Bounds are obtained for the efficiency or mean to peak ratio E(Ω) for the first Dirichlet eigenfunction (positive) for open, connected sets Ω 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 Ωn is any quadrilateral with perpendicular diagonals of lengths 1 and n respectively, then the sequence of first Dirichlet eigenfunctions localises, and E(Ωn) = O n−2/3 log n. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.
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Acknowledgements. The authors acknowledge support by the Leverhulme Trust through Emeritus Fellowship EM-2018-011-9, by GNAMPA of INdAM, and by a MIUR-PRIN 2017 grant “Qualitative and quantitative aspects of nonlinear PDE’s.” Michiel van den Berg wishes to thank Thomas Kappeler for helpful references to the literature.
© 2021 European Mathematical Society.
- first Dirichlet eigenfunction