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Regularity for Hamiltonian stationary equations in $\mathbb{R}_{n\leq 4}^{n}$

Arunima Bhattacharya

TL;DR

This paper addresses regularity for the Hamiltonian stationary equation governing Lagrangian submanifolds in Euclidean space, proving that in dimensions $n\le 4$ any $C^{1,1}$ solution is smooth and enjoys $C^{k,\alpha}$ estimates for all $k\ge 2$. The authors leverage a two-second-order decomposition into the Lagrangian mean curvature equation, establish Hölder regularity of the Lagrangian phase $\Theta$ via De Giorgi–Nash, and then apply viscosity solutions with VMO coefficients to bootstrap to $C^{2,\alpha}$, followed by iterative bootstrapping to higher regularity. The result extends prior 2D and special cases to $n=3,4$ without extra assumptions, clarifying the role of phase regularity and offering interior estimates that feed into compactness analyses in the Hamiltonian stationary setting. The work also discusses variational versus geometric formulations and notes how a critical/supercritical phase condition $|\Theta| \ge (n-2)\frac{\pi}{2}$ can enable extensions to all dimensions.

Abstract

In this paper, we study the regularity of solutions to the Hamiltonian stationary equation in complex Euclidean space. We show that in dimensions $n\leq 4$, for all values of the Lagrangian phase, any $C^{1,1}$ solution is smooth and derive a $C^{k,α}$ estimate for it, where $k \geq 2$.

Regularity for Hamiltonian stationary equations in $\mathbb{R}_{n\leq 4}^{n}$

TL;DR

This paper addresses regularity for the Hamiltonian stationary equation governing Lagrangian submanifolds in Euclidean space, proving that in dimensions any solution is smooth and enjoys estimates for all . The authors leverage a two-second-order decomposition into the Lagrangian mean curvature equation, establish Hölder regularity of the Lagrangian phase via De Giorgi–Nash, and then apply viscosity solutions with VMO coefficients to bootstrap to , followed by iterative bootstrapping to higher regularity. The result extends prior 2D and special cases to without extra assumptions, clarifying the role of phase regularity and offering interior estimates that feed into compactness analyses in the Hamiltonian stationary setting. The work also discusses variational versus geometric formulations and notes how a critical/supercritical phase condition can enable extensions to all dimensions.

Abstract

In this paper, we study the regularity of solutions to the Hamiltonian stationary equation in complex Euclidean space. We show that in dimensions , for all values of the Lagrangian phase, any solution is smooth and derive a estimate for it, where .

Paper Structure

This paper contains 3 sections, 3 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. Comparison with variational formulation

Key Result

Theorem 1.1

Suppose that $u\in C^{1,1}(B_1)$ is a weak solution of hstat0 on the unit ball $B_1\subset \mathbb{R}^n$ where $n\leq 4$. Then for $k\geq 2$, we have

Theorems & Definitions (7)

  • Theorem 1.1
  • Corollary 1.1
  • Definition 1: VMO
  • Proposition 2.1
  • proof
  • proof
  • proof