Revisiting the probe and enclosure methods

TL;DR

This paper revisits and extends the probe and enclosure methods for inverse obstacle problems governed by the stationary Schrödinger equation, providing foundational estimates and demonstrating the method's applicability to penetrable obstacles with various boundary regularities.

Contribution

It establishes new estimates for the indicator functions and extends the enclosure method to penetrable obstacles with Lipschitz boundaries in absorbing media.

Findings

Derived natural estimates for Dirichlet and Neumann data

Established the enclosure method for penetrable obstacles

Demonstrated applicability to obstacles with Lipschitz boundaries

Abstract

This paper is concerned with reconstruction issue of inverse obstacle problems governed by partial differential equations and consists of two parts. (i) The first part considers the foundation of the probe and enclosure methods for an impenetrable obstacle embedded in a medium governed by the stationary Schr\"odinger equation. Under a general framework, some natural estimates for a quantity computed from a pair of the Dirichlet and Neumann data on the outer surface of the body occupied by the medium are given. The estimates enables us to derive almost immediately the necessary asymptotic behaviour of indicator functions for both methods. (ii) The second one considers the realization of the enclosure method for a penetrable obstacle embedded in an absorbing medium governed by the stationary Schr\"odinger equation. The unknown obstacle considered here is modeled by a perturbation term added to the background complex-valued $L^{\infty}$-potential. Under a jump condition on the term across the boundary of the obstacle and some kind of regularity for the obstacle surface including Lipschitz one as a special case, the enclosure method using infinitely many pairs of the Dirichlet and Neumann data is established.