Abstract
The aim of this work is to provide a description of the corner asymptotics for the solutions of Maxwell equations in and outside a conductor body and to investigate the limit as the ratio permittivity/conductivity tends to zero (the eddy current limit). Corner singularities of the Maxwell transmission problem and also of the eddy current model have been described elsewhere [[6], [7]]. Here we concentrate on the uniform behavior with respect to the small parameter describing the eddy current limit — analyticity of the singular functions and stability of the decomposition of the fields into regular and singular parts.
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References
H. Ammari, A. Buffa and J.-C. Nédélec, A justification of eddy currents model for the Maxwell equations, SIAM J. Appl. Math. 60 (2000), 1805–1823.
A. Bossavit, Two dual formulations of the 3D eddy-current problem, COMPEL 4 (1985), 103–116.
A. Bossavit, Electromagnétisme en vue de la modélisation, Springer Verlag, 1993.
M. Costabel and M. Dauge, Singularités d’arêtes pour les problèmes aux limites elliptiques. In Journées “Equations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1992). École Polytech., Palaiseau, 1992, Exp. No. 4, 12p.
M. Costabel and M. Dauge, Singularities of Maxwell’s equations on polyhedral domains, Arch. Rational Mech. Anal. 151 (2000), 221–276.
M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems, RAIRO Modél. Math. Anal. Numér. 33 (1999), 627–649.
M. Costabel, M. Dauge and S. Nicaise, Singularities of eddy current problems, Preprint of the Newton Institute 03019, Cambridge (2003).
I. Gohberg and E. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Math. USSR Sbornik 13 (4) (1971), 603–625.
R. Hiptmair, Symmetric coupling for eddy currents problems, SIAM J. Numer. Anal. 40 (2002), 41–65.
T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322.
V.A. Kondrat’ev, Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. 16 (1967), 227–313.
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris, 1968.
V.G. Maz’ya, B.A. Plamenevskii, On the coefficients in the asymptotic of solutions of the elliptic boundary problem in domains with conical points, Amer. Math. Soc. Trans. (2) 123 (1984), 57–88.
V.G. Maz’ya and J. Rossmann, On a problem of Babuska (Stable asymptotics of the solution to the Dirichlet problem for elliptic equations of second order in domains with angular points), Math. Nachr. 155 (1992), 199–220.
S. Nicaise, Polygonal interface problems, Peter Lang, Berlin, 1993.
B. Schmutzler, Branching asymptotics for elliptic boundary value problems in a wedge, in “Boundary value problems and integral equations in nonsmooth domains (Luminy, 1993)”, M. Costabel, M. Dauge, and S. Nicaise, eds., Dekker, New York, (1995), 255–267.
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Costabel, M., Dauge, M., Nicaise, S. (2004). Corner Singularities of Maxwell Interface and Eddy Current Problems. In: Gohberg, I., Wendland, W., Ferreira dos Santos, A., Speck, FO., Teixeira, F.S. (eds) Operator Theoretical Methods and Applications to Mathematical Physics. Operator Theory: Advances and Applications, vol 147. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7926-2_28
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DOI: https://doi.org/10.1007/978-3-0348-7926-2_28
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