A note on the area and coarea formulas for general volume densities and some applications
Daniel Cibotaru, Jorge de Lira
TL;DR
This paper extends the area and coarea formulas for Lipschitz maps to general volume densities, providing new proofs and applications in anisotropic Sobolev inequalities and hypersurface geometry.
Contribution
It introduces generalized formulas for volume densities, offers a Euclidean proof of the anisotropic Sobolev inequality, and discusses an anisotropic tube formula for hypersurfaces.
Findings
Generalized area and coarea formulas for Lipschitz maps with volume densities
A Euclidean proof of the anisotropic Sobolev inequality
An anisotropic tube formula for hypersurfaces in bR^n
Abstract
We present the area and coarea formulas for Lipschitz maps, valid for general volume densities. As applications, we give a short, "euclidean" proof of the anisotropic Sobolev inequality and describe an anisotropic tube formula for hypersurfaces in . A discussion about the first variation of the anisotropic area is also included.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Advanced Mathematical Modeling in Engineering