Monotonicity and local uniqueness for the Helmholtz equation

By Bastian von Harrach, Valter Pohjola, Mikko Salo. Analysis & PDE, 2017

58 citations recorded by the source corpus.

This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schrodinger) 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 $q_1\not\equiv q_2$.

View Paper: https://www.semanticscholar.org/paper/8ee1b10c5c2fdf13eedb9f9e87208a287e7556bc

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1. Bastian von Harrach
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Citations in the corpus

• Monotonicity-based inversion of the fractional Schr\"odinger equation (2017, methodology): “Note that if H1 = H2 = H and L1, L2 are self-adjoint and compact, this is the same extended Loewner order as in [45].”