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

Lili Wang, Tao Wang

TL;DR

This work extends Maz'ya's capacity framework to the $p$-Laplacian on compact manifolds by defining Steklov and Neumann $(p,\alpha)$-isocapacitary constants and associated $(p,\alpha)$-Sobolev constants. It provides explicit two-sided bounds relating these isocapacitary constants to the Sobolev constants, yielding two-sided, Maz'ya-type estimates for the first nontrivial Steklov and Neumann eigenvalues (at $\alpha=1$). The results are obtained via a detailed capacity analysis and a new representation of $p$-capacity, along with variational characterizations (Rayleigh quotients) for the eigenvalues. The findings offer quantitative control of eigenvalues in terms of geometric-capacity data on manifolds, broadening the scope of spectral geometry for the $p$-Laplacian.

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 $(p,α)$-Sobolev constants, which degenerate to upper and lower bounds for the first nontrivial Steklov and Neumann eigenvalues of the $p$-Laplacian when $α= 1$.

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

TL;DR

This work extends Maz'ya's capacity framework to the -Laplacian on compact manifolds by defining Steklov and Neumann -isocapacitary constants and associated -Sobolev constants. It provides explicit two-sided bounds relating these isocapacitary constants to the Sobolev constants, yielding two-sided, Maz'ya-type estimates for the first nontrivial Steklov and Neumann eigenvalues (at ). The results are obtained via a detailed capacity analysis and a new representation of -capacity, along with variational characterizations (Rayleigh quotients) for the eigenvalues. The findings offer quantitative control of eigenvalues in terms of geometric-capacity data on manifolds, broadening the scope of spectral geometry for the -Laplacian.

Abstract

In this paper, we introduce Steklov and Neumann isocapacitary constants for the -Laplacian on compact manifolds. These constants yield two-sided bounds for the -Sobolev constants, which degenerate to upper and lower bounds for the first nontrivial Steklov and Neumann eigenvalues of the -Laplacian when .
Paper Structure (4 sections, 9 theorems, 76 equations)

This paper contains 4 sections, 9 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two-sided estimates of Steklov (p,alpha)-Sobolev constants and the first eigenvalue via isocapacity
  4. Two-sided estimates of Neumann (p, alpha)-Sobolev constants and the first eigenvalue via isocapacity

Key Result

Theorem 1.1

Assume that $(M,g)$ is a compact Riemannian manifold with smooth boundary $\partial M$. Then for any $\alpha\geq \frac{1}{p}$, we have

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 7 more