Partial regularity for Lipschitz solutions to the minimal surface system

TL;DR

This paper investigates the partial regularity of various Lipschitz solutions to the minimal surface system, providing new interior gradient estimates under specific conditions.

Contribution

It introduces new partial regularity results for different notions of Lipschitz solutions and establishes interior gradient estimates assuming area-decreasing conditions.

Findings

Partial regularity results for stationary, weak, and viscosity solutions.

Interior gradient estimates for classical solutions under area-decreasing assumptions.

Small infinity norm of components leads to improved regularity.

Abstract

In this paper, we study the regularity of several notions of Lipschitz solutions to the minimal surface system with an emphasis on partial regularity results. These include stationary solutions, integral weak solutions, and viscosity solutions. We also prove an interior gradient estimate for classical solutions to the system using the maximum principle, assuming the area-decreasing condition and that all but one component have small infinity norm.