Monotonicity and local uniqueness for the Helmholtz equation

Harrach, Bastian; Pohjola, Valter; Salo, Mikko

doi:10.2140/apde.2019.12.1741

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Monotonicity and local uniqueness for the Helmholtz equation

Bastian Harrach, Valter Pohjola and Mikko Salo

Vol. 12 (2019), No. 7, 1741–1771

DOI: 10.2140/apde.2019.12.1741

Abstract

This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schrödinger) equation $(Δ + k^{2} q) u = 0$ 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 $q_{1}$ and $q_{2}$ can be distinguished by partial boundary data if there is a neighborhood of the boundary part where $q_{1} \geq q_{2}$ and $q_{1} ≢ q_{2}$ .

Keywords

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

Mathematical Subject Classification 2010

Primary: 35R30

Secondary: 35J05

Milestones

Received: 27 November 2017

Revised: 20 July 2018

Accepted: 20 November 2018

Published: 22 July 2019

Authors

Bastian Harrach
Institute of Mathematics Goethe University Frankfurt Frankfurt Germany

Valter Pohjola
Research Unit of Mathematical Sciences University of Oulu Oulu Finland

Mikko Salo
Department of Mathematics and Statistics University of Jyväskylä Jyväskylä Finland

Vol. 12, No. 7, 2019