# The mean curvature measure

### Qiuyi Dai

Hunan University, Changsa, China### Neil S. Trudinger

Australian National University, Canberra, Australia### Xu-Jia Wang

Australian National University, Canberra, Australia

## 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.

## Cite this article

Qiuyi Dai, Neil S. Trudinger, Xu-Jia Wang, The mean curvature measure. J. Eur. Math. Soc. 14 (2012), no. 3, pp. 779–800

DOI 10.4171/JEMS/318