van den Berg: Efficiency and localisation for the first Dirichlet eigenfunction
van den Berg: Bounds are obtained for the efficiency or mean to peak ratio $E(\Omega)$ for the first Dirichlet eigenfunction 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 a wide class of elongating bounded, open, convex and planar sets, (iv) the efficiency of any quadrilateral with perpendicular diagonals of lengths $1$, and $n$ respectively is $O(n^{-2/3}\log n)$ as $n\rightarrow\infty$, and (v) the efficiency of $\big\{(x_1,x_2):(2|n^{-1}x_1|\big)^{\alpha}+(2|x_2|)^{\alpha}<1\big\}, 1\le \alpha<\infty,$ is $O\big(n^{-2/(\alpha+2)}(\log n)^{\max\{1/\alpha,1/2\}}\big),\, n\rightarrow\infty.$ This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.Joint work with F. Della Pietra, G. di Blasio, N. Gavitone. http://cvgmt.sns.it/seminar/724/
When
Wed Jan 8, 2020 9am – 10am Coordinated Universal Time
