Calculus of Variations and Geometric Measure Theory

U. Menne - C. Scharrer

A novel type of Sobolev-Poincaré inequality for submanifolds of Euclidean space

created by scharrer on 09 Mar 2020

[BibTeX]

preprint

Inserted: 9 mar 2020

Year: 2017

ArXiv: 1709.05504 PDF

Abstract:

For functions on generalised connected surfaces (of any dimensions) with boundary and mean curvature, we establish an oscillation estimate in which the mean curvature enters in a novel way. As application we prove an a priori estimate of the geodesic diameter of compact connected smooth immersions in terms of their boundary data and mean curvature. These results are developed in the framework of varifolds. For this purpose, we establish that the notion of indecomposability is the appropriate substitute for connectedness and that it has a strong regularising effect; we thus obtain a new natural class of varifolds to study. Finally, our development leads to a variety of questions that are of substance both in the smooth and the nonsmooth setting.