Asymptotic expansions for the conductivity problem with nearly touching inclusions with corner
Annales Henri Lebesgue, Volume 3 (2020), pp. 381-406.


Keywords Asymptotic expansion, corner singularity


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.


[AGG97] Amrouche, Chérif; Girault, Vivette; Giroire, Jean Dirichlet and Neumann exterior problems for the n–dimensional Laplace operator: an approach in weighted Sobolev spaces, J. Math. Pures Appl., Volume 76 (1997) no. 1, pp. 55-81 | DOI | MR

[AKT06] Ammari, Habib; Kang, Hyeonbae; Touibi, Karim 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] Bao, Ellen Shiting; Li, Yan Yan; Yin, Biao Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal., Volume 193 (2009) no. 1, pp. 195-226 | DOI | MR | Zbl

[BT13] Bonnetier, Eric; Triki, Faouzi 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] Bonnetier, Eric; Triki, Faouzi; Tsou, Chun-Hsiang Eigenvalues of the Neumann–Poincaré operator for two inclusions with contact of order m: a numerical study, SMAI J. Comput. Math., Volume 36 (2018) no. 1, pp. 17-28 | DOI | MR | Zbl

[CCDV06] Caloz, Gabriel; Costabel, Martin; Dauge, Monique; Vial, Grégory 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] Dauge, Monique Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions, Lecture Notes in Mathematics, Volume 1341, Springer, 1988, viii+259 pages | MR | Zbl

[Dau96] Dauge, Monique 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 (Progress in Nonlinear Differential Equations and their Applications) Volume 22 (1996), pp. 93-110 | MR | Zbl

[Gri85] Grisvard, Pierre Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, Volume 24, Pitman Advanced Publishing Program, 1985, xiv+410 pages | MR | Zbl

[KLY13] Kang, Hyeonbae; Lim, Mikyoung; Yun, KiHyun 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] Kondrat’ev, Vladimir Alexandrovich 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] Movchan, Alexander B.; Nazarov, Sergey A. Asymptotic behavior of the stress-strained state near sharp inclusions, Dokl. Akad. Nauk SSSR, Volume 290 (1986) no. 1, pp. 48-51 | MR

[MNP00] Mazʼya, Vladimir; Nazarov, Sergey A.; Plamenevskij, Boris A. Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Operator Theory: Advances and Applications, Volume 111-112, Birkhäuser, 2000 (translated from the German by Boris Plamenevski (Vol. I), Georg Heinig and Christian Posthoff, (Vol. II))

[NP18] Nazarov, Sergey A.; Pérez, María-Eugenia 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] Nazarov, Sergey A.; Taskinen, Jari 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] Nédélec, Jean-Claude Acoustic and electromagnetic equations. Integral representations for harmonic problems, Applied Mathematical Sciences, Volume 144, Springer, 2001, ix+316 pages | Zbl