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Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

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    0479662 - ÚJF 2018 RIV DE eng J - Journal Article
    Antunes, P. R. S. - Freitas, P. - Krejčiřík, David
    Bounds and extremal domains for Robin eigenvalues with negative boundary parameter.
    Advances in Calculus of Variations. Roč. 10, č. 4 (2017), s. 357-379. ISSN 1864-8258
    R&D Projects: GA ČR(CZ) GA14-06818S
    Institutional support: RVO:61389005
    Keywords : Eigenvalue optimisation * Robin Laplacian * negative boundary parameter * Bareket's conjecture
    Subject RIV: BA - General Mathematics
    OBOR OECD: Pure mathematics
    Impact factor: 1.676, year: 2017

    We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in two and three dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).
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