Journal of Convex Analysis 20 (2013), No. 1, 253--264
Copyright Heldermann Verlag 2013
Best Constants in Poincaré Inequalities for Convex Domains
Luca Esposito
Dip. di Matematica e Informatica, Università di Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy
luesposi@unisa.it
Carlo Nitsch
Dip. di Matematica e Applicazioni, Università di Napoli, Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy
c.nitsch@unina.it
Cristina Trombetti
Dip. di Matematica e Applicazioni, Università di Napoli, Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy
cristina@unina.it
We prove a Payne-Weinberger type inequality for the p-Laplacian Neumann eigenvalues (p ≥ 2). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in Poincaré inequality. The key point is the implementation of a refinement of the classical Pólya-Szegö inequality for the symmetric decreasing rearrangement which yields an optimal weighted Wirtinger inequality.
Keywords: Poincare inequality, p-Laplacian eigenvalues, Neumann boundary conditions.
MSC: 35B05, 47J05, 26D15
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