On the generalized mean curvature

Abstract

We study some properties of graphs whose mean curvature (in distributional sense) is a vector Radon measure. In particular, we prove that the distributional mean curvature of the graph of a Lipschitz continuous function u is a measure if and only if the distributional divergence of T u is a measure. This equivalence fails to be true if Lipschitz continuity is relaxed, as it is shown in a couple of examples. Finally, we prove a theorem of approximation in W ^sup (1,1)^ and in the sense of mean curvature of C ^sup 2^ graphs by polyhedral graphs. A number of examples illustrating different situations which can occur complete the work.[PUBLICATION ABSTRACT]