Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 7 (2019), 1741-1771.

Monotonicity and local uniqueness for the Helmholtz equation

Bastian Harrach, Valter Pohjola, and Mikko Salo

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Abstract

This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schrödinger) equation (Δ+k2q)u=0 in a bounded domain for fixed nonresonance frequency k>0 and real-valued scattering coefficient function q. We show a monotonicity relation between the scattering coefficient q and the local Neumann-to-Dirichlet operator that holds up to finitely many eigenvalues. Combining this with the method of localized potentials, or Runge approximation, adapted to the case where finitely many constraints are present, we derive a constructive monotonicity-based characterization of scatterers from partial boundary data. We also obtain the local uniqueness result that two coefficient functions q1 and q2 can be distinguished by partial boundary data if there is a neighborhood of the boundary part where q1q2 and q1q2.

Article information

Source
Anal. PDE, Volume 12, Number 7 (2019), 1741-1771.

Dates
Received: 27 November 2017
Revised: 20 July 2018
Accepted: 20 November 2018
First available in Project Euclid: 31 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.apde/1564538423

Digital Object Identifier
doi:10.2140/apde.2019.12.1741

Mathematical Reviews number (MathSciNet)
MR3986540

Subjects
Primary: 35R30: Inverse problems
Secondary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]

Keywords
inverse coefficient problems Helmholtz equation stationary Schrödinger equation monotonicity localized potentials

Citation

Harrach, Bastian; Pohjola, Valter; Salo, Mikko. Monotonicity and local uniqueness for the Helmholtz equation. Anal. PDE 12 (2019), no. 7, 1741--1771. doi:10.2140/apde.2019.12.1741. https://projecteuclid.org/euclid.apde/1564538423


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