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Perturbing the principal Dirichlet eigenfunction

Brian Chao, Laurent Saloff-Coste

TL;DR

This work analyzes how perturbations of a bounded inner uniform domain affect the principal Dirichlet eigenfunction in a strictly local regular Dirichlet space. By combining domain monotonicity, intrinsic ultracontractivity, and parabolic Harnack inequalities, the authors establish two-sided bounds $\varphi_V \lesssim \varphi_U \lesssim \varphi_{V_c}$ for domains $V\subseteq U\subseteq V_c$, with constants depending on doubling, Poincaré, and geometric data. They develop caricature methods $\Phi_U$ that are uniformly comparable to $\varphi_U$ in several Euclidean-domain families (triangles, polygons, rounded shapes, ellipsoids), enabling explicit two-sided eigenfunction and heat-kernel estimates. The results extend explicit understanding of diffusion with killing in perturbed domains and have broad applicability to second-order elliptic operators on Euclidean spaces, manifolds with nonnegative Ricci curvature, and Lie groups with polynomial growth. Overall, the paper provides a robust perturbation framework for principal Dirichlet eigenfunctions, linking geometric domain perturbations to sharp spectral and heat-kernel behavior with practical, computable expressions.

Abstract

We study the principal Dirichlet eigenfunction $\varphi_U$ when the domain $U$ is a perturbation of a bounded inner uniform domain in a strictly local regular Dirichlet space. We prove that if $U$ is suitably contained in between two inner uniform domains, then $\varphi_U$ admits two-sided bounds in terms of the principal Dirichlet eigenfunctions of the two approximating domains. The main ingredients of our proof include domain monotonicity properties associated to Dirichlet boundary conditions, intrinsic ultracontractivity estimates, and parabolic Harnack inequality. As an application of our results, we give explicit expressions comparable to $\varphi_U$ for certain domains $U\subseteq \mathbb{R}^n$, as well as improved Dirichlet heat kernel estimates for such domains. We also prove that under a uniform exterior ball condition on $U$, a point achieving the maximum of $\varphi_U$ is separated away from the boundary, complementing a result of Rachh and Steinerberger arXiv:1608.06604. Our principal Dirichlet eigenfunction estimates are applicable to second-order uniformly elliptic operators in Euclidean space, Riemannian manifolds with nonnegative Ricci curvature, and Lie groups of polynomial volume growth.

Perturbing the principal Dirichlet eigenfunction

TL;DR

This work analyzes how perturbations of a bounded inner uniform domain affect the principal Dirichlet eigenfunction in a strictly local regular Dirichlet space. By combining domain monotonicity, intrinsic ultracontractivity, and parabolic Harnack inequalities, the authors establish two-sided bounds for domains , with constants depending on doubling, Poincaré, and geometric data. They develop caricature methods that are uniformly comparable to in several Euclidean-domain families (triangles, polygons, rounded shapes, ellipsoids), enabling explicit two-sided eigenfunction and heat-kernel estimates. The results extend explicit understanding of diffusion with killing in perturbed domains and have broad applicability to second-order elliptic operators on Euclidean spaces, manifolds with nonnegative Ricci curvature, and Lie groups with polynomial growth. Overall, the paper provides a robust perturbation framework for principal Dirichlet eigenfunctions, linking geometric domain perturbations to sharp spectral and heat-kernel behavior with practical, computable expressions.

Abstract

We study the principal Dirichlet eigenfunction when the domain is a perturbation of a bounded inner uniform domain in a strictly local regular Dirichlet space. We prove that if is suitably contained in between two inner uniform domains, then admits two-sided bounds in terms of the principal Dirichlet eigenfunctions of the two approximating domains. The main ingredients of our proof include domain monotonicity properties associated to Dirichlet boundary conditions, intrinsic ultracontractivity estimates, and parabolic Harnack inequality. As an application of our results, we give explicit expressions comparable to for certain domains , as well as improved Dirichlet heat kernel estimates for such domains. We also prove that under a uniform exterior ball condition on , a point achieving the maximum of is separated away from the boundary, complementing a result of Rachh and Steinerberger arXiv:1608.06604. Our principal Dirichlet eigenfunction estimates are applicable to second-order uniformly elliptic operators in Euclidean space, Riemannian manifolds with nonnegative Ricci curvature, and Lie groups of polynomial volume growth.

Paper Structure

This paper contains 19 sections, 37 theorems, 158 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Examples
  3. Preliminaries
  4. Notation
  5. Dirichlet space $(X,d,\mu,\mathcal{E},\mathcal{D}(\mathcal{E}))$
  6. Inner uniform domains
  7. $R$-doubling, $R$-Poincaré inequality, and $R$-parabolic Harnack inequality
  8. Heat kernel, the generator $\mathcal{L}$, and Dirichlet eigenvalues
  9. The associated Hunt process
  10. The domain $V$ and the family of perturbations $V_c$
  11. Proofs of main results
  12. Upper bound for $\varphi_U$
  13. Lower bound for $\varphi_U$ when $c\geq 1$ is close to $1$
  14. Lower bound for $\varphi_U$ when $f(c)$ is bounded above
  15. Proof sketch of results from Section \ref{['examples']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Consider Euclidean space $\mathbb{R}^n$ equipped with the Laplacian and Lebesgue measure $d\mu$. Let $V\subseteq \mathbb{R}^n$ be a bounded $(C_0,c_0)$-inner uniform star-shaped domain with respect to the origin $0\in \mathbb{R}^n$. Let $\varphi_V>0$ denote the principal Dirichlet Laplacian eigenfun

Figures (10)

  • Figure 1: A regular $7$-gon $P$ is perturbed in two different ways; the perturbed domain is $U$.
  • Figure 2: $U$ is obtained by adding three bounded cylinders to a ball.
  • Figure 3: Triangles in $\mathbb{R}^2$ or $\mathbb{S}^2$ with angles $\alpha_1,\alpha_2,\alpha_3$.
  • Figure 4: A triangle $T\subseteq \mathbb{R}^2$ is perturbed by adding a small equilateral triangle of side length $\varepsilon$.
  • Figure 5: A triangle $T$ is perturbed by adding $6$ smaller triangles to one of its sides.
  • ...and 5 more figures

Theorems & Definitions (86)

  • Theorem 1.1
  • Example 2.1
  • Theorem 2.1
  • Example 2.2
  • Theorem 2.2
  • Theorem 2.3
  • Example 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Example 2.4
  • ...and 76 more