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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.
