Compactness of $M$-uniform domains and optimal thermal insulation problems

TL;DR

This paper investigates optimal shape design for heat insulation within $M$-uniform domains, proving existence of solutions and stability of spherical shapes, contributing to the mathematical understanding of thermal insulation optimization.

Contribution

It establishes the existence of optimal shapes in $M$-uniform domains and demonstrates that balls are stable solutions for the heat insulation problem.

Findings

Existence of optimal shapes in $M$-uniform domains.

Balls are stable solutions for the heat insulation problem.

Provides mathematical foundation for shape optimization in thermal insulation.

Abstract

In this paper, we will consider an optimal shape problem of heat insulation introduced by Bucur-Buttazzo-Nitsch. We will establish the existence of optimal shapes in the class of $M$-uniform domains. We will also show that balls are stable solutions of the optimal heat insulation problem.