2017/05/01 by Marina Ghisi, Massimo Gobbino, Ghisi, Marina +3 · 2 citations
Mathematics · #26D10 #49R05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:26D10 #msc:49R05
paper · pdf · doi:10.48550/arxiv.1705.00427
17 pages
arxiv created 2017/05/01 · arxiv updated 2017/05/02
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.