Calculus of Variations and Geometric Measure Theory
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S. Costea

Sobolev-Lorentz capacity and its regularity in the Euclidean setting

created by costea on 28 Jul 2017

[BibTeX]

preprint

Inserted: 28 jul 2017
Last Updated: 28 jul 2017

Year: 2017

ArXiv: 1707.08873 PDF

Abstract:

This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for $n \ge 1$ integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for $n \ge 2.$ Moreover, for $n \ge 2$ integer we obtain the exact value of the $n,1$ capacity of a point relative to all its bounded open neighborhoods from ${\mathbf{R}}^n,$ improving another previous result of ours. We show that this constant is also the value of the $n,1$ global capacity of any point from ${\mathbf{R}}^n,$ $n \ge 2.$ We also prove the embedding $H_{0}^{1,(n,1)}(\Omega) \hookrightarrow C(\bar{\Omega}) \cap L^{\infty}(\Omega),$ where $\Omega \subset {\mathbf{R}}^n$ is open and $n \ge 2$ is an integer. In the last section of the paper we show that the relative and the global $(p,1)$ and $p,1$ capacities are Choquet whenever $1 \le n<p<\infty$ or $1<n=p<\infty.$

Keywords: Sobolev spaces, Lorentz spaces, capacity

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