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Journal of Convex Analysis 15 (2008), No. 3, 507--521
Copyright Heldermann Verlag 2008

The Minimal Gap Between Λ2(Ω) and Λ¥(Ω) in a Class of Convex Domains

Marino Belloni
Dip. di Matematica, Università di Parma, Viale G. P. Usberti 53/A, 43100 Parma, Italy

Edouard Oudet
Laboratoire de Mathematiques, Université de Savoie, Campus Scientifique, 73376 Le-Bourget-du-Lac, France


We consider the minimization problem \begin{eqnarray*} \min_{\Omega\in X}\left(\Lambda_2-\Lambda_\infty\right)(\Omega), \end{eqnarray*} where $\Lambda_2(\Omega)$\ and $\Lambda_\infty(\Omega)$\ are the (square root of the) first eigenvalue of the Laplacian and the first eigenvalue of the $\infty-$Laplacian respectively. $X$ is the class of convex domains with prescribed diameter. We prove existence of a solution, and we provide several geometrical properties of minimizers.

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