Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets

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dc.contributor.areaMathematicsen_US
dc.contributor.authorMazzoleni, Dario
dc.contributor.authorZucco, Davide
dc.date.accessioned2015-12-09T15:55:27Z
dc.date.available2015-12-09T15:55:27Z
dc.date.issued2015
dc.description.abstractWe consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open set of $R^N$ of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it is possible to obtain informations on the geometry of the minimizers of convex combinations: we study, in particular, when these minimizers are no longer convex, and the optimality of balls. As an application of our results we study the boundary of the attainable set for the Dirichlet spectrum. Our techniques involve symmetry results à la Serrin, explicit constants in quantitative inequalities, as well as a purely geometrical problem: the minimization of the Fraenkel 2-asymmetry among convex sets of fixed measure.en_US
dc.identifier.sissaPreprint60/2015/MATE
dc.identifier.urihttps://openscience.sissa.it/handle/1963/35140
dc.language.isoenen_US
dc.miur.area1en_US
dc.publisherSISSA
dc.publisher
dc.subjectEigenvaluesen_US
dc.subjectDirichlet Laplacianen_US
dc.subjectFraenkel asymmetryen_US
dc.subjectattainable seten_US
dc.subject.miurMAT/05en_US
dc.titleConvex combinations of low eigenvalues, Fraenkel asymmetries and attainable setsen_US
dc.typePreprinten_US
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