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Estimates of variational eigenvalues on metric measure spaces

Prashanta Garain, Alexander Ukhlov

TL;DR

This work extends the theory of Neumann variational eigenvalues to general doubling metric measure spaces, addressing nonlinear $(p,q)$-Laplacian-type operators. It formulates the first nontrivial variational Neumann $(p,q)$-eigenvalue as a minimization over the Newtonian-Sobolev space $N^{1,p}(\Omega)$ with a zero-$L^q$-mean constraint and proves existence of minimizers under Sobolev $(p,p^*)$-embedding domain assumptions. A key insight is identifying $\lambda_{p,q}(\Omega)^{-1/p}$ with the best constant in the $(p,q)$-Sobolev-Poincaré inequality, and the work leverages extension operators and compact embeddings to establish the variational framework. The resulting eigenvalue estimates relate spectral data to extension norms, enabling comparisons across nested domains and generalizing Euclidean results to metric-measure spaces.

Abstract

In the article, we study variational eigenvalues on doubling metric measure spaces. We prove existence of minimizers of variational Neumann $(p,q)$-eigenvalues on metric measure spaces and on this base we obtain estimates of Neumann eigenvalues.

Estimates of variational eigenvalues on metric measure spaces

TL;DR

This work extends the theory of Neumann variational eigenvalues to general doubling metric measure spaces, addressing nonlinear -Laplacian-type operators. It formulates the first nontrivial variational Neumann -eigenvalue as a minimization over the Newtonian-Sobolev space with a zero--mean constraint and proves existence of minimizers under Sobolev -embedding domain assumptions. A key insight is identifying with the best constant in the -Sobolev-Poincaré inequality, and the work leverages extension operators and compact embeddings to establish the variational framework. The resulting eigenvalue estimates relate spectral data to extension norms, enabling comparisons across nested domains and generalizing Euclidean results to metric-measure spaces.

Abstract

In the article, we study variational eigenvalues on doubling metric measure spaces. We prove existence of minimizers of variational Neumann -eigenvalues on metric measure spaces and on this base we obtain estimates of Neumann eigenvalues.
Paper Structure (4 sections, 9 theorems, 57 equations)

This paper contains 4 sections, 9 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Sobolev spaces on metric measure spaces and embedding theorems
  3. Variational eigenvalues on metric measure spaces
  4. Estimates of variational eigenvalues

Key Result

Theorem 2.1

Let $\Omega\subset X = (X, \rho, \mu)$ be a domain. Suppose that $\{f_n\}_{n\in\mathbb{N}}$ is a sequence in $L^q(\Omega)$, $1<q<\infty$, and let $f\in L^q(\Omega)$ such that $\|f_n-f\|_{L^q(\Omega)}\to 0$ as $n\to\infty$. Then there exists a subsequence $\{f_{n_k}\}_{n_k\in\mathbb{N}}$ and a functi

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • ...and 2 more