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

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.