Partial Differential Equations
Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron
[Régularité anisotrope pour le Laplacien et l'opérateur de Maxwell dans un polyèdre]

Présenté par : Philippe G. Ciarlet

Buffa, Annalisa1 ; Costabel, Martin2 ; Dauge, Monique2
1 IMATI-CNR, Pavia, Italy
2 Irmar, Université de Rennes 1, France
Comptes Rendus. Mathématique, Tome 336 (2003) no. 7, pp. 565-570.

Nous choisissons d'étudier le problème de Dirichlet pour le Laplacien et le problème de Maxwell électrique, comme représentants de classes plus larges de problèmes intéressant la modélisation de phénomènes physiques stationnaires. Nous énonçons des résultats de régularité dans deux familles d'espaces de Sobolev à poids : l'une, classique, isotrope, et l'autre, nouvelle, anisotrope, où l'on tient compte de l'hypoellipticité le long des arêtes d'un domaine polyédral.

As representatives of a larger class of elliptic boundary value problems of mathematical physics, we study the Dirichlet problem for the Laplace operator and the electric boundary problem for the Maxwell operator. We state regularity results in two families of weighted Sobolev spaces: A classical isotropic family, and a new anisotropic family, where the hypoellipticity along an edge of a polyhedral domain is taken into account.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00138-9
Buffa, Annalisa 1 ; Costabel, Martin 2 ; Dauge, Monique 2

1 IMATI-CNR, Pavia, Italy
2 Irmar, Université de Rennes 1, France 
@article{CRMATH_2003__336_7_565_0,
     author = {Buffa, Annalisa and Costabel, Martin and Dauge, Monique},
     title = {Anisotropic regularity results for {Laplace} and {Maxwell} operators in a polyhedron},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {565--570},
     publisher = {Elsevier},
     volume = {336},
     number = {7},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00138-9},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/S1631-073X(03)00138-9/}
}
TY  - JOUR
AU  - Buffa, Annalisa
AU  - Costabel, Martin
AU  - Dauge, Monique
TI  - Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 565
EP  - 570
VL  - 336
IS  - 7
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/S1631-073X(03)00138-9/
DO  - 10.1016/S1631-073X(03)00138-9
LA  - en
ID  - CRMATH_2003__336_7_565_0
ER  - 
%0 Journal Article
%A Buffa, Annalisa
%A Costabel, Martin
%A Dauge, Monique
%T Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron
%J Comptes Rendus. Mathématique
%D 2003
%P 565-570
%V 336
%N 7
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/S1631-073X(03)00138-9/
%R 10.1016/S1631-073X(03)00138-9
%G en
%F CRMATH_2003__336_7_565_0
Buffa, Annalisa; Costabel, Martin; Dauge, Monique. Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron. Comptes Rendus. Mathématique, Tome 336 (2003) no. 7, pp. 565-570. doi : 10.1016/S1631-073X(03)00138-9. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00138-9/

[1] Apel, T.; Nicaise, S. The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges, Math. Methods Appl. Sci., Volume 21 (1998), pp. 519-549

[2] Assous, F.; Ciarlet, P. Jr.; Raviart, P.-A.; Sonnendrücker, E. Characterization of the singular part of the solution of Maxwell's equations in a polyhedral domain, Math. Methods Appl. Sci., Volume 22 (1999), pp. 485-499

[3] Birman, M.; Solomyak, M. L2-theory of the Maxwell operator in arbitrary domains, Russian Math. Surveys, Volume 42 (1987) no. 6, pp. 75-96

[4] Bonnet-Ben Dhia, A.-S.; Hazard, C.; Lohrengel, S. A singular field method for the solution of Maxwell's equations in polyhedral domains, SIAM J. Appl. Math., Volume 59 (2000), pp. 2028-2044

[5] A. Buffa, M. Costabel, and M. Dauge, Algebraic convergence for anisotropic edge elements in polyhedral domains, Technical report, IMATI-CNR, 2003, in preparation

[6] Costabel, M.; Dauge, M. General edge asymptotics of solutions of second order elliptic boundary value problems I & II, Proc. Roy. Soc. Edinburgh Sect. A, Volume 123 (1993), pp. 109-184

[7] Costabel, M.; Dauge, M. Singularities of electromagnetic fields in polyhedral domains, Arch. Rational Mech. Anal., Volume 151 (2000), pp. 221-276

[8] Costabel, M.; Dauge, M. Weighted regularization of Maxwell equations in polyhedral domains, Numer. Math., Volume 93 (2002), pp. 239-277

[9] Dauge, M. Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Math., Springer-Verlag, Berlin, 1988

[10] Guo, B.Q. The h-p version of the finite element method for solving boundary value problems in polyhedral domains (Costabel, M.; Dauge, M.; Nicaise, S., eds.), Boundary Value Problems and Integral Equations in Nonsmooth Domains, Luminy, 1993, Dekker, New York, 1995, pp. 101-120

[11] Kondrat'ev, V.A. Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., Volume 16 (1967), pp. 227-313

[12] Maz'ya, V.G.; Rossmann, J. Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten, Math. Nachr., Volume 138 (1988), pp. 27-53

[13] Nazarov, S.A.; Plamenevsky, B.A. Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Gruyter Exp. Math., 13, de Gruyter, Berlin, 1994

Cité par Sources :