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On the inner radius of the nonvanishing set for eigenfunctions of complex elliptic operators

Henrik Ueberschaer, Omer Friedland

Abstract

Let $Ω\subset\mathbb{R}^d$ be any open set. We consider solutions of $Hψ_λ=λψ_λ$, $λ\in\mathbb{C}$, where $H$ is an $m$th order complex constant-coefficient elliptic partial differential operator. We prove that either the eigenfunctions satisfy a lower bound on the inner radius of the complement of the zero set of $ψ_λ$ in $Ω$ of order $|λ|^{-1/m}$, or 100% of the $L^2$ mass of $ψ_λ$ concentrates in a boundary layer of width $|λ|^{-1/m}$, as $|λ|\to+\infty$.

On the inner radius of the nonvanishing set for eigenfunctions of complex elliptic operators

Abstract

Let be any open set. We consider solutions of , , where is an th order complex constant-coefficient elliptic partial differential operator. We prove that either the eigenfunctions satisfy a lower bound on the inner radius of the complement of the zero set of in of order , or 100% of the mass of concentrates in a boundary layer of width , as .
Paper Structure (12 sections, 10 theorems, 47 equations)

This paper contains 12 sections, 10 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Motivation and context
  3. Setting and notation
  4. Main results
  5. Two elementary geometric lemmas
  6. A nonvanishing ball from Lipschitz control
  7. Finding a point of large amplitude from an $L^2$ lower bound
  8. A uniform local Lipschitz bound at bounded spectral parameter
  9. A local derivative estimate from FU24
  10. A uniform $C^1$ bound on $B(0,3/4)$
  11. A bounded overlap cover and a mass ratio lemma
  12. Proofs of the main results

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^d$ be open and let $H$ be a homogeneous constant--coefficient elliptic operator of order $m$ satisfying eq:ellipticity. Then there exists a constant $c_{d,H}>0$ (depending only on $d$ and $H$) such that for every $|\lambda|\ge 1$ and every nonzero solution $\psi_\lambda\

Theorems & Definitions (19)

  • Theorem 1.1: Quantitative inradius bound
  • Theorem 1.2: Localized inradius bound
  • Corollary 1.3: Boundary layer concentration
  • Lemma 2.1: Lipschitz nonvanishing ball
  • proof
  • Lemma 2.2: Pointwise lower bound from $L^2$ mass
  • proof
  • Theorem 3.1: Local derivative bound FU24
  • Lemma 3.2: Uniform boundedness of $\mathcal{N}_\lambda$ on compact sets
  • proof
  • ...and 9 more