Some properties of the torsion function with Robin boundary conditions

Abstract

In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the L^{\infty} and L^p norms, for p\ge 1 , of the torsion function, seen as a functional on a bounded simply connected open set \Omega\subset\mathbb{R}^n , and prove that the balls are critical shapes for these functionals, when the volume of \Omega is preserved.

A Rigidity Result for the Robin Torsion Problem

“…and, so, (13) follows. Finally, we notice that, if v is the solution to (5), then all the inequalities above are equalities, and, consequently, we have (14).…”

“…As a particular case of the above result, if we take r = 0, we have Corollary 1 Let ⊂ R 2 be an open, bounded and Lipschitz set and let be the ball, centered at the origin, having the same measure of . Let u be the solution to (1) and let v be the solution to (5). If u m = v m , and u M = v M , then…”

“…It is still an open problem to establish if, for p ∈ (1, +∞), the ball maximizes the L p norm of the torsion function among open, bounded and Lipschitz sets (see [3,Open Problem 1]). A first evidence in this direction is provided in [5], where it is proved that the ball is a critical shape for every L p norm in dimension n > 2.…”

“…Theorem 1 Let ⊂ R 2 be an open, bounded and Lipschitz set and let be the ball centered at the origin and having the same measure as . Let u be the solution to (1) and let v be the solution to (5). If u (x) = v(x) for all x ∈ , then…”