A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions

TL;DR

This paper introduces an integer-valued degree theory for second order fully nonlinear elliptic operators with nonlinear oblique boundary conditions and demonstrates its applications to existence problems in nonlinear elliptic equations.

Contribution

It develops a new degree theory for complex elliptic operators with nonlinear boundary conditions, expanding tools for solving nonlinear elliptic PDEs.

Findings

Established an integer-valued degree for nonlinear elliptic operators

Applied the degree to prove existence of solutions in Yamabe and reflector problems

Extended classical degree theory to nonlinear boundary conditions

Abstract

In this paper we introduce an integer-valued degree for second order fully nonlinear elliptic operators with nonlinear oblique boundary conditions. We also give some applications to the existence of solutions of certain nonlinear elliptic equations arising from a Yamabe problem with boundary and reflector problems.