Hokkaido Mathematical Journal


Two sides of probe method and obstacle with impedance boundary condition.

Hokkaido Mathematical Journal, 35 (2006) pp.659-681




An inverse boundary value problem for the Helmholtz equation in a bounded domain is considered. The problem is to extract information about an unknown obstacle embedded in the domain with unknown impedance boundary condition (the Robin condition) from the associated Dirichlet-to-Neumann map. The main result is a characterization of the unknown obstacle via the sequences that are constructed by the Dirichletto- Neumann map, under smallness conditions on the wave number and the upper bound of the impedance. Moreover two alternative simple proofs of a previous result of Cheng- Liu-Nakamura which are based on only some energy estimates, an analysis of the blowup of the energy of so-called reflected solutions and an application of the enclosure method to the problem are also given.

Uncontrolled Keywordsinverse obstacle scattering problem, probe method, Poincar\'e inequality, enclosure method, impedance boundary condition, blowup, obstacle, indicator function