Optimal elliptic regularity: a comparison between local and nonlocal equations

Given $L\geq 1$, we discuss the problem of determining the highest $\alpha=\alpha(L)$ such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by $L$ is in $C^\alpha_{\rm loc}$. This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $\alpha(L)\gtrsim {\rm exp}(-CL^\beta)$, for some $C, \beta\geq 1$ depending on the dimension $N\geq 3$. We show that in the non-local case, $\alpha(L)\gtrsim L^{-1-\delta}$ for all $\delta>0$.


Introduction
The aim of this short note is to highlight some analogies and differences between the Hölder regularity theory for elliptic equations in divergence form with measurable coefficients in the classical and non-local setting respectively. To this end, let us recall some well known results.
Suppose that A(x) = (a ij (x)) is an N × N symmetric matrix with measurable coefficients, satisfying for some 0 < λ ≤ Λ the ellipticity condition If u ∈ W 1,2 (B 1 ), B 1 being the unit ball centered at 0, weakly solves By the linearity and homogeneity of the equation, we can multiply (1.2) by λ −1 , so that we can assume λ = 1 in (1.1) (and then Λ = L). The best Hölder exponent for solutions of elliptic equations with a given ellipticity ratio L is α(N, L) := sup α : H L ⊆ C α loc (B 1 ) . In [17], the following is proved: It is conjectured thatᾱ(N, L) = C(N )/L, but the problem of determiningᾱ(N, L) for N ≥ 3 is open up to now. The best available lower bound onᾱ can be found keeping trace of the 1 Technically, is should be called worst Hölder exponent, however the adjective "best" is more used in the literature. which is of course (exponentially) far from the known upper bound. Let us mention that after the work of Moser [14], many Hölder regularity results are nowadays derived from the Harnack inequality. Indeed, any positive solution of (1.2) in a ball B r satisfies sup u for a constant c H independent of r. By a standard argument, this implies that any solution of (1.2) satisfies , which in turn gives the oscillation estimate In other words, the best (scale invariant) Harnack constant for (1.2) bounds the inverse of the best Hölder exponent. Unfortunately, it is well known that the best Harnack constant is in fact exponential in the ellipticity ratio: for example, the function u(x, y) = e √ Lx cos y satisfies (1.2) for This example shows that one cannot get rid of the exponential in (1.3) by proving a scale invariant Harnack inequality 2 and it is not so surprising that the methods in [5] [16] and [13] give the bound (1.3) since each technique actually proves a Harnack inequality as well (see [6], [8] and [14], respectively). Our main result shows that the nonlocal analogue of (1.2) seems to enjoy better properties than (1.2) itself. By a non-local analogue of solutions to (1.2) we mean local minimizers of the non-local functional In turn, isotropic functionals (i.e. (1.5) with a ij (x) = a(x)δ ij ) are dense with respect to Mosco-convergence in the set of general elliptic functionals (1.5) by [15] (see also [4] for a more general density result and references). Notice that, when looking at local minimizers of (1.4), we can always suppose that a(x, y) is symmetric, since In this framework, the ellipticity condition (1.1) reads λ ≤ a(x, y) ≤ Λ, which implies the same condition of the symmetrized coefficient Local minimizers of (1.4) in, say, B 1 , satisfy Normalizing, we will suppose henceforth We can now state our main result, in a simplified version (for the full result see Theorem 4.3).
Let us notice that the De Giorgi-Nash-Moser regularity theory for equations like (1.6) has already been developed in [11], [7]. The method of proof in these papers is usually a (nontrivial) modification of the De Giorgi-Moser approach, and therefore provides a much smaller Hölder exponent than ours, namely, the one given in (1.3). The main point of our result is therefore the improvement, in the non-local case, of the (inverse of the) best Hölder exponent from exponential in L to almost linear in L. This is mainly due to a weak Harnack inequality which is proved, much in the spirit of [9], [10], through a non-local barrier argument, a method which avoids the usual De Giorgi-Moser arguments. We remark that the obtained Hölder exponent blows down as s → 1 − , i.e. our estimates are exclusively non-local.
Let us finally mention that, starting from the seminal work of Bass and Levine [1], a huge literature has grown around the regularity theory for some related non-local equations, which, very roughly speaking, may be seen as the non-local counterpart of non-divergence form elliptic equations, see [18, Section 3.6] for a discussion. The interested reader can consult the bibliographic references in [19].
The structure of the paper is as follows. In section 2 we will describe the functional analytic framework we will work in. Section 3 is devoted to a lower bound of the torsion function, i.e., Despite v being, strictly speaking, a supersolution of (1.6), the non-local nature of the equation allows to use it as a lower barrier to (1.6), when suitably modified away from B 1 . This idea will be realized in section 4, giving a weak Harnack inequality and, eventually, the claimed Hölder exponent.

Preliminaries
In all the paper N ≥ 1, s ∈ ]0, 1[, L ≥ 1 will be fixed. We also fix k(x, y) satisfying (1.8), and for simplicity we let All constants, unless otherwise specified, will depend on N and s only, and are allowed to change value from line to line keeping the same symbol, as long as this dependance is unaltered. When a constant depends also on some other parameter δ, it will be denoted as, say, C δ , i.e.
For any A ⊆ R N we will set A c = R N \ A and B r ⊆ R N will denote the open ball of radius r > 0, centered at 0. For simplicity, sup A u and inf A u will denote the essential supremum and infimum of u on a measurable set A, and u ∞ will be the L ∞ (R N ) of u. For any a ∈ R, we also let a + = max{a, 0}, a − = min{a, 0}.
Given any 3 open Ω ⊆ R N and measurable u : Ω → R, we define the Gagliardo semi-norms and the corresponding functional spaces Clearly, as long as u ∈ L ∞ (R N ), the second condition in the previous definition is trivially satisfied. The space W s,2 0 (Ω) is defined as in (1.7), substituting B 1 with Ω, and its dual is denoted by W −s,2 (Ω). Suitably modifying [9, Lemma 2.3], we see that for any u ∈ W s,2 (Ω), the functional is linear and continuous in W s,2 0 (Ω), where µ is defined in (2.1) and k fulfills (1.8). Therefore the notation used for K above is well posed when , denotes the duality bracket of W s,2 0 (Ω) and K : W s,2 (Ω) → W −s,2 (Ω) is clearly a linear operator. Using the linearity of the equation it is therefore possible to solve any Dirichlet problem of the form for any given u 0 ∈ W s,2 (Ω), simply minimizing the convex coercive functional . This setting also gives a meaning to inequalities such as Ku ≥ f in Ω for u ∈ W s,2 (Ω), f ∈ W −s,2 (Ω) by testing with nonnegative ϕ ∈ W s,2 0 (Ω). We will make use of the following comparison principle, obtained by a slight modification of [9, Proposition 2.10].

Lower bound on the torsion function
This section is devoted to obtain a lower bound, in terms of the ellipticity ratio L, for the torsion function, namely the solution to Ku = 1 with Dirichlet boundary conditions. We do not know wether the following proposition holds with δ = 0, a fact that would eventually provide a Hölder exponent of order 1/L in Theorem 1.1 Proposition 3.1. Assume that (1.8) holds and let u weakly solve Then for any δ > 0 there exists a constant c δ = c(N, s, δ) such that Proof. First we show that by a scaling argument we can suppose r = 1. Let u solve (3.1) and define u (r) (x) = u(rx), k (r) (x, y) = k(rx, ry) and K (r) the corresponding linear operator. Then by changing variables it follows that u (r) ∈ W s,2 0 (B 1 ) satisfies K (r) u (r) = r 2s , while clearly k (r) still satisfies the bounds (1.8). The linearity of the equation then gives that u (r) r −2s satisfies (3.1) in B 1 , which gives the general statement (3.2). Therefore we will suppose henceforth that r = 1.
Step 1: Caccioppoli inequality for negative powers of u. By the weak minimum principle it holds u ≥ 0 in B 1 . We fix ε > 0 and let u ε = u + ε. For any ρ 2 < ρ 1 < 1, we choose a cutoff function and for any β > 1 we test the equation with η β+1 u −β dµ.
We employ [11,Lemma 2.5], to obtain the pointwise inequality The right hand side can be bounded using the symmetry and structure of µ as and estimating the integral in y through [12, Lemma 2.3], which gives (3.5) We bound the left hand side of (3.4) from below using Sobolev-Poincaré inequality as follows: (3.6) Inserting (3.5) and (3.6) into (3.4) and rearranging, we obtain the following Caccioppoli inequality: Step 2: The iteration. We first iterate (3.7) neglecting the second term on the left. We fix a large m ∈ N to be chosen later and let

From (3.7) we infer
where the constant on the left hand side satisfies thus giving, raising the previous estimate to the 1/m-power, ) . Next, we follow the standard Moser iteration starting from the power m and ignoring the first term on the left of (3.7). We let γ = 2 * 2 > 1, The latter can be rewritten as This can be iterated estimating the infinite product (we precisely compute only the relevant one) as Applying (3.8) on the right hand side, we obtain (3.9) sup Step 3: Conclusion.

Remark 3.2.
The previous proof is completely local in nature and a similar estimate holds true for the torsion function associated to elliptic equations with measurable coefficients. In this more classical framework it may be interesting to know wether an optimal bound from below is true without the δ mentioned in the proposition. More precisely, if A = (a ij ) is measurable and symmetric and u is the weak solution of  s). Moreover, due to [2,3] the Sobolev-Poincaré embedding can be written as so that the second part of the iteration is independent of s. Following the proof one eventually gets c δ = (1 − s)c (N, δ). This is consistent with the case k(x, y) ≡ 1 and K = (−∆) s : indeed, if u ∈ W s,2 0 (B r ) solves (−∆) s u s = 1, then (1 − s)u s → u as s → 1 − , where u is the classical torsion function for the Laplacian.

Proof of the main result
We start with the core nonlocal estimate providing a weak Harnack inequality with best constant of order L 1+δ , δ > 0. The fact that we allow arbitrary K ≥ 0 in the statement is essential in order to deal with tail terms later in Corollary 4.2, where we localize the condition u ≥ 0 in R N .
Then, for any δ > 0 there exists σ δ = σ(N, s, δ) ∈ (0, 1) such that Proof. By the scaling argument described at the beginning of Proposition 3.1, we can assume R = 1. Let v solve (3.1) in B 1 , choose λ ∈ (0, 1) (to be determined later) and set Since χ B 2 \B 3/2 u ∈ W s,2 (B 1 ), we can compute K(χ B 2 \B 3/2 u), setting for simplicity χ = where we used the symmetry of µ and the fact that supp(χu) ⊆ B 2 \ B c 3/2 . Since both u and ϕ are nonnegative and |x − y| ≤ 7/2 for x ∈ B 1 and y ∈ B c 3/2 , we have Therefore, by linearity and Kv = 1, We choose λ = θ in the definition of w and, for δ > 0, let c δ ∈ ]0, 1] be the constant given in Proposition 3.1. We claim that (4.2) holds with σ δ = θ c δ , C = 1/θ. Indeed, if K = 0 in (4.1), we observe that , so that the weak comparison principle implies u ≥ w in B 1 . In particular (4.3) inf by Proposition 3.1. If K > 0, we can use linearity to reduce to K = 1 and prove Clearly we can suppose that M ≥ 1/θ, since otherwise the right hand side of (4.4) is negative and the inequality is trivially satisfied since u ≥ 0 in B 1 by assumption. In this case Noting that |x − y| ≥ |y|/2 for x ∈ B R and y ∈ B 2R , we see that |u − | |y| N +2s dy and applying (4.2) to u + gives the claim. Now we are ready to prove the main result. The condition u ∈ L ∞ (R N ) is assumed only for simplicity, allowing the bound Tail(u, R) ≤ C u ∞ .

Remark 4.4.
It is worth outlining the dependance of a δ on s as s → 1. As already pointed out, our exponent blows down as s → 1 − . Indeed, using remark 3.3, one gets σ δ , C δ ≃ (1 − s) in (4.5). The only relevant place where (1 − s) is involved in the choice of the Hölder exponent is therefore (4.10), which forces a δ (s) =ã(N, δ)(1 − s).