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  Sobolev functions on varifolds

Menne, U. (2016). Sobolev functions on varifolds. Proceedings London Mathematical Society, 113(5): 023. doi:10.1112/plms/pdw023.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0028-5059-7 Version Permalink: http://hdl.handle.net/21.11116/0000-0002-F76C-F
Genre: Journal Article

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 Creators:
Menne, Ulrich1, Author              
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1Geometric Measure Theory, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_1753352              

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Free keywords: Mathematics, Classical Analysis and ODEs, math.CA,Mathematics, Analysis of PDEs, math.AP,Mathematics, Differential Geometry, math.DG,
 Abstract: This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily H\"older continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding new results for those classes as well. Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions.

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 Dates: 2015-09-03201520162016
 Publication Status: Published in print
 Pages: 51 pages, no figures
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 Rev. Method: -
 Identifiers: arXiv: 1509.01178
URI: http://arxiv.org/abs/1509.01178
DOI: 10.1112/plms/pdw023
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Title: Proceedings London Mathematical Society
Source Genre: Journal
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Pages: - Volume / Issue: 113 (5) Sequence Number: 023 Start / End Page: - Identifier: -