A shape optimization problem in cylinders and related overdetermined problems

TL;DR

This paper investigates shape optimization in cylindrical domains, proving the existence of minimizers for torsional energy under volume constraints and revealing new geometric properties and solutions to related overdetermined problems.

Contribution

It establishes the existence of minimizers in cylindrical shape optimization and explores their properties, challenging previous assumptions about trivial solutions in overdetermined problems.

Findings

Existence of minimizers for all fixed volumes in cylindrical domains.

The trivial cylindrical domain is not always the optimal shape.

Solutions with flat level sets are not necessarily optimal for torsional energy.

Abstract

In this paper, we study a shape optimization problem for the torsional energy associated with a domain contained in an infinite cylinder, under a volume constraint. We prove that a minimizer exists for all fixed volumes and show some of its geometric and topological properties. As this issue is closely related to the question of characterizing domains in cylinders that admit solutions to an overdetermined problem, our minimization result allows us to deduce interesting consequences in that direction. In particular, we find that, for some cylinders and some volumes, the ``trivial" domain given by a bounded cylinder is not the only domain where the overdetermined problem has a solution. Moreover, it is not even a minimizer, which indicates that solutions with flat level sets are not always the best candidates for optimizing the torsional energy.