Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces

TL;DR

This paper proves that certain minimal surfaces are infinitely smooth by developing a new bootstrap regularity theory for a broad class of integro-differential equations, with implications for nonlocal geometric problems.

Contribution

It introduces a novel bootstrap regularity framework for integro-differential equations, enabling the proof of smoothness for nonlocal minimal surfaces.

Findings

$C^{1,eta}$ $s$-minimal surfaces are $C^0$ smooth.

Developed a new general regularity theory for integro-differential equations.

Applied the theory to establish smoothness of nonlocal minimal surfaces.

Abstract

We prove that $C^{1,\alpha}$ $s$-minimal surfaces are automatically $C^\infty$. For this, we develop a new bootstrap regularity theory for solutions of integro-differential equations of very general type, which we believe is of independent interest.