Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the geometry of star domains and the spectra of Hodge-Laplace operators under domain approximation

Martin Werner Licht

Abstract

We study the eigenvalues of Hodge--Laplace operators over bounded convex domains. Generalizing contributions by Guerini and Savo, we show that the Poincaré--Friedrichs constants of the Sobolev de~Rham complexes increase with the degree of the forms. We establish that the spectra of Hodge--Laplace operators converge when the domain is approximated by a sequence of convex domains up to a bi-Lipschitz deformation of the boundary. As preparatory work with independent relevance, we study the gauge function and the reciprocal radial function of convex sets, providing new proofs for Lipschitz estimates by Vrećica and Toranzos for the reciprocal radial function and improving a Lipschitz estimate for the gauge function due to Beer.

On the geometry of star domains and the spectra of Hodge-Laplace operators under domain approximation

Abstract

We study the eigenvalues of Hodge--Laplace operators over bounded convex domains. Generalizing contributions by Guerini and Savo, we show that the Poincaré--Friedrichs constants of the Sobolev de~Rham complexes increase with the degree of the forms. We establish that the spectra of Hodge--Laplace operators converge when the domain is approximated by a sequence of convex domains up to a bi-Lipschitz deformation of the boundary. As preparatory work with independent relevance, we study the gauge function and the reciprocal radial function of convex sets, providing new proofs for Lipschitz estimates by Vrećica and Toranzos for the reciprocal radial function and improving a Lipschitz estimate for the gauge function due to Beer.

Paper Structure

This paper contains 13 sections, 24 theorems, 207 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Domains and their convex kernel
  3. Star domains as graph domains
  4. Positively homogeneous scalars
  5. Bi-Lipschitz parametrizations
  6. Oriented distance functions of star domains
  7. Smooth approximation of star domains
  8. Analysis of the de Rham complex
  9. Closed range condition and potential operators
  10. Gaffney inequalities and consequences
  11. Sobolev--Poincaré inequalities
  12. Approximation of Poincaré--Friedrichs constants and Spectra
  13. Vector calculus

Key Result

Lemma 2.1

Suppose that ${\Omega} \subseteq {\mathbb R}^n$ is a bounded domain that is star-shaped with respect to a ball $B$. Then ${\Omega}$ is a Lipschitz domain.

Figures (3)

  • Figure 1: An example for domain that is star-shaped with respect to an interior ball, centered at $x_{0}$ and of radius $\rho$, and which is contained within some ball of radius $R$. Note that these two balls are not necessarily concentric.
  • Figure 2: Illustration of the geometric situation in the proof of Lemma \ref{['lemma:stardomainsarestargraphs']}. The angle $\gamma$ is the angle of convex cone spanned by $x$ and the interior ball $B_{\rho}({0})$.
  • Figure 3: Illustration of the geometric situation in the proof of Lemma \ref{['lemma:magnitudefunctionsarelipschitz']}

Theorems & Definitions (49)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 39 more