A method for establishing upper bounds for singular perturbation problems

Arkady Poliakovsky

doi:10.1016/j.crma.2005.06.009

Partial Differential Equations

A method for establishing upper bounds for singular perturbation problems
[Une méthode de construction des bornes supérieures pour des problèmes de perturbation singulières]

Présenté par : Haïm Brezis

Arkady Poliakovsky ¹

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

Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 97-102.

Résumé
Abstract

On présente deux applications d'une nouvelle méthode pour construire des bornes supérieures pour des problèmes de perturbation singulière où interviennent des applications à variation bornée. On applique cette méthode à deux problèmes, l'un du premier ordre et l'autre du second. Le premier est un problème de minimisation lié à la question de relèvement optimal pour des applications à variation bornée à valeurs dans $S^{1}$ . Pour ce problème on démontre un théorème de Γ-convergence. Le second problème concerne la fonctionnelle d'Aviles–Giga, $ɛ \int_{Ω} {| \nabla^{2} v |}^{2} d x + \frac{1}{ɛ} \int_{Ω} {(1 - {| \nabla v |}^{2})}^{2} d x$ , pour laquelle on construit une borne supérieure via une suite de fonctions ayant comme limite une fonction dont le gradient est dans BV.

We present two applications of a new method for proving upper bounds for singular perturbation problems involving maps of bounded variation. The two problems are of first and second order, respectively. The first is a minimization problem, related to the question of optimal lifting for BV-maps with values in $S^{1}$ , for which we prove a Γ-convergence result. The second problem involves the Aviles–Giga functional, $ɛ \int_{Ω} {| \nabla^{2} v |}^{2} d x + \frac{1}{ɛ} \int_{Ω} {(1 - {| \nabla v |}^{2})}^{2} d x$ , for which we construct upper bounds via a sequence of functions whose limit has gradient in BV.

Reçu le : 2005-05-30
Publié le : 2005-07-05

DOI : 10.1016/j.crma.2005.06.009

Affiliations des auteurs :

Arkady Poliakovsky ¹

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

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     title = {A method for establishing upper bounds for singular perturbation problems},
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     pages = {97--102},
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     doi = {10.1016/j.crma.2005.06.009},
     language = {en},
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Arkady Poliakovsky. A method for establishing upper bounds for singular perturbation problems. Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 97-102. doi : 10.1016/j.crma.2005.06.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.009/

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