no. 3
The boundary behavior of a composite material
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 3, pp. 407-435.

On étudie ici le comportement au voisinage de la frontière du domaine de solutions de problèmes elliptiques à coefficients oscillant périodiquement. Les résultats, connus pour des frontières plannes, sont étendus au cas de frontières courbes et pour un milieu stratifié. On généralise pour cela la notion de couche limite et on définit des correcteurs de frontière qui conduisent à une approximation d’ordre ε dans la norme énergie.

In this paper, we study how solutions to elliptic problems with periodically oscillating coefficients behave in the neighborhood of the boundary of a domain. We extend the results known for flat boundaries to domains with curved boundaries in the case of a layered medium. This is done by generalizing the notion of boundary layer and by defining boundary correctors which lead to an approximation of order ε in the energy norm.

Classification : 35B27, 35B40
Mots clés : homogenization, generalized boundary layers, energy error estimates 
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Neuss-Radu, Maria. The boundary behavior of a composite material. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 3, pp. 407-435. http://www.numdam.org/item/M2AN_2001__35_3_407_0/

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York, San Francisco, London (1975). | MR | Zbl

[2] R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, tensor analysis, and applications. 2nd edn. Appl. Math. Sci. 75 Springer-Verlag, New York (1988). | MR | Zbl

[3] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | Zbl

[4] G. Allaire and M. Amar, Boundary layer tails in periodic homogenization. ESAIM: COCV 4 (1999) 209-243. | EuDML | Numdam | Zbl

[5] I. Babuska, Solution of interface problems by homogenization I. SIAM J. Math. Anal. 7 (1976) 603-634. | Zbl

[6] I. Babuska, Solution of interface problems by homogenization II. SIAM J. Math. Anal. 7 (1976) 635-645. | Zbl

[7] N. Bakhvalov and G. Panasenko, Homogenization: Averaging processes in periodic media. Mathematics and its Applications 36, Kluwer Academic Publishers, Dordrecht (1990). | MR | Zbl

[8] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland, Amsterdam (1978). | MR | Zbl

[9] A. Bensoussan, J.L. Lions and G. Papanicolau, Boundary layer analysis in homogenization of diffusion equations with Dirichlet conditions on the half space, in Proc. Internat. Symposium SDE, K. Ito Ed. J. Wiley, New York (1978) 21-40. | Zbl

[10] F. Blanc and S.A. Nazarov, Asymptotics of solutions to the Poisson problem in a perforated domain with corners. J. Math. Pures Appl. 76 (1997) 893-911. | Zbl

[11] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin, Heidelberg, New York (1983). | MR | Zbl

[12] W. Jäger and A. Mikelic, On the boundary conditions at the contact interface between a porous medium and a free fluid. Ann. Sci. Norm. Sup. Pisa, Serie IV 23 (1996) 404-465. | Numdam | Zbl

[13] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin Heidelberg, New York (1994). | MR | Zbl

[14] J.L. Lions, Some methods in mathematical analysis of systems and their Control. Science Press, Beijing, Gordon and Breach, New York (1981). | MR | Zbl

[15] S. Moskow and M. Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, in Proc. Roy. Soc. Edinburgh., Sect A 127 6 (1997) 1263-1299. | Zbl

[16] N. Neuss, W. Jäger and G. Wittum, Homogenization and Multigrid. Preprint 1998-04, SFB 359, University of Heidelberg (1998). | MR | Zbl

[17] M. Neuss-Radu, A result on the decay of the boundary layers in the homogenization theory. Asympto. Anal. 23 (2000) 313-328. | Zbl

[18] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | Zbl

[19] O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical problems in elasticity and Homogenization. Studies in Mathematics and its Applications 26, North-Holland, Amsterdam (1992). | MR | Zbl

[20] J. Sanchez-Huber and E. Sanchez-Palencia, Exercices sur les méthodes asymptotiques et l'homogénéisation. Masson, Paris (1993).

[21] E. Sanchez-Palencia, Non-homogenous media and vibration theory. Lect. Notes Phys. 127, Springer-Verlag, Berlin (1980). | MR | Zbl

[22] J. Wloka, Partielle differentialgleichungen. Teubner-Verlag, Stuttgart (1982). | MR | Zbl