Monge-Amp\`ere type equations with Neumann boundary conditions on Riemannian manifolds

TL;DR

This paper extends the classical solvability results of Monge-Ampère type equations with Neumann boundary conditions from Euclidean spaces to Riemannian manifolds, establishing global regularity under certain assumptions.

Contribution

It generalizes the main theorem for Monge-Ampère equations with Neumann boundary conditions from Euclidean spaces to Riemannian manifolds.

Findings

Established global regularity for Monge-Ampère equations on Riemannian manifolds.

Extended classical solvability results from Euclidean space to curved manifolds.

Provided conditions under which the Neumann problem is solvable on Riemannian manifolds.

Abstract

In this paper, we consider the global regularity for Monge-Amp\`ere type equations with the Neumann boundary conditions on Riemannian manifolds. It is known that the classical solvability of the Neumann boundary value problem is obtained under some necessary assumptions. Our main result extends the main theorem from the case of Euclidean space $R^n$ in [11] to Riemannian manifolds.