Calculus of Variations and Geometric Measure Theory
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Efficiency and localisation for the first Dirichlet eigenfunction

Michiel van den Berg (University of Bristol)

created by malchiodi on 31 Dec 2019

8 jan 2020 -- 10:00   [open in google calendar]

Scuola Normale Superiore, Aula Bianchi

Abstract.

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.

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