[Formulations variationnelles pour les inverses des opérateurs intégraux logarithmiques définis sur un intervalle]
Nous présentons des formulations variationnelles explicites et exactes pour les opérateurs intégraux faiblement singulier et hyper-singulier définis sur un interval borné ainsi que pour leurs inverses. En décomposant les solutions en parties symétriques et anti-symétriques, nous caractérisons les espaces de Sobolev associés et retrouvons des identités du type Calderón dans chaque cas.
We present explicit and exact variational formulations for the weakly singular and hypersingular operators over an interval as well as for their corresponding inverses. By decomposing the solutions in symmetric and antisymmetric parts, we precisely characterize the associated Sobolev spaces. Moreover, we are able to define novel Calderón-type identities in each case.
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@article{CRMATH_2011__349_9-10_547_0,
author = {Carlos Jerez-Hanckes and Jean-Claude N\'ed\'elec},
title = {Variational forms for the inverses of integral logarithmic operators over an interval},
journal = {Comptes Rendus. Math\'ematique},
pages = {547--552},
publisher = {Elsevier},
volume = {349},
number = {9-10},
year = {2011},
doi = {10.1016/j.crma.2011.01.016},
language = {en},
}
TY - JOUR AU - Carlos Jerez-Hanckes AU - Jean-Claude Nédélec TI - Variational forms for the inverses of integral logarithmic operators over an interval JO - Comptes Rendus. Mathématique PY - 2011 SP - 547 EP - 552 VL - 349 IS - 9-10 PB - Elsevier DO - 10.1016/j.crma.2011.01.016 LA - en ID - CRMATH_2011__349_9-10_547_0 ER -
Carlos Jerez-Hanckes; Jean-Claude Nédélec. Variational forms for the inverses of integral logarithmic operators over an interval. Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 547-552. doi : 10.1016/j.crma.2011.01.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.016/
[1] The numerical solution of first-kind logarithmic kernel integral equations on smooth open arcs, Math. Comp., Volume 56 (1991) no. 193, pp. 119-139
[2] Singular Integral Equations, Birkhäuser, Boston, USA, 2000
[3] Numerical solution of an exterior Neumann problem using a double layer potential, Math. Comp., Volume 32 (1978), pp. 973-990
[4] Operator preconditioning, Comput. Math. Appl., Volume 52 (2006) no. 5, pp. 699-706
[5] C. Jerez-Hanckes J.C. Nédélec, Regularization of the inverse integral logarithmic operator, Technical Report 21, Seminar for Applied Mathematics, ETH Zurich, August 2010.
[6] Second kind integral equations for the classical potential theory on open surfaces II, J. Comput. Phys., Volume 195 (2004), pp. 1-16
[7] Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, FL, USA, 2003
[8] Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, New York, USA, 2000
[9] An Introduction to the Numerical Analysis of Spectral Methods, Lecture Notes in Physics, vol. 318, Springer-Verlag, Heidelberg, Germany, 1989
[10] Singular Integral Equations, Noordhoff International Publishing, Groningen, 1977
[11] Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Applied Mathematical Sciences, vol. 44, Springer-Verlag, Berlin, 2001
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