Calculus of Variations and Geometric Measure Theory
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M. Ghisi - M. Gobbino - G. Rovellini

Symmetry-breaking in a generalized Wirtinger inequality

created by gobbino on 04 Aug 2017

[BibTeX]

preprint

Inserted: 4 aug 2017

Year: 2017

ArXiv: 1705.00427 PDF

Abstract:

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.

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