Vol. 12, No. 7, 2019

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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
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 q1 q2 and q1q2.

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