We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet–Laplacian among open sets 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.

Accepted:

DOI: 10.1051/cocv/2016017

Keywords: Eigenvalues, Dirichlet Laplacian, Fraenkel asymmetry, attainable set

^{1}; Zucco, Davide

^{2, 3}

@article{COCV_2017__23_3_869_0, author = {Mazzoleni, Dario and Zucco, Davide}, title = {Convex combinations of low eigenvalues, {Fraenkel} asymmetries and attainable sets}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {869--887}, publisher = {EDP-Sciences}, volume = {23}, number = {3}, year = {2017}, doi = {10.1051/cocv/2016017}, mrnumber = {3660452}, zbl = {1422.47024}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016017/} }

TY - JOUR AU - Mazzoleni, Dario AU - Zucco, Davide TI - Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 869 EP - 887 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016017/ DO - 10.1051/cocv/2016017 LA - en ID - COCV_2017__23_3_869_0 ER -

%0 Journal Article %A Mazzoleni, Dario %A Zucco, Davide %T Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 869-887 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016017/ %R 10.1051/cocv/2016017 %G en %F COCV_2017__23_3_869_0

Mazzoleni, Dario; Zucco, Davide. Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 869-887. doi : 10.1051/cocv/2016017. http://archive.numdam.org/articles/10.1051/cocv/2016017/

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