Dimension bounds in monotonicity methods for the Helmholtz equation

Bastian Harrach; Valter Pohjola; Mikko Salo

doi:10.1137/19M1240708

paper

Dimension bounds in monotonicity methods for the Helmholtz equation

arXiv:1901.08495 · doi:10.1137/19M1240708

Abstract

The article [HPS] 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 $q_{1} \leq q_{2}$ , then the corresponding Neumann-to-Dirichlet operators satisfy $Λ (q_{1}) \leq Λ (q_{2})$ up to a finite dimensional subspace. Here we improve the bounds for the dimension of this space. In particular, if $q_{1}$ and $q_{2}$ have the same number of positive Neumann eigenvalues, then the finite dimensional space is trivial.