Convergence of Sobolev spaces on varying manifolds |
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Authors: | Guy Bouchitté Giuseppe Buttazzo Ilaria Fragalà |
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Institution: | (1) UFR des Sciences et Techniques, Université de Toulon et du Var, BP 132, 83957 La Garde, Cedex, France;(2) Dipartimento di Matematica, Università di Pisa, Via Buonarroti, 2, 56127 Pisa, Italy;(3) Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20132 Milano, Italy |
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Abstract: | We deal with variational problems on varying manifolds in ℝn. We represent each manifold by a positive measure μ, to which we associate a suitable notion of tangent space Tμ, of mean
curvature H(μ), and of Sobolev spaces with respect to μ on an open subset Ω ⊆ ℝn. We introduce the notions of weak and strong convergence for functions defined on varying manifolds, that is defined μh -a.e., being {μh} a weakly convergent sequence of measures. In this setting, we prove a strong-weak type compactness theorem for the pairs
(Pμ
h H(μh)), where Pμ
h are the projectors onto the tangent spaces Tμ
h. When μh belong to a suitable class of k-dimensional measures, having in particular a prescribed (k−1)-manifold as a boundary, we
enforce this result to study the convergence of energy functionals, possibly with a Dirichlet condition on ∂Ω. We also address
a perspective for optimization problems where the control variable is represented by a manifold with a prescribed boundary. |
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Keywords: | Math Subject Classifications" target="_blank">Math Subject Classifications 49J45 28A33 28A35 28A50 49Q10 |
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