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Optimal regularity for Lagrangian mean curvature type equations

Arunima Bhattacharya, Ravi Shankar

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 Hölder continuous phases, convex solutions are regular if they are $C^{1,β}$ for sufficiently large $β$. Singular solutions are given to show that each condition is optimal and that the Hölder exponent is sharp. Along the way, we generalize the constant rank theorem of Bian and Guan to include arbitrary dependence on the Legendre transform.

Optimal regularity for Lagrangian mean curvature type equations

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 and convex in the gradient variable. We next show that for merely Hölder continuous phases, convex solutions are regular if they are for sufficiently large . Singular solutions are given to show that each condition is optimal and that the Hölder exponent is sharp. Along the way, we generalize the constant rank theorem of Bian and Guan to include arbitrary dependence on the Legendre transform.

Paper Structure

This paper contains 11 sections, 12 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Regularity of the rotated Lagrangian potential
  3. $C^{2+\alpha}$ regularity on the rotated domain.
  4. Lewy-Yuan rotation $\bar{u}(\bar{x})$ solves a better equation
  5. Rotated regularity $\bar{u}(\bar{x})\in C^{2+\alpha-}$
  6. $C^{2,\alpha}$ estimates for VMO solutions of fully non-linear elliptic equations
  7. Generalizing the constant rank theorem
  8. Strong convexity without constant rank theorems
  9. Proof of Theorems \ref{['thm1']} and \ref{['thm2']}
  10. Singular solution to self similar equation \ref{['self']}
  11. Singular solution to rotator equation \ref{['rotator']}

Key Result

Theorem 1.1

Let $u$ be a convex viscosity solution of slag on $B_1\subset\mathbb R^n$. If $\psi\in C^2(B_1\times\mathbb{R}\times\mathbb{R}^n)$ satisfies the partial convexity condition convex then $u\in C^{4-}(B_1)$.

Theorems & Definitions (28)

  • Definition 1: Partial convexity
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • ...and 18 more