Abstract
Bounds are obtained for the efficiency or mean to peak ratio $E(\Omega)$ for the first Dirichlet eigenfunction (positive) for open, connected sets $\Omega$ with finite measure in Euclidean space $\R^m$. 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 $E(\Omega_n)=O\big(n^{-2/3}\log n\big)$. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.
Highlights
Let be a non-empty open set in Euclidean space Rm; m 2; with boundary @ and finite measure jj
Bounds are obtained for the efficiency or mean to max ratio E./ for the first Dirichlet eigenfunction for open, connected sets with finite measure in Euclidean space Rm
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 n is any quadrilateral with perpendicular diagonals of lengths 1 and n respectively, the sequence of first Dirichlet eigenfunctions localises and E.n/ D O.n 2=3 log n/
Summary
Let be a non-empty open set in Euclidean space Rm; m 2; with boundary @ and finite measure jj. More general results have been obtained in [6] It follows from inequality (3) and the main theorem in that paper that if is a bounded region in Rm, jBj .B/ m=2. It is straightforward to construct sequences .n/ for which .un / is localising and, as a consequence of Lemma 3 and (6), have vanishing efficiency. If R2 is open, bounded and convex, it is always possible to find an isometry of such that this isometric set is horn-shaped: let p and q be points on @ such that jp qj D w./, and p q is perpendicular to the pair of straight parallel lines tangent to @ at both p and q which define the width w./.
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