Escobar's Type Theorems for elliptic fully nonlinear degenerate equations

TL;DR

This paper extends Escobar's type theorems to classify solutions of elliptic fully nonlinear degenerate equations on spherical subdomains, providing non-existence and classification results for prescribed boundary mean curvature.

Contribution

It generalizes previous results to fully nonlinear degenerate equations on various spherical domains, including punctured balls and annuli.

Findings

Non-existence of solutions under certain conditions

Classification of solutions on specific domains

Extension of Escobar's results to nonlinear degenerate cases

Abstract

In this paper we prove non-existence and classification results for elliptic fully nonlinear elliptic degenerate conformal equations on certain subdomains of the sphere with prescribed constant mean curvature along its boundary. We also consider non-degenerate equations. Such subdomains are the hemisphere (or a geodesic ball in $\mathbb{S}^m$), punctured balls and annular domains. Our results extend those of Escobar in when $m\geq 3$, and Hang-Wang and Jimenez in when $m=2$.