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The Weyl law for the Dirichlet Laplacian

Alessandro Pietro Contini

Abstract

The purpose of this paper is to review the asymptotic distribution of eigenvalues of the Dirichlet Laplacian. We introduce and recall all the relevant spectral quantities and provide a proof based on the Fourier Tauberian Theorem.

The Weyl law for the Dirichlet Laplacian

Abstract

The purpose of this paper is to review the asymptotic distribution of eigenvalues of the Dirichlet Laplacian. We introduce and recall all the relevant spectral quantities and provide a proof based on the Fourier Tauberian Theorem.

Paper Structure

This paper contains 7 sections, 18 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Sobolev spaces
  3. The Laplace operator on a Riemannian manifold with boundary
  4. The spectral function
  5. The Hadamard parametrix
  6. The Tauberian theorem and the Weyl law
  7. Historical notes and outlook

Key Result

Lemma 1

If $s=n/2+k+\gamma$ for $k\in\mathbb{N}$ and $\gamma\in(0,1)$, and if $u\in \mathsf{H}^s(\mathbb{R}^n)$, then the equivalence class of $u$ contains an elements of $\mathit{C}^{k,\gamma}(\mathbb{R}^n)$. In other words, the identity map extends to a continuous embedding

Theorems & Definitions (26)

  • Lemma 1: Sobolev embeddings
  • Lemma 2: Rellich compactness
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • Theorem 1
  • proof
  • Lemma 6
  • proof
  • ...and 16 more