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.

Accepted:

DOI: 10.1051/cocv/2017059

Keywords: Generalized Wirtinger inequality, generalized Poincaré inequality, best constant in Sobolev inequalities, symmetry of minimizers, variable changes

^{1}; Gobbino, Massimo

^{1}; Rovellini, Giulio

^{1}

@article{COCV_2018__24_4_1381_0, author = {Ghisi, Marina and Gobbino, Massimo and Rovellini, Giulio}, 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}, zbl = {1416.26033}, mrnumber = {3922449}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017059/} }

TY - JOUR AU - Ghisi, Marina AU - Gobbino, Massimo AU - Rovellini, Giulio TI - Symmetry-breaking in a generalized Wirtinger inequality JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1381 EP - 1394 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017059/ DO - 10.1051/cocv/2017059 LA - en ID - COCV_2018__24_4_1381_0 ER -

%0 Journal Article %A Ghisi, Marina %A Gobbino, Massimo %A Rovellini, Giulio %T Symmetry-breaking in a generalized Wirtinger inequality %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1381-1394 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017059/ %R 10.1051/cocv/2017059 %G en %F COCV_2018__24_4_1381_0

Ghisi, Marina; Gobbino, Massimo; Rovellini, Giulio. Symmetry-breaking in a generalized Wirtinger inequality. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1381-1394. doi : 10.1051/cocv/2017059. http://archive.numdam.org/articles/10.1051/cocv/2017059/

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