A uniqueness result for a two-dimensional variational problem
[Un résultat d’unicité pour un problème variationnel en dimension deux]
Lledos, Benjamin1
1 Laboratoire MIPA, Université de Nîmes, Site des Carmes, Place Gabriel Péri, 30000 Nîmes (France)
Annales de l'Institut Fourier, Online first, 33 p.

We investigate the uniqueness of the solutions for a non-strictly convex problem in the Calculus of Variations of the form ∫φ(∇v)-λv. Here, φ is a convex function define on ℝ 2 and λ is Lipschitz continuous. We establish the uniqueness of the solutions when the gradient of λ is small and give some counterexamples when that is not the case. The proof is based on the global Lipschitz regularity of the minimizers and on the study of their level sets.

Nous étudions l’unicité des solutions d’un problème non strictement convexe en calcul des variations de la forme ∫φ(∇v)-λv. Ici, φ est une fonction convexe définie sur ℝ 2 et λ est une fonction lipschitzienne. Nous établissons l’unicité des solutions lorsque le gradient de λ est petit et donnons des contre-exemples lorsque ce n’est pas le cas. La preuve est basée sur la régularité lipschitzienne globale des minimiseurs et sur l’étude de leurs ensembles de niveau.

Reçu le :
Révisé le :
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Première publication :
DOI : 10.5802/aif.3734
Classification : 35A02, 49J45, 49N99
Keywords: uniqueness, level sets, Lipschitz regularity
Mots-clés : unicité, ensembles de niveau, régularité lipschitzienne

Lledos, Benjamin 1

1 Laboratoire MIPA, Université de Nîmes, Site des Carmes, Place Gabriel Péri, 30000 Nîmes (France) 
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Lledos, Benjamin. A uniqueness result for a two-dimensional variational problem. Annales de l'Institut Fourier, Online first, 33 p.

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