On a minimization problem related to lifting of BV functions with values in $ {S}^{1}$

Arkady Poliakovsky

doi:10.1016/j.crma.2004.09.030

Partial Differential Equations

On a minimization problem related to lifting of BV functions with values in

S^{1}

Presented by: Haïm Brezis

Arkady Poliakovsky ¹

¹ Department of Mathematics, Technion – I.I.T., 32000 Haifa, Israel

Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 855-860

Abstract (Original language)
Résumé

For $u \in W^{1, 1} (Ω, S^{1})$ denote by K the set of minimizers of the problem $\min \int_{Ω} | u \land \nabla u - D ϕ |$ , over $ϕ \in BV (Ω)$ satisfying $\int_{Ω} ϕ = 0$ . 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 $BV (Ω, S^{1})$ .

Pour $u \in W^{1, 1} (Ω, S^{1})$ on désigne par K l'ensemble des minimiseurs pour le problème $\min \int_{Ω} | u \land \nabla u - D ϕ |$ sur l'ensemble des fonctions $ϕ \in BV (Ω)$ vérifiant $\int_{Ω} ϕ = 0$ . 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 $u \in BV (Ω, S^{1})$ .

Received: 2004-09-28
Published online: 2004-11-11

DOI: 10.1016/j.crma.2004.09.030

Author's affiliations:

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

References
Cited by

[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, $W^{1, 1}$ maps with values into $S^{1}$ , 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 $S^{1}$ , 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 $BV (S^{2}; S^{1})$ : minimal connection and optimal lifting, preprint

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