Renormalized solutions of the fractional Laplace equation

Nathaël Alibaud; Boris Andreianov; Mostafa Bendahmane

doi:10.1016/j.crma.2010.05.006

Partial Differential Equations

Renormalized solutions of the fractional Laplace equation
[Solutions renormalisées de l'équation de Laplace fractionnaire]

Présenté par : Jean-Michel Bony

Nathaël Alibaud ^1,² ; Boris Andreianov ¹ ; Mostafa Bendahmane ³

¹ Laboratoire de mathématiques, UMR CNRS 6623, 16, route de Gray, 25030 Besançon cedex, France
² École nationale supérieure de mécanique et des microtechniques, 26 chemin de l'Épitaphe, 25030 Besançon cedex, France
³ Institut de mathématiques de Bordeaux, université Bordeaux 2, 3ter, place de la Victoire, 33076 Bordeaux, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 759-762.

Résumé
Abstract

Nous introduisons une notion de solution renormalisée pour les problèmes du genre $β (u) + {(- Δ)}^{s / 2} u ∋ f$ in $R^{n}$ , $f \in L^{1} (R^{n})$ . Ici β est un graphe maximal monotone dans $R$ , et ${(- Δ)}^{s / 2}$ , $s \in (0, 2)$ , est l'opérateur de Laplace fractionnaire qui est un représentant type des diffusions de Lévy. Nous montrons que le problème est bien posé dans le cadre des solutions renormalisées. Le problème de Cauchy pour l'équation d'évolution associée peut alors se traiter par les techniques de semigroupes.

We define renormalized solutions for the problems of the kind $β (u) + {(- Δ)}^{s / 2} u ∋ f$ in $R^{n}$ , $f \in L^{1} (R^{n})$ . Here β is a maximal monotone graph in $R$ , and ${(- Δ)}^{s / 2}$ , $s \in (0, 2)$ , is the fractional Laplace operator which is a particular case of Lévy diffusions. We prove well-posedness in the framework of renormalized solutions. Then the Cauchy problem for the associated evolution equations can be solved using the Crandall–Liggett semigroup technique.

Reçu le : 2010-05-11
Accepté le : 2010-05-31
Publié le : 2010-06-23

DOI : 10.1016/j.crma.2010.05.006

Affiliations des auteurs :

Nathaël Alibaud ^{1, 2} ; Boris Andreianov ¹ ; Mostafa Bendahmane ³

¹ Laboratoire de mathématiques, UMR CNRS 6623, 16, route de Gray, 25030 Besançon cedex, France
² École nationale supérieure de mécanique et des microtechniques, 26 chemin de l'Épitaphe, 25030 Besançon cedex, France
³ Institut de mathématiques de Bordeaux, université Bordeaux 2, 3ter, place de la Victoire, 33076 Bordeaux, France

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     title = {Renormalized solutions of the fractional {Laplace} equation},
     journal = {Comptes Rendus. Math\'ematique},
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     doi = {10.1016/j.crma.2010.05.006},
     language = {en},
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Nathaël Alibaud; Boris Andreianov; Mostafa Bendahmane. Renormalized solutions of the fractional Laplace equation. Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 759-762. doi : 10.1016/j.crma.2010.05.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.05.006/

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