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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2005-05-30
Published online: 2005-07-05

DOI: 10.1016/j.crma.2005.06.009

Author's affiliations:

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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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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