On the placement of an obstacle so as to optimize the Dirichlet heat content

TL;DR

This paper proves that for doubly connected domains bounded by two spheres, the heat content is minimized when the spheres are concentric, and extends this to optimal obstacle placement for heat content optimization.

Contribution

It establishes the minimal heat content configuration for spherical domains and generalizes to optimal placement of convex obstacles within larger domains.

Findings

Concentric spheres minimize heat content among spherical domains.

Optimal obstacle placement can maximize or minimize heat content.

Results apply to convex obstacles in various domains.

Abstract

We prove that among all doubly connected domains of R^n (n>=2) bounded by two spheres of given radii, the Dirichlet heat content at any fixed time achieves its minimum when the spheres are concentric. This is shown to be a special case of a more general theorem concerning the optimal placement of a convex obstacle inside some larger domain so as to maximize or minimize the Dirichlet heat content.