Faber-Krahn and Lieb-type inequalities for the composite membrane problem

Giovanni Cupini; Eugenio Vecchi

doi:10.3934/cpaa.2019119

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2019, Volume 18, Issue 5: 2679-2691. Doi: 10.3934/cpaa.2019119

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Faber-Krahn and Lieb-type inequalities for the composite membrane problem

Giovanni Cupini^1, , and
Eugenio Vecchi^2,

1.
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
2.
Dipartimento di Matematica "Guido Castelnuovo", Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Roma, Italy

^* Corresponding author
^* Corresponding author

Received: May 31, 2018

Revised: September 30, 2018

Published: March 2019

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Abstract

The classical Faber-Krahn inequality states that, among all domains with given measure, the ball has the smallest first Dirichlet eigenvalue of the Laplacian. Another inequality related to the first eigenvalue of the Laplacian has been proved by Lieb in 1983 and it relates the first Dirichlet eigenvalues of the Laplacian of two different domains with the first Dirichlet eigenvalue of the intersection of translations of them. In this paper we prove the analogue of Faber-Krahn and Lieb inequalities for the composite membrane problem.

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Mathematics Subject Classification: Primary: 35B06; Secondary: 35P30, 49Q10.

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References

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