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The BV-energy of maps into a manifold : relaxation and density results

Mariano Giaquinta;  Domenico Mucci — 2006

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let  𝒴   be a smooth compact oriented riemannian manifoldwithout boundary, and assume that its 1 -homology group has notorsion. Weak limits of graphs of smooth maps  u k : B n → 𝒴   with equibounded total variation give riseto equivalence classes of cartesian currents in  cart 1 , 1 ( B n 𝒴 )   for which we introduce a natural B V -energy.Assume moreover that the first homotopy group of   𝒴   iscommutative. In any dimension   n   we prove that every element  T   in   cart 1 , 1 ( B n 𝒴 )   can be approximatedweakly in the sense of currents by a sequence of graphs...

Weak and strong density results for the Dirichlet energy

Mariano Giaquinta;  Domenico Mucci — 2004

Journal of the European Mathematical Society

Let 𝒴 be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in B n × 𝒴 with equibounded Dirichlet energies, B n being the unit ball in ℝ n . More precisely, weak limits of graphs of smooth maps u k : B n → 𝒴 with equibounded Dirichlet integral give rise to elements of the space cart 2 , 1 ( B n × 𝒴 ) (cf. [4], [5], [6]). In this paper we prove that every element T in cart 2 , 1 ( B n × 𝒴 ) is the weak limit...

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