TY - RPRT
T1 - Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets
Y1 - 2015
A1 - Dario Mazzoleni
A1 - Davide Zucco
AB - We 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.
PB - SISSA
UR - http://urania.sissa.it/xmlui/handle/1963/35140
U1 - 35378
U2 - Mathematics
U4 - 1
U5 - MAT/05
ER -