An inverse problem for a three-dimensional heat equation in bounded regions with several convex cavities

TL;DR

This paper investigates an inverse problem for the 3D heat equation in a bounded region with convex cavities, aiming to determine the minimal broken path length connecting points inside and outside the region based on boundary observations.

Contribution

It introduces a method to find the minimal broken path length in a 3D heat inverse problem with convex cavities using boundary temperature and heat flux data.

Findings

Derived a formula for the minimum broken path length.

Established uniqueness of the inverse problem solution.

Provided a theoretical framework for cavity detection.

Abstract

In this paper, an inverse initial-boundary value problem for the heat equation in three dimensions is studied. Assume that a three-dimensional heat conductive body contains several cavities of strictly convex. In the outside boundary of this body, a single pair of the temperature and heat flux is given as an observation datum for the inverse problem. It is found the minimum length of broken paths connecting arbitrary fixed point in the outside, a point on the boundary of the cavities and a point on the outside boundary in this order, if the minimum path is not line segment.