On the relaxation of variational integrals in metric Sobolev spaces

Omar Anza Hafsa (1, 2) , Jean-Philippe Mandallena (1, 2)

1 M3 - Mathématiques et Modélisations en Mécanique
2 MIPA - Mathématiques, Informatique, Physique, et Applications / Université de Nîmes

We give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral representation for relaxed variational functionals of variational integrals of the calculus of variations in the setting of metric measure spaces. We prove integral representation theorems, both in the convex and non-convex case, which extend and complete previous results in the setting of euclidean measure spaces to the setting of metric measure spaces. We also show that these integral representation theorems can be applied in the setting of Cheeger-Keith's differentiable structure.

Mots clés: Relaxation; variational integral; Sobolev spaces with respect to a metric measure space; integral representation; quasiconvexification with respect to a measure; differentiable structure for metric measure spaces

Domaines: Analyse classique [math.CA]

Omar Anza Hafsa, Jean-Philippe Mandallena. On the relaxation of variational integrals in metric Sobolev spaces. Advances in Calculus of Variation, 2014, 8 (1), pp.69-91. ⟨10.1515/acv-2013-0207⟩. ⟨hal-00959117⟩