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 -∇·a∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\nabla \cdot a\nabla $$\end{document} in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d$$\end{document}, with d>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d>2$$\end{document}. The tensor field a=a(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=a(x)$$\end{document} is only assumed to be bounded 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}-coercive. For almost every point y∈Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \in {\mathbb {R}}^d$$\end{document}, the existence of a Green’s function G(·,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(\cdot , y)$$\end{document} centered in y has been proven in Conlon et al. (Calc Var PDEs 56(6), (2017))[2]. In this paper we show that the set of points y∈Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \in {\mathbb {R}}^d$$\end{document} for which G(·,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G( \cdot , y)$$\end{document} does not exist has zero p-capacity, for an exponent p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p >2$$\end{document} 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 of m equations, 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 ∀x ∈ R d , ∀ ξ ∈ R m×d |a(x)ξ | |ξ |. (0.1) In [2], Conlon and the authors show that a Green's function G(·, 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 of points y ∈ R d for which G(a; ·, y) does not exist has p-capacity zero [4,Definition 4.10], 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 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 = G(·, ·) in both variables x, y ∈ R d . This yields optimal estimates for the L 2 -norm in y and x of G, ∇ x G and ∇ x ∇ y G 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(·, 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 and extend the set of Lebesgue points y where G(·, y) is well-defined up to the set having zero p-capacity.
We remark that in the case of elliptic systems the set may indeed 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(·, y). A classical example of a discontinuous a-harmonic vector field is due to 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 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 if This implies, in particular, that 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 Soucek [10], which exhibits an a-harmonic vector field discontinuous on a dense countable set, and the one by John et al. [6], in which, for every countable union of closed sets (i.e. F σ -set), 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 exist sets, of arbitrarily small 2-capacity, outside of which the function considered is continuous [4,Definition 4.11]. This argument is oblivious to the difference between scalar and vectorial functions. Using the equation and appealing to Meyers' [9] or Gehring's [5] estimates, this notion of continuity may be upgraded from 2quasicontinuity 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 exist common sets of arbitrarily small p-capacity outside of which F is equicontinuous (see Corollary 1). These sets are universal in the sense that they depend only on the coefficient a and on the dimension d, but not on the family F.

Notation and previous results
For notational convenience, as in [2] we assume that a is symmetric, i.e. that for almost every x ∈ R d , the tensor a(x) ∈ L(R m×d ; R m×d ) is symmetric. Throughout this paper the expression "almost every" is meant with respect to the Lebesgue measure and all the PDEs considered are assumed to hold only in the distributional sense.
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 W 1, p (R d ). The same criteria are employed for all the other standard functions spaces used in the paper. If we represent 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 write and for C and C with a constant C depending only on the dimension d, the ellipticity ratio λ and the dimension of the target space m.
For an open (not necessarily bounded) set D ⊆ R d , we may define the space The main theorem of [2, Theorem 1] provides the existence of a map and for almost every y ∈ R d , the tensor field G(·, y) ∈ Y 1,2 Here, the notation α means that the constant depends also on the exponent α.
We also remark that in this paper we restrict ourselves to the case d > 2 as in dimension d = 2 the map G does not exist. We refer to [2] for further details and for an existence result for G if R 2 is replaced by any domain D ⊆ R 2 having at least a bounded direction [2, Theorem 1, (b)].

Main result
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 = (a) ⊆ R d with the following properties: In addition, as a corollary we have: Let the coefficient field a and the exponent p > 2 be as in Theorem 1. Consider Then for every ε > 0 there exists an open set U ε ⊆ {|x| < 1} having such that F is equicontinuous in {|x| < 1}\U ε and the modulus of continuity is uniform in {|x| < 1}\U ε .

Proofs
For the sake of simplicity, throughout this section we use a scalar notation and language by pretending that a is a field of d × d matrices and that the Green function is scalar. For a detailed discussion about this abuse of notation, we refer to [2,Section 2]. Moreover, for a function f ∈ L 1 loc (R d ), we introduce the notation Before giving the proof of Theorem 1, we recall some further properties of G(·, ·) obtained in [2] which will be used in our proofs: • [2, Definition (11) and Theorem 1]: For every R > 0, z ∈ R d and almost every (2.12) may be written as the identity (up to a set of Lebesgue-measure zero) Proof of Theorem 1. We divide the proof into five steps: in Step 1 we give a formulation of the standard Gehring's estimate tailored to our needs. It allows to upgrade estimate (0.8) into an L 2 -estimate in x and Lp in y, for the Gehring exponentp > 2.
Steps 2-4 contain the main capacitary estimates for the exceptional set , which is closely related to the set of points y ∈ R d where G(·, y) and ∇ x G(·, y) have infinite W 1,q These estimates on the capacity of crucially rely on the upgraded version of (0.8) 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 .
Step 1. Gehring's estimate Let u ∈ H 1 ({|x| < 2R}) be a solution to (2.14) Then, there exists an exponentp =p(d, λ) > 2 such that 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 we may assumep 2d d−2 , the Poincaré-Sobolev inequality yields Step 2. Capacity estimates: First reduction Recall the definition (2.10) of M 0 which we always think as acting on the y-variable. Let p 1. We claim that if for an exponent 0 then we may also find an exponent q = q(d, λ) > 1 such that for every R > 0 Similarly, if for every R > 0 and z ∈ R d it holds Since we may cover the whole space R d with a countable number of unit balls, the subadditivity of the capacity and (2.19) immediately imply (2.20). Analogously, the subadditivity of the capacity and (2.17) yield that for each R > 0 Since for α < d 2 , Hölder's inequality implies that there exists 1 < q = q(α) < 2 such that for any û , estimate (2.18) is implied by this inequality together with identity (2.21) and the monotonicity of the capacity.
Step 3. Capacity estimates: Second reduction Let 2 < p <p be fixed, withp as in Step 1. We now argue that in order to prove (2.17) and (2.19) with this choice of exponent p, it suffices to show that for every R > 0, z ∈ R d , and all λ > 0 Without loss of generality, we argue (2.17) and (2.19) in the case z = 0 and for R = 1. We begin by observing that, since p p, definition (2.10) for M 0 together with Jensen's inequality yield that for every This and the first inequality in (2.22) imply, after relabelling the parameter λ p p as λ, that for every R > 0 and λ > 0 it holds  and send λ ↑ +∞. We now derive (2.17) for ∇ x G from (2.25). We begin by smuggling R α into the left-hand side of (2.25) and redefining R α λ as λ so that we conclude that also We now define then the inclusion of the sequence spaces 1 ⊆ 2 , the subadditivity of the operator M 0 , and assumption (2.28) yield We thus established (2.30).
By ( Choosing in (2.28) ω n = 6 (π n) 2 , the sum on the right-hand side converges provided that α > From this inequality we conclude (2.17) for G as we did in the case of ∇ x G from (2.25). This concludes the proof of Step 3.
Step 4. Maximal function estimate We now prove (2.22) and begin with the first estimate. Without loss of generality, we focus on the case z = 0. For any R > 0 and y ∈ R d , let (2.31) We first claim that it suffices to show that for every R > 0 We thus apply the maximal function estimate [7, Inequality (3.1)] to η R F R and infer that where M is defined in (2.9). Since by the assumption on η R and the definitions (2.9) and (2.10) we have we infer that we conclude (2.22) for ∇ x G from (2.33).

Sincep 2, by Minkowski's inequality this in turn yieldŝ
. (2.40) Since for almost every y ∈ R d we have that [see the line above (0.5) and the one below ( i.e. estimate (2.39). The proof of Step 4 is complete.
Step 5. Construction of G * (a; ·, ·) Wrapping up Steps 2-4, we have that G and ∇ x G satisfy (2.22) and therefore also (2.19) and (2.17) and (2.18) and (2.20) with an exponent p that may be chosen strictly bigger than 2. Equipped with these estimates, we now proceed to prove the existence of G * (·, y) for y outside an exceptional set satisfying (a) in the statement of Theorem 1.
For a test function ζ ∈ C ∞ 0 (R d ), we consider the function u(y) =ˆζ(x)G(x, y) dx.
By the representation formula (2.13), u ∈ Y 1,2 (R d ) and solves exists as an element of R m for all y ∈ R d outside a set of zero p-capacity. Select a countable subset {ζ n } n∈N ⊆ C ∞ 0 (R d ) dense with respect to the C 1 -topology. Hence, there exists a set˜ with p-cap(˜ ) = 0 such that (2.42) Let 1 and 2 be the p-capacity zero sets of (2.18), (2.20) in Step 2 and define With this definition, satisfies (a) of Theorem 1. By (2.19) and (2.17) of Step 2, we remark that for every y / ∈ , it also holds so that by (2.18) and (2.20) and weak compactness, we infer that for every y ∈ R d \ there exists a subsequence δ k ↓ 0 (a priori depending on y) and a limit G * (·, y) such that for all R > 0 . Moreover, inequality (2.43) and weak lower-semicontinuity also yield min{|x − y| 2α , 1}|∇ x G * (x, y)| 2 dx < +∞. (2.45) We now show that (2.44) holds for the entire family δ ↓ 0: Let us assume that this were not the case, i.e. that there exist two sequences {δ k } k along which we obtain two different limits G (1) (·, y), G (2) (·, y) in (2.44). Appealing to (2.42), to Fubini's theorem to exchange the order of the integrals, and to (2.44) we infer that for every n ∈ N ζ n (x)G (1) Since the subset {ζ n } n∈N is chosen to be dense, we conclude that G (1) For every point y outside , we thus constructed G * (·, y) which, by (2.44) and (2.45), satisfies (b) and the last inequality in (c) of Theorem 1. To conclude the proof of Theorem 1, it thus remains to show that G * (·, y) solves Eq. (0.5). Since G(·,ỹ) solves Eq. (0.5) for almost everyỹ ∈ R d , for every ζ ∈ C ∞ 0 (R d ), y ∈ R d and δ > 0 we have so that, by Fubini's theorem, (2.46) Taking the limit δ ↓ 0, the assumption on ζ , the boundedness (0.1) of a and (2.44) yield that for all y ∈ R d \ it holdŝ is arbitrary, we conclude that G * (·, y) solves Eq. (0.5).
By the triangle inequality, (2.52) and the definition of U ε , we know indeed that if we fix j j 0 such that 2 − j < κ 3 , then 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. Inserting this into (2.59) we conclude (2.52). The proof of Corollary 1 is complete.