The Mean Curvature Measure

2009/12/02 by Qiuyi Dai, Dai, Qiuyi, Neil S. Trudinger +3 · 1 citation
Mathematics · Physics and Astronomy · #35J66 #35J93 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.0912.0341

openalex publication_date 2009/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Existence theory for linear-growth variational integrals with signed measure data

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