The Bernstein technique for integro-differential equations

TL;DR

This paper extends Bernstein's technique to integro-differential equations, providing new derivative estimates for fractional and nonlinear equations, with implications for extremal and obstacle problems.

Contribution

It introduces a robust method for derivative estimates in fractional equations, including nonlinear and convex cases, and explores applications to extremal and obstacle problems.

Findings

First and one-sided second derivative estimates for fractional equations.

Uniform estimates as the order approaches two.

New results even in the linear case.

Abstract

We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully nonlinear equations of order smaller than two -- for which we prove uniform estimates as their order approaches two. Our method is robust enough to be applied to some Pucci-type extremal equations and to obstacle problems for fractional operators, although several of the results are new even in the linear case. We also raise some intriguing open questions, one of them concerning the "pure" linear fractional Laplacian, another one being the validity of one-sided second derivative estimates for Pucci-type convex equations associated to linear operators with general kernels.