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Some results on minimizers and stable solutions of a variational problem

Published online by Cambridge University Press:  10 June 2011

ALBERTO FARINA and
ENRICO VALDINOCI
ALBERTO FARINA
Affiliation:
LAMFA – CNRS UMR 6140, Université de Picardie Jules Verne, Faculté des Sciences, 33, rue Saint-Leu, 80039 Amiens CEDEX 1, France (email: alberto.farina@u-picardie.fr)
ENRICO VALDINOCI
Affiliation:
Università di Roma Tor Vergata, Dipartimento di Matematica, via della ricerca scientifica, 1, I-00133 Rome, Italy (email: enrico@mat.uniroma3.it)
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Abstract

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We consider the functional \[ \int \frac {|\nabla u|^2}{2}+F(x,u)\,dx \] in a periodic setting. We discuss whether the minimizers or the stable solutions satisfy some symmetry or monotonicity properties, with special emphasis on the autonomous case when F is x-independent. In particular, we give an answer to a question posed by Victor Bangert when F is autonomous in dimension n≤3 and in any dimension for non-zero rotation vectors.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012, pp. 1302 - 1312
Copyright
Copyright © Cambridge University Press 2011

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