Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A minimization problem with free boundary and its application to inverse scattering problems

Pu-Zhao Kow, Mikko Salo, Henrik Shahgholian

Abstract

We study a minimization problem with free boundary, resulting in hybrid quadrature domains for the Helmholtz equation, as well as some application to inverse scattering problem.

A minimization problem with free boundary and its application to inverse scattering problems

Abstract

We study a minimization problem with free boundary, resulting in hybrid quadrature domains for the Helmholtz equation, as well as some application to inverse scattering problem.
Paper Structure (17 sections, 34 theorems, 330 equations)

This paper contains 17 sections, 34 theorems, 330 equations.

Table of Contents

  1. Introduction
  2. Minimization problem
  3. Quadrature domains via minimization
  4. Real analytic quadrature domains
  5. Applications to inverse scattering problems
  6. Existence of minimizers
  7. The Euler-Lagrange equation
  8. Comparison of minimizers
  9. Some properties of local minimizers
  10. Relation with hybrid quadrature domains
  11. Functions of bounded variation and sets with finite perimeter
  12. Further properties of local minimizers
  13. Computations related to Bessel functions
  14. Examples of hybrid k-quadrature domains with real-analytic boundary
  15. Some remarks on null k-quadrature domains
  16. ...and 2 more sections

Key Result

Theorem 1.5

Let $n \ge 2$, and assume $h$ and $g$ are sufficiently regular. If $\mu$ is a non-negative measure on $\mathbb{R}^{n}$ with mass concentrated near a point and $R > 0$, then for each sufficiently small $k>0$ there exists a bounded open domain $D$ in $\mathbb{R}^{n}$ with the boundary $\partial D$ hav

Theorems & Definitions (90)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: See Theorem \ref{['thm:main-quadrature-detail']} for a more detailed statement
  • Remark 1.6
  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['thm:non-scattering-result1']}
  • Lemma 2.3
  • ...and 80 more