## Analysis & PDE

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

### Monotonicity and local uniqueness for the Helmholtz equation

#### 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 $q1≢q2$.

#### Article information

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

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

https://projecteuclid.org/euclid.apde/1564538423

Digital Object Identifier
doi:10.2140/apde.2019.12.1741

Mathematical Reviews number (MathSciNet)
MR3986540

#### 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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