Efficiency and localisation for the first Dirichlet eigenfunction

Abstract

Bounds are obtained for the efficiency or mean to max ratio $E(\Omega )$ for the first Dirichlet eigenfunction (positive) for open, connected sets $\Omega$ with finite measure in Euclidean space ${\mathbb{R}}^m$. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if ${\Omega }_n$ is any quadrilateral with perpendicular diagonals of lengths $1$ and $n$ respectively, then the sequence of first Dirichlet eigenfunctions localises and $E({\Omega }_n)=O(n^{-2/3}\log n)$. This disproves some claims in the literature. A key technical tool is the Feynman–Kac formula.