Geodesic
The Minimal Gap Between Λ2(Ω) and Λ¥(Ω) in a Class of Convex Domains
Journal of convex analysis, Tome 15 (2008) no. 3, pp. 507-521.

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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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     author = {M. Belloni and E. Oudet},
     title = {The {Minimal} {Gap} {Between} {\ensuremath{\Lambda}\protect\textsubscript{2}(\ensuremath{\Omega})} and {\ensuremath{\Lambda}\protect\textsubscript{<font} {face="Symbol">{\textyen}</font>}(\ensuremath{\Omega})} in a {Class} of {Convex} {Domains}},
     journal = {Journal of convex analysis},
     pages = {507--521},
     publisher = {mathdoc},
     volume = {15},
     number = {3},
     year = {2008},
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M. Belloni; E. Oudet. The Minimal Gap Between Λ2(Ω) and Λ¥(Ω) in a Class of Convex Domains. Journal of convex analysis, Tome 15 (2008) no. 3, pp. 507-521. http://geodesic.mathdoc.fr/item/JCA_2008_15_3_JCA_2008_15_3_a4/