Comptes Rendus
Partial Differential Equations
On a minimization problem related to lifting of BV functions with values in S1
Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 855-860.

For uW1,1(Ω,S1) denote by K the set of minimizers of the problem minΩ|uuDϕ|, over ϕBV(Ω) satisfying Ωϕ=0. We show that an extreme point of K must be a lifting of u, up to an additive constant. We also prove a more general result for the case of u in BV(Ω,S1).

Pour uW1,1(Ω,S1) on désigne par K l'ensemble des minimiseurs pour le problème minΩ|uuDϕ| sur l'ensemble des fonctions ϕBV(Ω) vérifiant Ωϕ=0. On démontre que chaque point extrême de K est un relèvement de u, à une constante additive près. On démontre ainsi une généralisation pour le cas uBV(Ω,S1).

Received:
Published online:
DOI: 10.1016/j.crma.2004.09.030
Arkady Poliakovsky 1

1 Department of Mathematics, Technion – I.I.T., 32000 Haifa, Israel
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Arkady Poliakovsky. On a minimization problem related to lifting of BV functions with values in $ {S}^{1}$. Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 855-860. doi : 10.1016/j.crma.2004.09.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.030/

[1] L. Ambrosio; L. Fusco; D. Pallara Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000

[2] H. Brezis, P. Mironescu, A.C. Ponce, W1,1 maps with values into S1, in: S. Chanillo, P. Cordaro, N. Hanges, J. Hounie, A. Meziani (Eds.), Geometric Analysis of PDE and Several Complex Variables, Contemp. Math., American Mathematical Society, in press

[3] J. Dávila; R. Ignat Lifting of BV functions with values in S1, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003), pp. 159-164

[4] M. Giaquinta; G. Modica; J. Soucek Cartesian Currents in the Calculus of Variations, vol. II, Springer, 1998

[5] R. Ignat, The space BV(S2;S1): minimal connection and optimal lifting, preprint

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