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On the generalized mean curvature. (English) Zbl 1207.49052

Summary: 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 \(Tu\) 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 \(W ^{(1,1)}\) and in the sense of mean curvature of \(C^2\) graphs by polyhedral graphs. A number of examples illustrating different situations which can occur complete the work.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

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