Optimal one-dimensional structures for the principal eigenvalue of two-dimensional domains

TL;DR

This paper investigates shape optimization problems for maximizing the principal eigenvalue of two-dimensional domains using one-dimensional structures, with constraints on length and connectivity, providing existence results for relaxed formulations.

Contribution

It introduces new shape optimization models involving one-dimensional stiffeners and establishes existence results for these problems under various constraints.

Findings

Existence of optimal structures for the eigenvalue maximization problem.

Relaxed formulations of the shape optimization problems are developed.

Analysis of constraints like total length and connectivity in the optimization.

Abstract

A shape optimization problem arising from the optimal reinforcement of a membrane by means of one-dimensional stiffeners or from the fastest cooling of a two-dimensional object by means of ``conducting wires'' is considered. The criterion we consider is the maximization of the first eigenvalue and the admissible classes of choices are the one of one-dimensional sets with prescribed total length, or the one where the constraint of being connected (or with an a priori bounded number of connected components) is added. The corresponding relaxed problems and the related existence results are described.