The Mean Curvature Measure

TL;DR

This paper introduces a measure for mean curvature subharmonic functions, proves its weak continuity, establishes a sharp Harnack inequality, and demonstrates the existence of weak solutions to the Dirichlet problem with measure data.

Contribution

It defines a new measure for mean curvature subharmonic functions and proves its weak continuity, along with a sharp Harnack inequality, advancing the understanding of minimal surface equations.

Findings

The measure agrees with mean curvature for smooth functions.

The measure is weakly continuous under almost everywhere convergence.

A sharp Harnack inequality for the minimal surface equation is established.

Abstract

We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the corresponding Dirichlet problem when the inhomogeneous term is a measure.