Semiconvexity estimates for nonlinear integro-differential equations

TL;DR

This paper introduces local semiconvexity estimates for fully nonlinear integro-differential equations and obstacle problems, using a novel Bernstein technique applicable to nonlocal operators, solving an open problem and advancing regularity theory.

Contribution

It develops the first local semiconvexity estimates for nonlinear integro-differential equations and obstacle problems, and extends the Bernstein technique to nonlocal and parabolic operators.

Findings

Established local semiconvexity estimates for nonlinear integro-differential equations.

Solved an open problem from Cabré-Dipierro-Valdinoci [CDV22].

Proved optimal regularity and free boundary smoothness near regular points.

Abstract

In this paper we establish for the first time local semiconvexity estimates for fully nonlinear equations and for obstacle problems driven by integro-differential operators with general kernels. Our proof is based on the Bernstein technique, which we develop for a natural class of nonlocal operators and consider to be of independent interest. In particular, we solve an open problem from Cabr\'e-Dipierro-Valdinoci [CDV22]. As an application of our result, we establish optimal regularity estimates and smoothness of the free boundary near regular points for the nonlocal obstacle problem on domains. Finally, we also extend the Bernstein technique to parabolic equations and nonsymmetric operators.