zbMATH — the first resource for mathematics

Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients. (English) Zbl 0894.35035
Authors’ summary: We consider a system of equations modelling a quasistationary induction heating process. Existence of a solution is obtained in Sobolev spaces using estimations in \(L^\infty\)-norm. Using a truncation technique, we build a sequence of truncated problems, the solutions of which converge to a solution of the initial unbounded coefficient problems.

35J60 Nonlinear elliptic equations
35A35 Theoretical approximation in context of PDEs
Full Text: DOI EuDML
[1] A. BENSOUSSAN, J. L. LIONS, G. PAPANICOLAOU, 1978, Asymptotic Analysis for Periodic Structures, North Holland. Zbl0404.35001 MR503330 · Zbl 0404.35001
[2] L. BOCCARDO, T. GALLOUET, 1989, Nonlinear Elliptic and Parabolic Equa tions Involving Measure Data, Journal of Functional Analysis, 87, No 1. Zbl0707.35060 MR1025884 · Zbl 0707.35060 · doi:10.1016/0022-1236(89)90005-0
[3] H. BREZIS, 1987, Analyse fonctionnelle, Masson, Paris. Zbl0511.46001 MR697382 · Zbl 0511.46001
[4] S. CLAIN, 1994, Analyse mathématique et numérique d’un modèle de chauffage par induction, PhD Thesis, EPFL, Lausanne.
[5] S. CLAIN, R. TOUZANI, A two-Dimensional Stationary Induction Heating Problem, Math. Meth. Appl. Sci., 1997, 20, 759-766. Zbl0870.35034 MR1446209 · Zbl 0870.35034 · doi:10.1002/(SICI)1099-1476(199706)20:9<759::AID-MMA879>3.0.CO;2-S
[6] S. CLAIN, J. RAPPAZ, M. SWIERKOSZ, R. TOUZANI, 1993, Numerical Modelling of Induction Heating for Two Dimensional Geometries, Math. Mod. Meth. Appl. Sci. 3, no 6, 905-822. Zbl0801.65120 MR1245636 · Zbl 0801.65120 · doi:10.1142/S0218202593000400
[7] T. GALLOUËT, R. HERBIN, 1994, Existence of a solution to a Coupled Elliptic System 7, No 2, Appl. Math. Lett., 49-55. Zbl0791.35043 MR1350145 · Zbl 0791.35043 · doi:10.1016/0893-9659(94)90030-2
[8] D. GILBARG, N. TRUDINGER, 1977, Elliptic partial Differential Equations of Second Order, Springer Verlag. Zbl0361.35003 MR473443 · Zbl 0361.35003
[9] R. LEWANDOWSKI, The Mathematical Analisys of a Coupling of a Turbulent Kinetic Energy Equation to the Bavier-Stokes Equation with an Addy Viscosity, Paper in preparation. Zbl0863.35077 · Zbl 0863.35077 · doi:10.1016/0362-546X(95)00149-P
[10] F. MURAT, 1994 Private Communication.
[11] C. G. SIMADER, 1972On Dirichlet’s Boundary Value Problem, vol. 268, Lecture Notes in Mathematics, Springer Verlag. Zbl0242.35027 MR473503 · Zbl 0242.35027
[12] G. STAMPACCHIA, 1966, Equations elliptiques du second ordre à coefficients discountinus, Presses Universitaires de Montréal. Zbl0151.15501 MR251373 · Zbl 0151.15501
[13] G. TALENTI, Best Constants in Sobolev inequality, vol. 110, Ann. Mat. Pura Appl., 1976. Zbl0353.46018 MR463908 · Zbl 0353.46018 · doi:10.1007/BF02418013
[14] V. VESRPI, Semigroup Theory and Application, Lecture notes in pure and applied mathematics, vol. 116, 1989.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.