: For any dimension d > 2, the vector field u : solves −∇ · a 0 ∇u = 0 in R d , with a 0 satisfying (0.1) and being smooth everywhere outside of the origin. We remark that the coefficient a 0 is not only λ-coercive as in (0.1), but also strongly elliptic: For almost every x ∈ R d and every matrix ξ ∈ R d×d , it satisfies ξ · a 0 (x)ξ λ|ξ| 2 , with λ depending on d.
In the case d = 3, the previous example implies that the exceptional set Σ for a 0 contains at least the origin. For higher dimensions d 3, the trivial extension of the vector field u for d = 3 is itselfā 0 -harmonic ifā This implies, in particular, that Σ forā 0 has Hausdorff dimension at least d − 3.
The previous counterexample also implies that for (locally) a-harmonic vector fields one may only aim at statements on their partial regularity as, for instance, their continuity outside of a singular set. We remark that there exist examples of discontinuous a-harmonic vector fields with discontinuity much larger than (0.2): We refer, for instance, to the paper by J. Soucek [10], which exhibits an a-harmonic vector field discontinuous on a dense countable set, and the one by O. John, J. Malý and J. Stará [6], in which, for every countable union of closed sets, an a-harmonic vector field discontinuous there is constructed. Without using the equation, the fact that a-harmonic functions are locally in H 1 immediately implies that they are 2-quasicontinuous. This means that there exists a set, having 2-capacity which can be chosen arbitrarily small, outside of which the function considered is continuous [4][ Definition 4.11]. This argument is oblivious to the difference between scalar and vectorial functions. By using the equation and appealing to Meyers's [9] or Gehring's [5] estimates, this notion of continuity may be upgraded from 2-quasicontinuity to p-quasicontinuity, for an exponent p > 2. The result of this paper provides an analogous statement for the solution operator for −∇ · a∇. By means of representation formulas, indeed, we prove that for any family F of locally a-harmonic functions that are uniformly bounded in the H 1 loc -norm, there exists a universal set of zero p-capacity outside of which F is equicontinuous (see Corollary 1). This set is universal in the sense that it depends only on the coefficient a and on the dimension d, but not on the family F . Notation and previous results. For the sake of simplicity, throughout the paper we use a scalar notation by pretending that a is a matrix field and that the Green function is scalar. For a detailed discussion about this abuse of notation, we refer to [2, Section 2]. Moreover, again for notational convenience, as in [2] we assume that a is symmetric, i.e. that for almost every x ∈ R d , the the tensor a(x) is symmetric. If we denote the elements of the product space R d × R d by (x, y), we use the notation W 1,q x (R d ) to specify in the lower index the differentiation and integration variable. Similarly, we write ∇ x , ∇ y or ∇ x,y when the gradient is taken with respect to x, y or both variables (x, y), respectively. We denote by W 1,p (R d , R m ), p 1 the Sobolev spaces of functions in R d taking values in R m ; if m = 1, we use the usual notation For an open set D ⊆ R d , we may define the space such that for all 1 q < d d−1 and r > 0 G(a; ·, ·) ∈ W 1,q and for almost every y ∈ R d it holds (in the weak sense) Furthermore, the matrix-field G(a; ·, ·) is unique in the class of fields F : R d × R d → R m×m solving (0.5) for almost every y ∈ R d and satisfying for some 1. Main result Theorem 1. Let a be symmetric and satisfy assumptions (0.1). Let G : R d × R d → R m×m be the Green function in the sense of [2] and constructed there. Then, there exists a (measurable) set Σ = Σ(a) ⊆ R d with the following properties: (a) There exists an exponent p = p(d, λ) > 2 such that In addition, as a corollary we have: Let the coefficient field a and the exponent p > 2 be as in Theorem 1. Let Then F is uniformly p-quasicontinuous in {|x| < 1}. More precisely, for every ε > 0 there exists an open set U ε ⊆ {|x| < 1} having

Proofs
Throughout this whole section we fix the coefficient field a and drop the argument a in the notation for G, ∇ x,y G and ∇ x ∇ y G. We write and for C and C with the constant depending only on the dimension d, the ellipticity ratio λ and the dimension of the target space m. Finally, for a function f ∈ L 1 loc (R d ), we introduce the notation Before giving the proof of Theorem 1, we recall some of the main properties of G(·, ·) obtained in [2, Section 2] which will be crucially used in our proofs: • For every R > 0, z ∈ R d and almost every y ∈ R d such that |y − z| > 2R, (2.14) may be written as the identity (up to a set of Lebesgue measure zero) Proof of Theorem 1. We divide the proof into steps: In Step 1 we give a formulation of the standard Gehring's estimate tailored for our needs. Roughly speaking, this allows to upgrade estimate (2.12) into an L 2 -estimate in x and L p in y, for the Gehring exponent p > 2. Steps 2-4 contain the main capacitary estimates for the exceptional set Σ, which is closely related to the set of points y ∈ R d where G(·, y) and ∇G(·, y) have infinite W 1,q loc -and Y 1,2 ({|x − y| > 1})norms. These estimates on the capacity of Σ crucially rely on the upgraded version of (2.12) and are combined with a maximal function estimate for Sobolev functions. Finally, in Step 5 we argue how to construct the representative G * (·, y) away from the singularity set Σ.
This is a standard result in elliptic regularity theory and we refer to [5, Chapter V, Theorem 2.1] for its proof. 1 We pick a (smooth) cut-off function η for {|x| < R} in {|x| < 2R}. Since for any p 1 In [5] the coefficients are assumed to be very strongly elliptic. However, the argument only relies on Caccioppoli's and Poincaré-Sobolev's inequality which hold true also if a is assumed to be only λ-coercive as in (0.1). Moreover, [5][Inequality (0.2)] corresponds to the standard case of a-harmonic functions; our case is an immediate adaptation of the Caccioppoli's inequality in the case of solutions to (2.16).
it follows that Step 2. Capacity estimates: First reduction. Let M 0 be as in definition (2.9). We claim that if there exists an exponent α(d, λ) < d 2 such that for every R > 0 and we may also find an exponent q = q(d, λ) > 1 such that for every R > 0 , estimate (2.22) is implied by this inequality together with both identities (2.24).
Step 3. Capacity estimates: Second reduction. We now further argue that for (2.19)-(2.20) it suffices to prove that for every r > 0 and all λ > 0 By redefining r α λ as λ and reducing the domain of integration from {|x − y| > 8r} to {8r < |x − y| < 16r}, we further obtain then the inclusion of the sequence spaces ℓ 1 ⊆ ℓ 2 , the sublinearity of the operator M 0 and assumption (2.31) yield We thus established (2.33).
By (2.33) and the subadditivity of the capacity we get and, recalling definition (2.32), we use estimate (2.29) with λ and r substituted by ω n λ and 2 −n+3 to bound By choosing in (2.31) ω n = 6 (πn) 2 , the sum on the right-hand side converges provided that the exponent α satisfies α > 3  Step 4. Maximal function estimate. We now prove (2.25) and begin with the first estimate. Without loss of generality, we focus on the case z = 0. For any 0 < r 1, let (2.34) We first claim that it suffices to show that for every r > 0 Indeed, if η r is a smooth cut-off function for {|y| < r 2 } in {|y| < r}, then by (2.35) the function η r F r satisfies We thus apply the maximal function estimate[7, Inequality (3.1)] to η r F r and infer that for every λ > 0 where M is defined in (2.8). Since by the assumption on η r and definitions (2.8)-(2.9) we have To complete the argument for the first line in (2.25) it remains to prove (2.35): The main ingredient for this are inequalities (2.11) and (2.12) which, by setting R = 2r and z = 0, we rewrite asˆ| Since by (2.14) and (2.13) the vector field ∇ x G(x, ·) is a-harmonic in {|y| < 2r} for almost every x such that |x| > 4r, we apply (2.18) of Step 1 and upgrade the previous estimate tô with p > 2 from Step 1.
If we differentiate the right-hand side of (2.34) in y, the chain rule and an application of Cauchy-Schwarz's inequality yield This concludes the proof of (2.35) and of (2.25) for ∇G. This concludes the proof of Step 4.
Step 5. Construction of G * (a; ·, ·). By wrapping up Steps 2-4, we have that G and ∇G satisfy (2.25) and therefore also (2.19)-(2.20) and (2.22)-(2.23). Equipped with these identities, we now proceed to prove the existence of G * (·, y) for y outside an exceptional set Σ satisfying (a) in the statement of Theorem 1. For a test function ζ ∈ C ∞ 0 (R d ; R m ), we consider the vector field u(y) =ˆG(x, y)ζ(x) dx.
Both g and f are supported in {2 < |x| < 4} and by the definition of η, the second inequality in (0.1), the bound on the Dirichlet energy of u and Poincaré's inequality satisfŷ |g(x)| 2 dx +ˆ|f (x)| 2 dx 1.
To make our notation leaner, we define v(y) :=ˆg(x) · ∇ x G * (x, y) dx, w(y) :=ˆf (x)G * (x, y) dx. (2.52) and prove the statement of the corollary for v. The vector field w may be treated analogously.
It thus remains to pick δ such that the last term on the right-hand-side is smaller than κ 3 . This concludes the statement of the corollary.