Monotonicity and local uniqueness for the Helmholtz equation
Harrach, B., Pohjola, V., & Salo, M. (2019). Monotonicity and local uniqueness for the Helmholtz equation. Analysis and PDE, 12(7), 2019. https://doi.org/10.2140/apde.2019.12.1741
Published inAnalysis and PDE
© 2019 Mathematical Sciences Publishers
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.
PublisherMathematical Sciences Publishers
Publication in research information system
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Related funder(s)Academy of Finland; European Commission
Funding program(s)Centre of Excellence, AoF; FP7 (EU's 7th Framework Programme)
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about fundingPohjola and Salo were supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Problems Research, grant number 284715) and by an ERC Starting Grant (grant number 307023).
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