Generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear equations with rough kernels of variable orders

Abstract

We establish the generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear elliptic equations with rough kernels of variable orders. In contrast to the fractional Laplacian type operators having a fixed order of differentiability $\sigma \in (0,2)$, the operators under consideration have variable orders of differentiability. Since the order is not characterized by a single number, we consider a function $\phi$ describing the variable orders of differentiability, which is allowed to oscillate between two functions $r^{{\sigma }_1}$ and $r^{{\sigma }_2}$ for some $0<{\sigma }_1\le {\sigma }_2<2$. By introducing the generalized Hölder spaces, we provide $C^{\phi \psi }$ estimates that generalizes the standard Evans--Krylov and Schauder type $C^{\sigma +\alpha }$ estimates.

References in corpus (2)

• Regularity for fully nonlinear integro-differential operators with kernels of variable orders
• Existence and non-existence results for a class of semilinear nonlocal operators with exterior condition