On some variational problems involving capacity, torsional rigidity, perimeter and measure

TL;DR

This paper studies the existence of optimal convex sets in higher dimensions that maximize the product of torsional rigidity and capacity under measure and perimeter constraints.

Contribution

It establishes the existence of maximizers for a class of variational problems involving capacity and torsional rigidity with geometric constraints.

Findings

Proved existence of maximizers under various constraints.

Extended classical results to higher dimensions and different capacities.

Provided conditions for optimal convex sets in the variational problems.

Abstract

We investigate the existence of a maximiser among open, bounded, convex sets in $\R^d,\,d\ge 3$ for the product of torsional rigidity and Newtonian capacity (or logarithmic capacity if $d=2$), with constraints involving Lebesgue measure or a combination of Lebesgue measure and perimeter.