Comptes Rendus
no. 9
Partial differential equations/Functional analysis
Norm-resolvent convergence for elliptic operators in domain with perforation along curve

Presented by: the Editorial Board

Denis Borisov1, 2; Giuseppe Cardone3; Tiziana Durante4
1 Institute of Mathematics CS USC RAS, Chernyshevsky str. 112, Ufa, 450008, Russian Federation
2 Bashkir State Pedagogical University, October St. 3a, Ufa, 450000, Russian Federation
3 University of Sannio, Department of Engineering, Corso Garibaldi, 107, 82100 Benevento, Italy
4 University of Salerno, Department of Information and Electrical Engineering and Applied Mathematics, Via Giovanni Paolo II, 132, 84084, Fisciano (SA), Italy
Comptes Rendus. Mathématique, Volume 352 (2014) no. 9, pp. 679-683.

We consider an infinite strip perforated along a curve by small holes. In this perforated domain, we consider a scalar second-order elliptic differential operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic, we describe possible homogenized problems and prove the norm-resolvent convergence of the perturbed operator to a homogenized one. We also provide estimates for the rate of the convergence.

On considère une bande infinie avec une famille de petits trous placés le long d'une courbe. Dans ce domaine perforé, on étudie un opérateur scalaire elliptique du second ordre, avec des conditions aux limites classiques aux bords des trous. En supposant que l'emplacement des trous n'est pas périodique, on décrit les problèmes homogénéisés possibles et on démontre la convergence au sens de la norme de la résolvante des opérateurs perturbés vers les opérateurs homogénéisés. On obtient également des estimées pour le taux de convergence.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.07.003

Denis Borisov 1, 2; Giuseppe Cardone 3; Tiziana Durante 4

1 Institute of Mathematics CS USC RAS, Chernyshevsky str. 112, Ufa, 450008, Russian Federation
2 Bashkir State Pedagogical University, October St. 3a, Ufa, 450000, Russian Federation
3 University of Sannio, Department of Engineering, Corso Garibaldi, 107, 82100 Benevento, Italy
4 University of Salerno, Department of Information and Electrical Engineering and Applied Mathematics, Via Giovanni Paolo II, 132, 84084, Fisciano (SA), Italy 
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Denis Borisov; Giuseppe Cardone; Tiziana Durante. Norm-resolvent convergence for elliptic operators in domain with perforation along curve. Comptes Rendus. Mathématique, Volume 352 (2014) no. 9, pp. 679-683. doi : 10.1016/j.crma.2014.07.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.07.003/

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