2015/09/30 by Ulrich Menne
Mathematics · #Advanced Harmonic Analysis Research #Bounded function #Class (philosophy) #Differentiable function #Embedding #Geodesic #Geometric Analysis and Curvature Flows #Interpolation space #Lp space #Nonlinear Partial Differential Equations #Sobolev inequality #Sobolev space #math.AP #math.CA #math.DG #msc:46E35 #msc:49Q15 #msc:53C22
paper · pdf · doi:10.1112/plms/pdw023
published as Proc. Lond. Math. Soc. (3), 113(6):725--774, 2016 · Version initially accepted by Proc. Lond. Math. Soc. (3). The final printed version will be different. 55 pages, no figures
arxiv created 2016/05/27 · openalex created_date 2016/06/24 · openalex publication_date 2016/11/11 · arxiv updated 2017/05/25 · openalex updated_date 2026/08/05
This paper introduces first-order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally non-linear 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 Hölder continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, then the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many 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.