The enclosure method for inverse obstacle scattering over a finite time interval: IV. Extraction from a single point on the graph of the response operator

Abstract

Abstract A final and maybe the simplest formulation of the enclosure method applied to inverse obstacle problems governed by partial differential equations in a spatial domain with an outer boundary over a finite time interval is presented. The method employs only a single pair of quite natural Neumann data prescribed on the outer boundary and the corresponding Dirichlet data on the same boundary of the solution of the governing equation over a finite time interval, that is, a single point on the graph of the so-called response operator. It is shown that the methods enables us to extract the distance of a given point outside the domain to an embedded unknown obstacle, that is, the maximum sphere centered at the point whose exterior encloses the unknown obstacle. To make the explanation of the idea clear only an inverse obstacle problem governed by the wave equation is considered.