Dimension bounds in monotonicity methods for the Helmholtz equation

Harrach, Bastian; Pohjola, Valter; Salo, Mikko (2019-07-25)
 
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https://doi.org/10.1137/19M1240708

Harrach, Bastian
Pohjola, Valter
Salo, Mikko
Society for Industrial and Applied Mathematics
25.07.2019

Harrach, Bastian; Pohjola, Valter; Salo, Mikko, Dimension bounds in monotonicity methods for the Helmholtz equation, SIAM J. Math. Anal., 51(4), 2995–3019. https://doi.org/10.1137/19M1240708

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© 2019, Society for Industrial and Applied Mathematics.
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doi:https://doi.org/10.1137/19M1240708
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https://urn.fi/URN:NBN:fi-fe2019092629903
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Abstract

The article [B. Harrach, V. Pohjola, and M. Salo, Anal. PDE] established a monotonicity inequality for the Helmholtz equation and presented applications to shape detection and local uniqueness in inverse boundary problems. The monotonicity inequality states that if two scattering coefficients satisfy q₁ ≤ q₂, then the corresponding Neumann-to-Dirichlet operators satisfy Λ(q₁) ≤ Λ(q₂) up to a finite-dimensional subspace. Here we improve the bounds for the dimension of this space. In particular, if q₁ and q₂ have the same number of positive Neumann eigenvalues, then the finite-dimensional space is trivial.

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