# The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell type initial data

## TL;DR

This paper introduces a new method for inverse obstacle scattering that uses shell-type initial data to determine the smallest enclosing sphere of unknown inclusions within a body, applicable to heat and elastic wave systems.

## Contribution

It presents a novel approach to identify the minimal enclosing domain of unknown inclusions using specially designed heat flux without large parameters, extending to elastic wave systems.

## Key findings

- Successfully determines the smallest enclosing sphere of unknown inclusions.
- Design of a parameter-free heat flux for inverse boundary problems.
- Extension of the method to elastic wave and heat systems.

## Abstract

A simple idea of finding a domain that encloses an unknown discontinuity embedded in a body is introduced by considering an inverse boundary value problem for the heat equation. The idea gives a design of a special heat flux on the surface of the body such that from the corresponding temperature field on the surface one can extract the smallest radius of the sphere centered at an arbitrary given point in the whole space and enclosing unknown inclusions. Unlike before, the designed flux is free from a large parameter. An application of the idea to a coupled system of the elastic wave and heat equations are also given.