Dimension bounds in monotonicity methods for the Helmholtz equation

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 $q_1\le q_2$, then the corresponding Neumann-to-Dirichlet operators satisfy $\Lambda (q_1)\le \Lambda (q_2)$ 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.

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• Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem
• Simultaneous recovery of piecewise analytic coefficients in a semilinear elliptic equation
• Monotonicity Principle in Tomography of Nonlinear Conducting Materials
• The Monotonicity Principle for Magnetic Induction Tomography
• The monotonicity method for the inverse crack scattering problem
• The factorization and monotonicity method for the defect in an open periodic waveguide
• Remarks on the factorization and monotonicity method for inverse acoustic scatterings
• The inverse obstacle problem for nonlinear inclusions