Optimal elliptic regularity: a comparison between local and nonlocal equations

Abstract

Given $L\ge 1$, 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 $C_{\mathrm{loc}}^{\alpha }$. This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $\alpha (L)\gtrsim \exp (-C{L}^{\beta })$, for some $C,\beta \ge 1$ depending on the dimension $N\ge 3$. We show that in the non-local case, $\alpha (L)\gtrsim L^{-1-\delta }$ for all $\delta >0$.

12 pages, comments welcome