no. 21-22
Mathematical Physics/Calculus of Variations
Shape optimization for the Maxwell equations under weaker regularity of the data
[Dérivée par rapport au domaine dans l'équation de Maxwell sous des hypothèses de plus faible régularité des données]

Présenté par : the Editorial Board

Cagnol, John1 ; Eller, Matthias2
1 École Centrale Paris, laboratoire MAS, grande voie des vignes, 92295 Chatenay-Malabry cedex, France
2 Georgetown University, Dept. of Mathematics, Washington, DC 20057, USA
Comptes Rendus. Mathématique, Tome 348 (2010) no. 21-22, pp. 1225-1230.

On considère un problème d'optimisation de forme dans le cadre des équations de Maxwell avec une condition de bord dissipative. On établit un résultat de dérivabilité par rapport au domaine dans le cas de faible régularité. Au détour de cette preuve, on établit la régularité « cachée » des traces du champ éléctrique et magnétique sur le bord du domaine.

We consider a shape optimization problem for Maxwell's equations with a strictly dissipative boundary condition. In order to characterize the shape derivative as a solution to a boundary value problem, sharp regularity of the boundary traces is critical. This Note establishes the Fréchet differentiability of a shape functional.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.10.021
Cagnol, John 1 ; Eller, Matthias 2

1 École Centrale Paris, laboratoire MAS, grande voie des vignes, 92295 Chatenay-Malabry cedex, France
2 Georgetown University, Dept. of Mathematics, Washington, DC 20057, USA 
@article{CRMATH_2010__348_21-22_1225_0,
     author = {Cagnol, John and Eller, Matthias},
     title = {Shape optimization for the {Maxwell} equations under weaker regularity of the data},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1225--1230},
     publisher = {Elsevier},
     volume = {348},
     number = {21-22},
     year = {2010},
     doi = {10.1016/j.crma.2010.10.021},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2010.10.021/}
}
TY  - JOUR
AU  - Cagnol, John
AU  - Eller, Matthias
TI  - Shape optimization for the Maxwell equations under weaker regularity of the data
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 1225
EP  - 1230
VL  - 348
IS  - 21-22
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2010.10.021/
DO  - 10.1016/j.crma.2010.10.021
LA  - en
ID  - CRMATH_2010__348_21-22_1225_0
ER  - 
%0 Journal Article
%A Cagnol, John
%A Eller, Matthias
%T Shape optimization for the Maxwell equations under weaker regularity of the data
%J Comptes Rendus. Mathématique
%D 2010
%P 1225-1230
%V 348
%N 21-22
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2010.10.021/
%R 10.1016/j.crma.2010.10.021
%G en
%F CRMATH_2010__348_21-22_1225_0
Cagnol, John; Eller, Matthias. Shape optimization for the Maxwell equations under weaker regularity of the data. Comptes Rendus. Mathématique, Tome 348 (2010) no. 21-22, pp. 1225-1230. doi : 10.1016/j.crma.2010.10.021. http://www.numdam.org/articles/10.1016/j.crma.2010.10.021/

[1] Cagnol, John; Zolésio, Jean-Paul Shape derivative in the wave equation with Dirichlet boundary conditions, J. Differential Equations, Volume 158 (1999) no. 2, pp. 175-210

[2] Eller, Matthias On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition, Discrete Contin. Dyn. Syst. Ser. S, Volume 2 (2009) no. 3, pp. 473-481

[3] Lagnese, John E.; Leugering, Günter Domain Decomposition Methods in Optimal Control of Partial Differential Equations, Internat. Ser. Numer. Math., vol. 148, Birkhäuser Verlag, Basel, 2004

[4] Leugering, Guenther; Novotny, Antonio André; Menzala, Gustavo Perla; Sokołowski, Jan Shape Sensitivity Analysis of a Quasi-Electrostatic Piezoelectric System in Multilayered Media, Math. Methods Appl. Sci., 2010

[5] Majda, Andrew; Osher, Stanley Initial–boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., Volume 28 (1975) no. 5, pp. 607-675

[6] Sokołowski, Jan; Zolésio, Jean-Paul Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Ser. Comput. Math., vol. 16, Springer-Verlag, Berlin, 1992

[7] Zolésio, Jean-Paul Hidden boundary shape derivative for the solution to Maxwell equations and non cylindrical wave equations, Optimal Control of Coupled Systems of Partial Differential Equations, Internat. Ser. Numer. Math., vol. 158, Birkhäuser, Basel, 2009, pp. 319-345

Cité par Sources :