Optimal elliptic regularity: a comparison between local and nonlocal equations

TL;DR

This paper compares the regularity of solutions to classical and non-local elliptic equations, establishing bounds on the highest Hölder exponent depending on the ellipticity ratio.

Contribution

It provides new quantitative bounds on the optimal regularity exponent for non-local elliptic equations, contrasting with classical results.

Findings

Classical case: lpha(L)\u2265xp(-CL^eta)

Non-local case: lpha(L)\u2265L^{-1-\u03b4} for all <

Shows sharper decay of regularity in the non-local setting

Abstract

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 $\delta>0$.