Efficiency and localisation for the first Dirichlet eigenfunction
| Autor: | Berg, Michiel van den, Della Pietra, Francesco, Di Blasio, Giuseppina, Gavitone, Nunzia |
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| Rok vydání: | 2019 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 18 pages |
| Databáze: | arXiv |
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