Continuity estimates for the gradient of solutions to the Monge-Amp\`ere equation with nature boundary conditions

TL;DR

This paper establishes optimal global log-Lipschitz continuity estimates for the gradient of solutions to the Monge-Ampère equation with natural boundary conditions, advancing understanding of regularity in nonlinear PDEs.

Contribution

It provides the first derivative estimates in terms of modulus of continuity for solutions to the Monge-Ampère equation with natural boundary conditions, achieving optimal regularity results.

Findings

Proved global log-Lipschitz continuity of the gradient.

Established optimal regularity estimates for solutions.

Enhanced understanding of boundary behavior in Monge-Ampère equations.

Abstract

We study the first derivative estimates for solutions to Monge-Amp\`ere equations in terms of modulus of continuity. As a result, we establish the optimal global log-Lipschitz continuity for the gradient of solutions to the Monge-Amp\`ere equation with natural boundary conditions.