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$L^p$ continuity of eigenprojections for 2-d Dirichlet Laplacians under perturbations of the domain

Ryan L. Acosta Babb, James C. Robinson

Abstract

We generalise results by Lamberti and Lanza de Cristoforis (2005) concerning the continuity of projections onto eigenspaces of self-adjoint differential operators with compact inverses as the (spatial) domain of the functions is perturbed in $\mathbb{R}^2$. Our main case of interest is the Dirichlet Laplacian. We extend these results from bounds from $H_0^1$ to $H_0^1$ to bounds from $L^p$ to $L^p$, under the assumption that $(-Δ^{-1}-z)^{-1}$ is $L^p$ bounded when $z$ lies outside of the spectrum of $-Δ^{-1}$. We show that this assumption is met if the initial domain is a square or a rectangle.

$L^p$ continuity of eigenprojections for 2-d Dirichlet Laplacians under perturbations of the domain

Abstract

We generalise results by Lamberti and Lanza de Cristoforis (2005) concerning the continuity of projections onto eigenspaces of self-adjoint differential operators with compact inverses as the (spatial) domain of the functions is perturbed in . Our main case of interest is the Dirichlet Laplacian. We extend these results from bounds from to to bounds from to , under the assumption that is bounded when lies outside of the spectrum of . We show that this assumption is met if the initial domain is a square or a rectangle.
Paper Structure (10 sections, 19 theorems, 102 equations, 2 figures)

This paper contains 10 sections, 19 theorems, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The inverse of the Laplacian
  4. Continuity and splitting of eigenvalues
  5. Bounds on the norms of the operators
  6. Auxiliary bounds
  7. Bounds on $T-T_\varphi$
  8. Continuity of projections under domain perturbations
  9. Resolvent bounds for the square
  10. Conclusions and open problems

Key Result

Theorem 2

Let $F\subset \mathbb{N}$ be a finite set of indices that does not split eigenvalues of $T$ or $\widetilde{T}$. Then, there is a positive constant $C = C(T,\widetilde{T},F)$ such that

Figures (2)

  • Figure 1: Graph of the smooth cut-off $\rho_z$ centred at $\left\vert\xi\right\vert=\frac{1}{\sqrt{z}}.$
  • Figure 2: The "box" $F_6$ splits the eigenvalue $\lambda=50\pi^2$. The region $\Gamma_6$ is denoted by the solid blue arc.

Theorems & Definitions (45)

  • Definition 1
  • Theorem 2: Lamberti2005
  • Remark
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Remark
  • Definition 6
  • Remark
  • Definition 7
  • ...and 35 more