[Développement asymptotique pour le problème de cavité sur un domaine contenant des inclusions avec coins proches]
Nous considérons le problème de Laplace pour un matériau contenant deux inclusions ou inhomogénéités avec coins proches. Nous présentons deux approches asymptotiques différentes pour décrire le phénomène sous certaines conditions géométriques. Ces développements asymptotiques sont étudiés et comparés dans un même cadre. Nous aboutissons à une formule de représentation qui caractérise l’éloignement des deux coins et proposons une relaxation des hypothèses géométriques.
We investigate the case of a medium with two inclusions or inhomogeneities with nearly touching corner singularities. We present two different asymptotic models to describe the phenomenon under specific geometrical assumptions. These asymptotic expansions are analysed and compared in a common framework. We conclude by a representation formula to characterise the detachment of the corners and we provide the possible extensions of the geometrical hypotheses.
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Mots clés : Asymptotic expansion, corner singularity
@article{AHL_2020__3__381_0,
author = {Bonnaillie-No\"el, Virginie and Poignard, Clair and Vial, Gr\'egory},
title = {Asymptotic expansions for the conductivity problem with nearly touching inclusions with corner},
journal = {Annales Henri Lebesgue},
pages = {381--406},
publisher = {\'ENS Rennes},
volume = {3},
year = {2020},
doi = {10.5802/ahl.36},
language = {en},
url = {http://www.numdam.org/articles/10.5802/ahl.36/}
}
TY - JOUR AU - Bonnaillie-Noël, Virginie AU - Poignard, Clair AU - Vial, Grégory TI - Asymptotic expansions for the conductivity problem with nearly touching inclusions with corner JO - Annales Henri Lebesgue PY - 2020 SP - 381 EP - 406 VL - 3 PB - ÉNS Rennes UR - http://www.numdam.org/articles/10.5802/ahl.36/ DO - 10.5802/ahl.36 LA - en ID - AHL_2020__3__381_0 ER -
%0 Journal Article %A Bonnaillie-Noël, Virginie %A Poignard, Clair %A Vial, Grégory %T Asymptotic expansions for the conductivity problem with nearly touching inclusions with corner %J Annales Henri Lebesgue %D 2020 %P 381-406 %V 3 %I ÉNS Rennes %U http://www.numdam.org/articles/10.5802/ahl.36/ %R 10.5802/ahl.36 %G en %F AHL_2020__3__381_0
Bonnaillie-Noël, Virginie; Poignard, Clair; Vial, Grégory. Asymptotic expansions for the conductivity problem with nearly touching inclusions with corner. Annales Henri Lebesgue, Tome 3 (2020), pp. 381-406. doi : 10.5802/ahl.36. http://www.numdam.org/articles/10.5802/ahl.36/
[AGG97] Dirichlet and Neumann exterior problems for the –dimensional Laplace operator: an approach in weighted Sobolev spaces, J. Math. Pures Appl., Volume 76 (1997) no. 1, pp. 55-81 | DOI | MR
[AKT06] An asymptotic formula for the voltage potential in the case of a near-surface conductivity inclusion, Z. Angew. Math. Phys., Volume 57 (2006) no. 2, pp. 234-243 | DOI | MR | Zbl
[BLY09] Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal., Volume 193 (2009) no. 1, pp. 195-226 | DOI | MR | Zbl
[BT13] On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal., Volume 209 (2013) no. 2, pp. 541-567 | DOI | MR | Zbl
[BTT18] Eigenvalues of the Neumann–Poincaré operator for two inclusions with contact of order : a numerical study, SMAI J. Comput. Math., Volume 36 (2018) no. 1, pp. 17-28 | DOI | MR | Zbl
[CCDV06] Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptotic Anal., Volume 50 (2006) no. 1-2, pp. 121-173 | MR | Zbl
[Dau88] Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions, Lecture Notes in Mathematics, 1341, Springer, 1988, viii+259 pages | MR | Zbl
[Dau96] Strongly elliptic problems near cuspidal points and edges, Partial differential equations and functional analysis. In memory of Pierre Grisvard. Proceedings of a conference held in November 1994 (Cea, Jean; al., eds.) (Progress in Nonlinear Differential Equations and their Applications), Volume 22, Birkhäuser (1996), pp. 93-110 | MR | Zbl
[Gri85] Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, 24, Pitman Advanced Publishing Program, 1985, xiv+410 pages | MR | Zbl
[KLY13] Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl., Volume 99 (2013) no. 2, pp. 234-249 | DOI | MR | Zbl
[Kon67] Boundary value problems for elliptic equations in domains with conical or angular points, Tr. Mosk. Mat. O.-va, Volume 16 (1967), pp. 209-292 | MR
[MN86] Asymptotic behavior of the stress-strained state near sharp inclusions, Dokl. Akad. Nauk SSSR, Volume 290 (1986) no. 1, pp. 48-51 | MR
[MNP00] Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Operator Theory: Advances and Applications, 111-112, Birkhäuser, 2000 translated from the German by Boris Plamenevski (Vol. I), Georg Heinig and Christian Posthoff, (Vol. II)
[NP18] On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary, Rev. Mat. Complut., Volume 31 (2018) no. 1, pp. 1-62 | DOI | MR | Zbl
[NT18] Singularities at the contact point of two kissing Neumann balls, J. Differ. Equations, Volume 264 (2018) no. 3, pp. 1521-1549 | DOI | MR | Zbl
[Néd01] Acoustic and electromagnetic equations. Integral representations for harmonic problems, Applied Mathematical Sciences, 144, Springer, 2001, ix+316 pages | Zbl
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