Monotonicity in inverse obstacle scattering on unbounded domains

We consider an inverse obstacle scattering problem for the Helmholtz equation with obstacles that carry mixed Dirichlet and Neumann boundary conditions. We discuss far field operators that map superpositions of plane wave incident fields to far field patterns of scattered waves, and we derive monotonicity relations for the eigenvalues of suitable modifications of these operators. These monotonicity relations are then used to establish a novel characterization of the support of mixed obstacles in terms of the corresponding far field operators. We apply this characterization in reconstruction schemes for shape detection and object classification, and we present numerical results to illustrate our theoretical findings.