Translator Disclaimer
2019 Monotonicity and local uniqueness for the Helmholtz equation
Bastian Harrach, Valter Pohjola, Mikko Salo
Anal. PDE 12(7): 1741-1771 (2019). 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+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.

Citation

Download Citation

Bastian Harrach. Valter Pohjola. Mikko Salo. "Monotonicity and local uniqueness for the Helmholtz equation." Anal. PDE 12 (7) 1741 - 1771, 2019. https://doi.org/10.2140/apde.2019.12.1741

Information

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

zbMATH: 07115633
MathSciNet: MR3986540
Digital Object Identifier: 10.2140/apde.2019.12.1741

Subjects:
Primary: 35R30
Secondary: 35J05

Rights: Copyright © 2019 Mathematical Sciences Publishers

JOURNAL ARTICLE
31 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.12 • No. 7 • 2019
MSP
Back to Top