Volume 24, Number 4, October–December 2018
|Page(s)||1381 - 1394|
|Published online||24 October 2018|
Symmetry-breaking in a generalized Wirtinger inequality
Università degli Studi di Pisa,
Dipartimento di Matematica,
2 Università degli Studi di Pisa, Dipartimento di Ingegneria Civile e Industriale, Pisa, Italy
3 Scuola Normale Superiore, Classe di Scienze, Pisa, Italy
* Corresponding author: firstname.lastname@example.org
Accepted: 30 August 2017
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.
Mathematics Subject Classification: 26D10 / 49R05
Key words: Generalized Wirtinger inequality / generalized Poincaré inequality / best constant in Sobolev inequalities / symmetry of minimizers / variable changes
© EDP Sciences, SMAI 2018
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