For a fixed bounded open set , a sequence of open sets and a sequence of sets , we study the asymptotic behavior of the solution of a nonlinear elliptic system posed on , satisfying Neumann boundary conditions on and Dirichlet boundary conditions on . We obtain a representation of the limit problem which is stable by homogenization and we prove that this representation depends on and locally.
Mots clés : homogenization, varying domains, nonlinear problems
@article{COCV_2009__15_1_49_0, author = {Calvo-Jurado, Carmen and Casado-D{\'\i}az, Juan and Luna-Laynez, Manuel}, title = {Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {49--67}, publisher = {EDP-Sciences}, volume = {15}, number = {1}, year = {2009}, doi = {10.1051/cocv:2008021}, mrnumber = {2488568}, zbl = {1170.35014}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2008021/} }
TY - JOUR AU - Calvo-Jurado, Carmen AU - Casado-Díaz, Juan AU - Luna-Laynez, Manuel TI - Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 49 EP - 67 VL - 15 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008021/ DO - 10.1051/cocv:2008021 LA - en ID - COCV_2009__15_1_49_0 ER -
%0 Journal Article %A Calvo-Jurado, Carmen %A Casado-Díaz, Juan %A Luna-Laynez, Manuel %T Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 49-67 %V 15 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008021/ %R 10.1051/cocv:2008021 %G en %F COCV_2009__15_1_49_0
Calvo-Jurado, Carmen; Casado-Díaz, Juan; Luna-Laynez, Manuel. Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 49-67. doi : 10.1051/cocv:2008021. http://www.numdam.org/articles/10.1051/cocv:2008021/
[1] The limit of Dirichlet systems for variable monotone operators in general perforated domains. J. Math. Pures Appl. 81 (2002) 471-493. | MR | Zbl
and ,[2] Homogenization of elliptic problems with the Dirichlet and Neumann conditions imposed on varying subsets. Math. Meth. Appl. Sci. 30 (2007) 1611-1625. | MR | Zbl
, and ,[3] Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains. J. Math. Pures Appl. 76 (1997) 431-476. | MR | Zbl
,[4] Homogenization of Dirichlet problems for monotone operators in varying domains. Proc. Roy. Soc. Edinburgh A 127 (1997) 457-478. | MR | Zbl
,[5] Asymptotic behavior of nonlinear elliptic systems on varying domains. SIAM J. Math. Anal. 31 (2000) 581-624. | MR | Zbl
and ,[6] Un terme étrange venu d'ailleurs, in Nonlinear partial differential equations and their applications, Collège de France seminar, Vols. II and III, H. Brézis and J.-L. Lions Eds., Research Notes in Math. 60 and 70, Pitman, London (1982) 98-138 and 154-78. | MR | Zbl
and ,[7] Limits of nonlinear Dirichlet problems in varying domains. Manuscripta Math. 61 (1988) 251-278. | EuDML | MR | Zbl
and ,[8] New results on the asymptotic behaviour of Dirichlet problems in perforated domains. Math. Mod. Meth. Appl. Sci. 3 (1994) 373-407. | MR | Zbl
and ,[9] Wiener-criterion and -convergence. Appl. Math. Optim. 15 (1987) 15-63. | MR | Zbl
and ,[10] Asymptotic behaviour and correctors for the Dirichlet problems in perforated domains with homogeneous monotone operators. Ann. Sc. Norm. Sup. Pisa 7 (1997) 765-803. | Numdam | MR
and ,[11] Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 445-486. | Numdam | MR | Zbl
and ,[12] A capacitary method for the asymptotic analysis of Dirichlet problems for monotone operators. J. Anal. Math. 71 (1997) 263-313. | MR | Zbl
, and ,[13] Boundary homogenization for elliptic problems. J. Math. Pures Appl. 66 (1987) 351-361. | MR | Zbl
and ,[14] Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). | MR | Zbl
and ,[15] The Lebesgue set of a function whose distribution derivaties are p-th power sumable. Indiana Univ. Math. J. 22 (1972) 139-158. | MR | Zbl
and ,[16] Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93 (1965) 97-107. | Numdam | MR | Zbl
and ,[17] Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). | MR
,[18] Local behaviour of solutions of quasilinear equations. Acta Math. 111 (1964) 302-347. | MR | Zbl
,[19] Asymptotic behaviour of solutions of nonlinear elliptic problems in perforated domains. Mat. Sb. 184 (1993) 67-90. | Zbl
,[20] Averaging of quasilinear parabolic problems in domains with fine-grained boundary. Diff. Equations 31 (1995) 327-339. | MR | Zbl
,[21] Weakly Differentiable Functions. Springer-Verlag, Berlin (1989). | MR | Zbl
,Cité par Sources :