Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation

TL;DR

This paper investigates the shape optimization of convex, rotationally symmetric domains for an eigenvalue problem related to optimal insulation, proposing an algorithm and analyzing the optimal shapes.

Contribution

It introduces a new algorithm for shape optimization under convexity and rotational symmetry constraints, and proves the existence of optimal domains for the eigenvalue problem.

Findings

Optimal domains exist under the given constraints.

The proposed algorithm effectively approximates optimal shapes.

Discussion of the shape characteristics of optimal solutions.

Abstract

We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in $\mathbb{R}^3$, the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for the eigenvalue problem in optimal insulation are discussed.