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Isocapacitary constants associated with $p$-Laplacian on graphs

Bobo Hua, Lili Wang

TL;DR

This work develops a capacity-based framework for the $p$-Laplacian on graphs by introducing discrete isocapacitary constants that quantify how subsets separate in the graph. A discrete coarea formula links $p$-energy to level-set capacities, enabling two-sided, order-matched bounds for the first Dirichlet, Neumann, and Steklov eigenvalues in terms of $\alpha_p^D$, $\overline{\alpha}_p^N$, and $\overline{\alpha}_p^S$, respectively. The results hold for finite and infinite graphs (via exhaustion) and yield a unified geometric characterization that improves upon classical Cheeger-type bounds, with a constructive Steklov theory through graph sequences. An explicit path-graph example demonstrates the sharpness of the bounds, illustrating the effectiveness of the capacity approach in discrete nonlinear spectral problems.

Abstract

In this paper, we introduce isocapacitary constants for the $p$-Laplacian on graphs and apply them to derive estimates for the first eigenvalues of the Dirichlet $p$-Laplacian, the Neumann $p$-Laplacian, and the $p$-Steklov problem.

Isocapacitary constants associated with $p$-Laplacian on graphs

TL;DR

This work develops a capacity-based framework for the -Laplacian on graphs by introducing discrete isocapacitary constants that quantify how subsets separate in the graph. A discrete coarea formula links -energy to level-set capacities, enabling two-sided, order-matched bounds for the first Dirichlet, Neumann, and Steklov eigenvalues in terms of , , and , respectively. The results hold for finite and infinite graphs (via exhaustion) and yield a unified geometric characterization that improves upon classical Cheeger-type bounds, with a constructive Steklov theory through graph sequences. An explicit path-graph example demonstrates the sharpness of the bounds, illustrating the effectiveness of the capacity approach in discrete nonlinear spectral problems.

Abstract

In this paper, we introduce isocapacitary constants for the -Laplacian on graphs and apply them to derive estimates for the first eigenvalues of the Dirichlet -Laplacian, the Neumann -Laplacian, and the -Steklov problem.

Paper Structure

This paper contains 7 sections, 12 theorems, 138 equations.

Table of Contents

  1. introduction
  2. preliminaries
  3. Coarea formula
  4. The first Dirichlet eigenvalues of the p-Laplacian
  5. The first nonzero Neumann eigenvalue of the p-Laplacian
  6. The first nonzero Steklov eigenvalue of the p-Laplacian
  7. Acknowledgments

Key Result

Theorem 1.1

For a bounded domain $\Omega$ in a Riemannian manifold, where $\lambda_{1,p}$ is the first eigenvalue for the Dirichlet $p$-Laplacian on $\Omega$, and $c_p=(p-1)^{p-1}p^{-p}.$

Theorems & Definitions (27)

  • Theorem 1.1: Mazya1960mazya1964Mazya1985mazya2009
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 2.1
  • Lemma 2.2: Green's Formula
  • Lemma 3.1
  • ...and 17 more