The Mean Curvature Measure

Details The Mean Curvature Measure The Mean Curvature Measure

2009-12-02

arxiv.org/abs/0912.0341v1

Mathematics

Analysis of PDEs

29pages

Scientific paper

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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