$C^{\sigma+\alpha}$ regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels

TL;DR

This paper proves interior regularity estimates of order $C^{\sigma+\alpha}$ for concave nonlocal fully nonlinear equations with rough kernels, extending known results to equations with variable coefficients.

Contribution

It introduces a new method combining Liouville theorem and blow-up techniques to establish regularity for nonlocal equations with rough kernels and variable coefficients.

Findings

Establishes $C^{\sigma+\alpha}$ regularity for solutions.

Extends regularity results to equations with $x$-dependent kernels.

Provides a flexible proof method applicable to other nonlocal equations.

Abstract

We establish $C^{\sigma+\alpha}$ interior estimates for concave nonlocal fully nonlinear equations of order $\sigma\in(0,2)$ with rough kernels. Namely, we prove that if $u\in C^{\alpha}(\mathbb R^n)$ solves in $B_1$ a concave translation invariant equation with kernels in $\mathcal L_0(\sigma)$, then $u$ belongs to $C^{\sigma+\alpha}(\overline{ B_{1/2}})$, with an estimate. More generally, our results allow the equation to depend on $x$ in a $C^\alpha$ fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs for nonlocal equations.