Isocapacitary constants for the $p$-Laplacian on compact manifolds

TL;DR

This paper introduces isocapacitary constants for the p-Laplacian on compact manifolds, providing bounds for Sobolev constants and eigenvalues, advancing understanding of spectral properties in geometric analysis.

Contribution

It defines Steklov and Neumann isocapacitary constants for the p-Laplacian, linking them to Sobolev constants and eigenvalues, which was not previously established.

Findings

Derived two-sided bounds for Sobolev constants

Established bounds for first nontrivial eigenvalues

Connected isocapacitary constants with spectral properties

Abstract

In this paper, we introduce Steklov and Neumann isocapacitary constants for the $p$-Laplacian on compact manifolds. These constants yield two-sided bounds for the $(p,\alpha)$-Sobolev constants, which degenerate to upper and lower bounds for the first nontrivial Steklov and Neumann eigenvalues of the $p$-Laplacian when $\alpha= 1$.