Hessian estimates for Lagrangian mean curvature equation with sharp Lipschitz phase

TL;DR

This paper develops sharp interior regularity estimates for convex solutions to the Lagrangian mean curvature equation with Lipschitz phase, highlighting the importance of phase regularity and providing new interior regularity results.

Contribution

It establishes sharp interior $C^{1,1}$ estimates for convex solutions with Lipschitz phase and proves interior $C^{2,eta}$ regularity for viscosity solutions, advancing understanding of phase regularity effects.

Findings

Counter-examples for Hölder continuous phase

Interior $C^{1,1}$ estimates for Lipschitz phase

Interior $C^{2,eta}$ regularity for viscosity solutions

Abstract

We establish a prior interior $C^{1,1}$ estimates for convex solutions and supercritical phase solutions to the Lagrangian mean curvature equation with sharp Lipschitz phase. Counter-examples exist when the phase is H\"{o}lder continuous but not Lipschitz. As an application we obtain interior $C^{2,\alpha}$ regularity for $C^0$ viscosity solutions on the first phase interval $((n-2)\frac\pi2,n\frac\pi2)$.