Reconstruction of a source domain from the Cauchy data: II. Three dimensional case

TL;DR

This paper develops a framework for reconstructing sources and obstacles in three-dimensional inverse problems governed by the Helmholtz equation, using the enclosure method and explicit formulas for conical singularities.

Contribution

It introduces a 3D enclosure method framework, provides explicit formulas for conical singularities, and applies these to inverse obstacle problems with inhomogeneous Helmholtz equations.

Findings

Framework for 3D inverse source reconstruction established

Explicit formula for circular cone singularity derived

Application to inverse obstacle problem demonstrated

Abstract

This paper is concerned with reconstruction issue of some typical inverse problems and consists of three parts. First a framework of the enclosure method for an inverse source problem governed by the Helmholtz equation at a fixed wave number in three dimensions is introduced. It is based on the nonvanishing of the coefficient of the leading profile of an oscillatory integral over a domain having a conical singularity. Second an explicit formula of the coefficient for a domain having a circular cone singularity and its implication under the framework are given. Third, an application under the framework to an inverse obstacle problem governed by an inhomogeneous Helmholtz equation at a fixed wave number in three dimensions is given.