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

Abstract

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