Boundary value problems for some fully nonlinear elliptic equations

TL;DR

This paper addresses boundary value problems for certain fully nonlinear elliptic equations, extending existing results by deriving boundary estimates directly from boundary bounds, with applications to a nonlinear Yamabe problem.

Contribution

It introduces a method to obtain boundary $C^2$ estimates from boundary $C^0$ estimates for fully nonlinear elliptic equations, generalizing Escobar's work on the Yamabe problem.

Findings

Derived boundary $C^2$ estimates from boundary $C^0$ estimates.

Generalized Escobar's results to a broader class of nonlinear equations.

Applied techniques to nonlinear Yamabe problem on conformally flat manifolds.

Abstract

We consider a nonlinear version of the Yamabe problem on locally conformally flat compact manifolds with boundary. The main technique we used is to derive boundary $C^2$ estimates directly from boundary $C^0$ estimates. In particular, the result is a generalization of the work by Escobar.