Monotonicity and local uniqueness for the Helmholtz equation

TL;DR

This paper develops a monotonicity-based approach for the Helmholtz equation to characterize and distinguish scattering coefficients using partial boundary data, extending inverse problem methods to stationary wave equations.

Contribution

It introduces a monotonicity relation for the Helmholtz equation and a constructive method for localizing scatterers from partial boundary measurements.

Findings

Monotonicity relation holds up to finitely many eigenvalues.

Constructive characterization of scatterers from partial data.

Local uniqueness of the coefficient functions under certain conditions.

Abstract

This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schr\"odinger) equation $(\Delta + k^2 q) u = 0$ in a bounded domain for fixed non-resonance frequency $k>0$ and real-valued scattering coefficient function $q$. We show a monotonicity relation between the scattering coefficient $q$ and the local Neumann-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 where $q_1\geq q_2$ and $q_1\not\equiv q_2$.