Regularity results for nonlocal equations and applications

Abstract

We introduce the concept of C m,α -nonlocal operators, extending the notion of second order elliptic operator in divergence form with C m,α -coefficients. We then derive the nonlocal analogue of the key existing results for elliptic equations in divergence form, notably the Hölder continuity of the gradient of the solutions in the case of C 0,α -coefficients and the classical Schauder estimates for C m+1,α -coefficients. We further apply the regularity results for C m,α -nonlocal operators to derive optimal higher order regularity estimates of Lipschitz graphs with prescribed Nonlocal Mean Curvature. Applications to nonlocal equation on manifolds are also provided.

On the Calderon-Zygmund property of Riesz-transform type operators arising in nonlocal equations

“…To our knowledge this is new, although results somewhat similar in spirit (in the context of L 2estimates) have been observed e.g. in [6,Section 3]. Observe that we obtain this estimate at all differentiability scales below 1.…”

“…Such equations have been studied by numerous authors, we name just a few [10,5,7,11,4,2,14,6,1,3,13].…”

Calderon-Zygmund theory for non-convolution type nonlocal equations with continuous coefficient

“…For t ≥ s, local W t,p -Calderon Zygmund theory has been studied in [11] for a different type of kernels by three of the authors. On the other hand the first author recently studied Hölder regularity theory for (1.1) in [13]. To some extent, with respect to motivation, techniques, and results, this paper is a combination of the ideas in [13] and [11], and we refer to those papers for a background and motivation of equations of the type we are considering.…”

Higher Hölder regularity for nonlocal equations with irregular kernel

“…Regarding other related regularity results, in [12] a similar result is proved in the linear case when Φ(t) = t, where A is required to be locally close enough to b x−y |x−y| for some even function b : S n−1 → R that is bounded between two positive constants, which in the limit is contained in our assumption (7), see also Remark 4.4 below. More results concerning higher Hölder regularity for various types of nonlocal equations are for instance contained in [11], [23], [5], [6] and [14]. Furthermore, results regarding basic Hölder regularity for nonlocal equations are proved for example in [8], [15], [25] and [19], while results concerning Sobolev regularity can be found for example in [18], [24], [1], [7], [21] and [22].…”