A constrained shape optimization problem in Orlicz-Sobolev spaces

TL;DR

This paper investigates an optimization problem in Orlicz-Sobolev spaces, focusing on finding optimal boundary and interior shapes with measure constraints, establishing existence and properties of solutions.

Contribution

It introduces a new shape optimization framework in Orlicz-Sobolev spaces, proving existence of minimizers and analyzing boundary and interior shape problems.

Findings

Existence of minimizer profiles and optimal sets.

Properties of extremal shapes and profiles.

Analysis of interior hole optimization with volume constraint.

Abstract

In this manuscript we study the following optimization problem: given a bounded and regular domain $\Omega\subset \mathbb{R}^N$ we look for an optimal shape for the "$\mathrm{W}-$vanishing window" on the boundary with prescribed measure over all admissible profiles in the framework of the Orlicz-Sobolev spaces associated to constant for the "Sobolev trace embedding". In this direction, we establish existence of minimizer profiles and optimal sets, as well as we obtain further properties for such extremals. Finally, we also place special emphasis on analyzing the corresponding optimization problem involving an "$\mathrm{A}-$vanishing hole" (inside the domain) with volume constraint.