Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part I: Shape Differentiability of Pseudo-homogeneous Boundary Integral Operators

Integral Equations and Operator Theory

Volumn 72, Issue 4, 2012, Pages 509-535

  Costabel, Martin ^a   Le Louër, Frédérique ^b  

^a UNIVERSITÉ DE RENNES 1   (France)

^b UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

Boundary integral operators; Fundamental solution; Pseudo homogeneous kernels; Shape derivatives; Sobolev spaces; Surface differential operators

Indexed keywords

EID: 84859445558     PISSN: 0378620X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00020-012-1954-z     Document Type: Article

References (26)

1. Charalambopoulos A.: On the Fréchet differentiability of boundary integral operators in the inverse elastic scattering problem. Inverse Problems 11, 1137-1161 (1995).
2. Colton, D., Kress, R.: Inverse acoustic and electromagnetic scattering theory. Applied Mathematical Sciences, 2nd edn, vol. 93. Springer, Berlin (1998).
3. Costabel, M., Le Louër, F.: Shape derivatives of boundary integral operators in electromagnetic scattering. Part II: Application to scattering by a homogeneous dielectric obstacle Integr. Equ. Oper. Theory. (2012). doi: 10. 1007/s00020-012-1955-y.
4. divΓet nature de l'opérateur de Steklov-Poincaré du problème extérieur de l'électromagnétisme. C. R. Acad. Sci. Paris Sér. I Math. 316, 369-372 (1993).
5. Delfour, M. C., Zolésio, J.-P.: Shapes and geometries. In: Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), vol. 4. Analysis, differential calculus, and optimization, Philadelphia (2001).
6. Delfour, M. C., Zolésio, J.-P.: Tangential calculus and shape derivatives. Lecture Notes in Pure and Appl. Math., vol. 216. Dekker, New York, (2001).
7. Eskin, G. I.: Boundary value problems for elliptic pseudodifferential equations. Translations of Mathematical Monographs, vol. 52. American Mathematical Society, Providence (1981).
8. Hadamard J.: Sur quelques questions du calcul des variations. Ann. Sci. École Norm. Sup. (3) 24, 203-231 (1907).
9. Henrot, A., Pierre, M.: Variation et optimisation de formes. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 48. Springer, Berlin (2005) (Une analyse géométrique. [A geometric analysis]).
10. Hörmander, L.: The analysis of linear partial differential operators. III. Classics in Mathematics. Springer, Berlin (2007) (Pseudo-differential operators, Reprint of the 1994 edition).
11. Hsiao, G. C., Wendland, W. L.: Boundary integral equations. In: Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008).
12. Ivanyshyn O.: Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Probl. Imaging 1, 609-622 (2007).
13. Ivanyshyn, O., Kress, R.: Nonlinear integral equations in inverse obstacle scattering, pp. 39-50 (2006).
14. Journé, J.-L.: Calderón-Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón. In: Lecture Notes in Mathematics, vol. 994. Springer, Berlin (1983).
15. Kress R., Rundell W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Problems 21, 1207-1223 (2005).
16. Kupradze, V. D., Gegelia, T. G., Basheleǐshvili, M. O., Burchuladze, T. V.: Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. In: North-Holland Series in Applied Mathematics and Mechanics, vol. 25. North-Holland Publishing Co., Amsterdam (1979) (russian ed., Edited by V. D. Kupradze).
17. Le Louër, F.: Optimisation de formes d'antennes lentilles intégrées aux ondes millimétriques, PhD in Numérical Analysis, Université de Rennes 1 (2009) http://tel. archives-ouvertes. fr/tel-00421863/fr/.
18. Leugering G., Novotny A. A., Menzala G. P., Sokołowski J.: On shape optimization for an evolution coupled system. Appl. Math. Optim 64, 441-466 (2011).
19. Meyer, Y.: Ondelettes et opérateurs. II, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, (1990) Opérateurs de Calderón-Zygmund. [Calderón-Zygmund operators].
20. Nédélec, J.-C.: Acoustic and electromagnetic equations. In: Applied Mathematical Sciences, vol. 144. Springer, New York (2001) (Integral representations for harmonic problems).
21. Potthast R.: Fréchet differentiability of boundary integral operators in inverse acoustic scattering. Inverse Problems 10, 431-447 (1994).
22. Potthast R.: Fréchet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain. J. Inverse IL Posed Probl. 4, 67-84 (1996).
23. Schwartz, J. T.: Nonlinear functional analysis. Gordon and Breach Science Publishers, New York (1969) (Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher, Notes on Mathematics and its Applications).
24. Sokołowski, J., Zolésio, J.-P.: Introduction to shape optimization: Shape sensitivity analysis. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992).
25. Taylor, M. E.: Partial differential equations. I. Basic theory. Applied Mathematical Sciences, vol. 115. Springer, New York (1996).
26. Taylor, M. E.: Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials. Mathematical Surveys and Monographs, vol. 81. American Mathematical Society, Providence (2000).