Optimal elliptic regularity: A comparison between local and nonlocal equations

Abstract

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