On efficiency and localisation for the torsion function

Abstract

We consider the torsion function for the Dirichlet Laplacian $-\Delta$, and for the Schrödinger operator $-\Delta +V$ on an open set $\Omega \subset {\mathbb{R}}^m$ of finite Lebesgue measure $0<|\Omega |<\infty$ with a real-valued, non-negative, measurable potential $V.$ We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.

33 pages. The published version in Potential Analysis (2022) 57, 571--600 has some typos: Theorem 3(i): the first exponent should read $(m-2)/m$; Example 2 Line 2: .... $B(p_{n+1};c{n}^{-\beta })$.....; Formula (109): ${\kappa }^{-1}$ missing after the second inequality