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On the Lowest Eigenvalue of Laplace Operators with Mixed Boundary Conditions

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Abstract

In this paper we consider a Robin-type Laplace operator on bounded domains. We study the dependence of its lowest eigenvalue on the boundary conditions and its asymptotic behavior in shrinking and expanding domains. For convex domains we establish two-sided estimates on the lowest eigenvalues in terms of the inradius and of the boundary conditions.

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Acknowledgements

I thank Enrico Serra and Paolo Tilli for numerous helpful discussions. The support from the MIUR-PRIN’08 grant for the project “Trasporto ottimo di massa, disuguaglianze geometriche e funzionali e applicazioni” is gratefully acknowledged.

Finally, I would like to thank the anonymous referee for useful remarks and comments that helped me improve the original text, and in particular for suggesting a simpler way of proving (4.2).

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Brescia, Via Branze, 38-25123, Bresica, Italy

    Hynek Kovařík

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  1. Hynek Kovařík
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Correspondence to Hynek Kovařík.

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Communicated by Steven G. Krantz.

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Kovařík, H. On the Lowest Eigenvalue of Laplace Operators with Mixed Boundary Conditions. J Geom Anal 24, 1509–1525 (2014). https://doi.org/10.1007/s12220-012-9383-4

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