Geodesic
Symmetry-breaking in a generalized Wirtinger inequality
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1381-1394

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The search of the optimal constant for a generalized Wirtinger inequality in an interval consists in minimizing the p-norm of the derivative among all functions whose q-norm is equal to 1 and whose (r − 1)-power has zero average. Symmetry properties of minimizers have attracted great attention in mathematical literature in the last decades, leading to a precise characterization of symmetry and asymmetry regions. In this paper we provide a proof of the symmetry result without computer assisted steps, and a proof of the asymmetry result which works as well for local minimizers. As a consequence, we have now a full elementary description of symmetry and asymmetry cases, both for global and for local minima. Proofs rely on appropriate nonlinear variable changes.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017059
Classification : 26D10, 49R05
Keywords: Generalized Wirtinger inequality, generalized Poincaré inequality, best constant in Sobolev inequalities, symmetry of minimizers, variable changes

Ghisi, Marina 1 ; Gobbino, Massimo 1 ; Rovellini, Giulio 1

1 
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     title = {Symmetry-breaking in a generalized {Wirtinger} inequality},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1381--1394},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {4},
     year = {2018},
     doi = {10.1051/cocv/2017059},
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     mrnumber = {3922449},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017059/}
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Ghisi, Marina; Gobbino, Massimo; Rovellini, Giulio. Symmetry-breaking in a generalized Wirtinger inequality. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1381-1394. doi: 10.1051/cocv/2017059

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