A1 Journal article (refereed)
Monotonicity and local uniqueness for the Helmholtz equation (2019)

Harrach, Bastian; Pohjola, Valter; Salo, Mikko (2019). Monotonicity and local uniqueness for the Helmholtz equation. Analysis and PDE, 12 (7), 2019, 1741-1771. DOI: 10.2140/apde.2019.12.1741

JYU authors or editors

Publication details

All authors or editors: Harrach, Bastian; Pohjola, Valter; Salo, Mikko

Journal or series: Analysis and PDE

ISSN: 2157-5045

eISSN: 1948-206X

Publication year: 2019

Volume: 12

Issue number: 7

Pages range: 2019

Article number: 1741-1771

Publisher: Mathematical Sciences Publishers

Publication country: United States

Publication language: English

DOI: http://doi.org/10.2140/apde.2019.12.1741

Open Access: Publication channel is not openly available

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/65183

Publication is parallel published: https://arxiv.org/abs/1709.08756


This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schrödinger) equation (1 + 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 monotonicitybased 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 q1 6≡ q2.

Keywords: inverse problems

Free keywords: inverse coefficient problems; Helmholtz equation; stationary Schrödinger equation; monotonicity, localized

Contributing organizations

Related projects

Centre of Excellence in Inverse Problems Research
Salo, Mikko
Academy of Finland

Ministry reporting: Yes

Reporting Year: 2019

JUFO rating: 3

Last updated on 2020-18-08 at 13:20