Boundary estimates for a fully nonlinear Yamabe problem on Riemannian manifolds

TL;DR

This paper establishes boundary second derivative estimates and existence results for fully nonlinear Yamabe equations on Riemannian manifolds with boundary, including uniform estimates across a family of related equations.

Contribution

It provides the first a priori boundary estimates for fully nonlinear Yamabe problems and demonstrates existence of smooth solutions under subsolution assumptions.

Findings

Derived boundary second derivative estimates for fully nonlinear Yamabe equations

Proved existence of smooth solutions assuming a subsolution exists

Provided an example of a boundary regularity failure in a specific case

Abstract

In this paper, we consider the Dirichlet boundary value problem for fully nonlinear Yamabe equations on Riemannian manifolds with boundary. Assuming the existence of a subsolution, we derive \emph{a priori} boundary second derivative estimates and consequently obtain the existence of a smooth solution. Moreover, with respect to a family of equations interpolating the fully nonlinear Yamabe equation and the classical semi-linear Yamabe equation, our estimates remain uniform. Finally, an example of a $C^1$ solution which is smooth in the interior but not smooth at the boundary is also given.