The first initial boundary value problem for Hessian equations of parabolic type on Riemannian manifolds

TL;DR

This paper addresses the existence and a priori estimates for solutions to a class of fully nonlinear parabolic Hessian equations on Riemannian manifolds, establishing nearly optimal conditions for classical solutions.

Contribution

It provides the first analysis of initial boundary value problems for Hessian-type parabolic equations on Riemannian manifolds, including key a priori estimates and existence results.

Findings

Established a priori C^2 estimates for solutions.

Proved existence of classical solutions under nearly optimal conditions.

Extended analysis to fully nonlinear parabolic equations on Riemannian manifolds.

Abstract

In this paper, we are concerned with the first initial boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. As usual, the establishment of the a priori C^2 estimates is our main part. Based on these estimates, the existence of classical solutions is proved under conditions which are nearly optimal.