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

Content Provider	Semantic Scholar
Author	Giaquinta, Mariano Mucci, Domenico
Copyright Year	2006
Abstract	Let Y be a smooth compact oriented Riemannian manifold without boundary, and assume that its 1-homology group has no torsion. Weak limits of graphs of smooth maps uk : B n → Y with equibounded total variation give rise to equivalence classes of Cartesian currents in cart1,1(Bn × Y) for which we introduce a natural BV -energy. Assume moreover that the first homotopy group of Y is commutative. In any dimension n we prove that every element T in cart1,1(Bn × Y) can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps uk : B n → Y with total variation converging to the BV -energy of T . As a consequence, we characterize the lower semicontinuous envelope of functions of bounded variations from Bn into Y . Mathematics Subject Classification (2000): 49Q15 (primary); 49Q20 (secondary). In this paper we deal with sequences of smooth maps uk : Bn → Y with equibounded total variation sup k E1,1(uk) < ∞ , E1,1(uk) := ∫ Bn |Duk | dx and their limit points. Here Bn is the unit ball in Rn and Y is a smooth oriented Riemannian manifold of dimension M ≥ 1, isometrically embedded in RN for some N ≥ 2. We shall assume that Y is compact, connected, without boundary. In addition, we assume that the integral 1-homology group H1(Y) := H1(Y;Z) has no torsion. Modulo passing to a subsequence the (n, 1)-currents Guk , integration over the graphs of uk of n-forms with at most one vertical differential, converge to a current T ∈ cart1,1(Bn × Y), see Section 2 below. To every T ∈ cart1,1(Bn × Y) it corresponds a function uT ∈ BV (Bn,Y), i.e., uT ∈ BV (Bn,RN ) such that uT (x) ∈ Y for Ln-a.e. x ∈ Bn , compare [14, Vol. I, Section 4.2] [14, Vol. II, Section 5.4]. Also, the weak convergence Guk ⇀ T yields the convergence uk ⇀ uT weakly in the BV -sense. Received June 12, 2006; accepted in revised form October 17, 2006.
Starting Page	483
Ending Page	548
Page Count	66
File Format	PDF HTM / HTML
Volume Number	5
Alternate Webpage(s)	http://annaliscienze.sns.it/public/pdf/abstracts/2006/4/Abstract_4-2006.pdf
Alternate Webpage(s)	http://archive.numdam.org/article/ASNSP_2006_5_5_4_483_0.pdf
Alternate Webpage(s)	http://cvgmt.sns.it/papers/giamuc05b/giamuc05b.pdf
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article