On efficiency and localisation for the torsion function

Abstract

We consider the torsion function for the Dirichlet Laplacian $-\Delta$, and for the Schr\"odinger operator $- \Delta + V$ on an open set $\Omega\subset \R^m$, with Lebesgue measure $0<|\Omega|<\infty$, with a real-valued, non-negative, measurable potential $V.$ We investigate the phenomena of vanishing efficiency and localisation, and large efficiency for the torsion function and the first Dirichlet eigenfunction.Comment: 34 pages, minor corrections and additions to v1 (May 2020