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Boundary deformation techniques for Neumann problems for the Helmholtz equation

Manuel Cañizares

TL;DR

This work develops boundary deformation techniques to ensure solvability of the Neumann Helmholtz problem with rough potentials by selecting a domain that avoids Neumann eigenvalues when the energy $λ$ exceeds a threshold $λ^V$. It leverages layer potentials for the forward problem, a calculus of domain perturbations via anticonvective derivatives, and Henry's genericity theorem to show that simple Neumann eigenvalues are generic, enabling perturbations that remove NEVs. The main result guarantees the existence of a bounded $C^3$ domain $Ω$ with ${\rm supp}\,V⊂Ω$ such that the Neumann problem has a unique solution for all admissible forcing when $λ>λ^V$, with a boundary subset $Σ'$ enabling Runge approximation in inverse scattering with partial data. Together, these results provide a principled path to Runge-type density arguments and expand the toolbox for boundary control in Helmholtz-type problems.

Abstract

We adapt boundary deformation techniques to solve a Neumann problem for the Helmholtz equation with rough electric potentials in bounded domains. In particular, we study the dependance of Neumann eigenvalues of the perturbed Laplacian with respect to boundary deformation, and we illustrate how to find a domain in which the Neumann problem can be solved for any energy, if there is some freedom in the choice of the domain. This work is motivated by a Runge approximation result in the context of an inverse problem in point-source scattering with partial data.

Boundary deformation techniques for Neumann problems for the Helmholtz equation

TL;DR

This work develops boundary deformation techniques to ensure solvability of the Neumann Helmholtz problem with rough potentials by selecting a domain that avoids Neumann eigenvalues when the energy exceeds a threshold . It leverages layer potentials for the forward problem, a calculus of domain perturbations via anticonvective derivatives, and Henry's genericity theorem to show that simple Neumann eigenvalues are generic, enabling perturbations that remove NEVs. The main result guarantees the existence of a bounded domain with such that the Neumann problem has a unique solution for all admissible forcing when , with a boundary subset enabling Runge approximation in inverse scattering with partial data. Together, these results provide a principled path to Runge-type density arguments and expand the toolbox for boundary control in Helmholtz-type problems.

Abstract

We adapt boundary deformation techniques to solve a Neumann problem for the Helmholtz equation with rough electric potentials in bounded domains. In particular, we study the dependance of Neumann eigenvalues of the perturbed Laplacian with respect to boundary deformation, and we illustrate how to find a domain in which the Neumann problem can be solved for any energy, if there is some freedom in the choice of the domain. This work is motivated by a Runge approximation result in the context of an inverse problem in point-source scattering with partial data.
Paper Structure (16 sections, 23 theorems, 144 equations, 1 figure)

This paper contains 16 sections, 23 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Application to a Runge approximation
  3. The result
  4. Outline of the paper
  5. Preliminaries
  6. Unique continuation properties
  7. The Scattering Problem
  8. Layer potentials
  9. Perturbation of domains
  10. The approach of $\mathcal{C}^m$ embeddings
  11. Perturbation of a simple eigenvalue
  12. Simple eigenvalues are generic
  13. Henry's genericity theorem
  14. Generic simplicity of eigenvalues
  15. Proof of Theorem \ref{['thm:nev']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $n\geq 3$ and $V$ be a potential of the form V, and fix $\lambda>\lambda^V$. We can find a bounded open domain $\Omega$ of class $\mathcal{C}^3$ such that $\mathrm{supp}\, V\subset \Omega$, in which there exists a unique solution $u\in H^1(\Omega)$ to the problem for any $f_1\in H^{-1}_0(\Omega)\cap L^2(\Omega\setminus\mathrm{supp}\, V)$ and $f_2\in L^2(\partial \Omega)$. Furthermore, let $\S

Figures (1)

  • Figure 1: Schematic representation of an admissible deformation

Theorems & Definitions (46)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Corollary 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 36 more