Regularity of viscosity solutions to fully nonlinear elliptic transmission problems

TL;DR

This paper develops a comprehensive regularity theory for viscosity solutions to fully nonlinear elliptic transmission problems, including existence, uniqueness, and regularity up to the interface for both flat and curved cases.

Contribution

It provides the first complete regularity results for viscosity solutions to fully nonlinear elliptic transmission problems with flat and curved interfaces.

Findings

Established existence and uniqueness of solutions.

Proved $C^{0,eta}$, $C^{1,eta}$, and $C^{2,eta}$ regularity up to the interface.

Extended regularity results to curved interfaces.

Abstract

We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and regularity of solutions to flat interface transmission problems; and the $C^{0,\alpha}$, $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity of viscosity solutions up to the transmission surface for the case of curved interfaces.