Optimal regularity for Lagrangian mean curvature type equations

TL;DR

This paper classifies the regularity of solutions to Lagrangian mean curvature equations, establishing conditions under which convex solutions are smooth and demonstrating the optimality of these conditions with explicit singular solutions.

Contribution

It provides new regularity criteria for convex solutions based on the smoothness of the Lagrangian phase and generalizes the constant rank theorem to broader settings.

Findings

Convex viscosity solutions are regular if the phase is $C^2$ and convex in the gradient.

Solutions are regular for H"older continuous phases if they are sufficiently smooth.

Explicit singular solutions show the sharpness of the regularity conditions.

Abstract

We classify regularity for Lagrangian mean curvature type equations, which include the potential equation for prescribed Lagrangian mean curvature and those for Lagrangian mean curvature flow self-shrinkers and expanders, translating solitons, and rotating solitons. Convex solutions of the second boundary value problem for certain such equations were constructed by Brendle-Warren 2010, Huang 2015, and Wang-Huang-Bao 2023. We first show that convex viscosity solutions are regular provided the Lagrangian angle or phase is $C^2$ and convex in the gradient variable. We next show that for merely H\"older continuous phases, convex solutions are regular if they are $C^{1,\beta}$ for sufficiently large $\beta$. Singular solutions are given to show that each condition is optimal and that the H\"older exponent is sharp. Along the way, we generalize the constant rank theorem of Bian and Guan to include arbitrary dependence on the Legendre transform.