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Reiterated ergodic algebras and applications. (English) Zbl 1228.46049

The authors generalize the framework of [G. Nguetseng, Z. Anal. Anwend. 22, No. 1, 73–107 (2003; Zbl 1045.46031)] by omitting the separability assumption. They show how this allows them to generalize the class of problems to which the previous theory applied. In particular, they give some compactness results with applications to Reynolds type equations and to some nonlinear hyperbolic equations.

MSC:

46J10 Banach algebras of continuous functions, function algebras
35L70 Second-order nonlinear hyperbolic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B15 Almost and pseudo-almost periodic solutions to PDEs

Citations:

Zbl 1045.46031
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References:

[1] Allaire G., Briane M.: Multi-scale convergence and reiterated homogenization. Proc. Roy. Soc. Edinb. Sect. A. 126, 297–342 (1996) · Zbl 0866.35017
[2] Barles G., Murat F.: Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions. Arch. Rat. Mech. Anal. 133, 77–101 (1995) · Zbl 0859.35031 · doi:10.1007/BF00375351
[3] Besicovitch A.S.: Almost periodic functions. Dover Publications, Cambridge (1954) · Zbl 0065.07102
[4] Boccardo, L., Murat F., Puel J.P.: Existence de solutions non born ées pour certaines équations quasi-linéaires. Port. Math. 41, 507–534 (1982) · Zbl 0524.35041
[5] Bohr H.: Almost periodic functions. Chelsea, New York (1947) · Zbl 0005.20303
[6] Bourbaki N.: Intégration. Chap. 1–4. Hermann, Paris (1966) · Zbl 0141.00602
[7] Bourbaki, N.: Intégration. Chap. 5. Paris: Hermann, 1967
[8] Bourbaki, N.: Topologie générale. Chap. 1–4. Paris: Hermann, 1971 · Zbl 0249.54001
[9] Carrillo J., Wittbold P.: Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems. J. Diff. Eq. 156, 93–121 (1999) · Zbl 0932.35129 · doi:10.1006/jdeq.1998.3597
[10] Casado Diaz J., Gayte I.: A derivation theory for generalized Besicovitch spaces and its application for partial differential equations. Proc. R. Soc. Edinb. A 132, 283–315 (2002) · Zbl 1028.35012 · doi:10.1017/S0308210500001633
[11] Casado Diaz J., Gayte I.: The two-scale convergence method applied to generalized Besicovitch spaces. Proc. R. Soc. Lond. A 458, 2925–2946 (2002) · Zbl 1099.35010 · doi:10.1098/rspa.2002.1003
[12] Cavalcanti M.M., Domingos Cavalcanti V.N., Soriano J.A.: Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients. Electron. J. Diff. Eq. 1998, 1–21 (1998) · Zbl 0892.35027
[13] Chou C.: Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups. Trans. Amer. Math. Soc. 274, 141–157 (1982) · Zbl 0505.43004 · doi:10.1090/S0002-9947-1982-0670924-2
[14] Chou C.: Weakly almost periodic functions and almost convergent functions on a group. Trans. Amer. Math. Soc. 206, 175–200 (1975) · Zbl 0303.43017 · doi:10.1090/S0002-9947-1975-0394062-8
[15] De Leeuw K., Glicksberg I.: Applications to almost periodic compactifications. Acta Math. 105, 63–97 (1961) · Zbl 0104.05501 · doi:10.1007/BF02559535
[16] Eberlein W.F.: Abstract ergodic theorems and weak almost periodic functions. Trans. Amer. Math. Soc. 67, 217–240 (1949) · Zbl 0034.06404 · doi:10.1090/S0002-9947-1949-0036455-9
[17] Eberlein W.F.: A note on Fourier-Stieltjes transforms. Proc. Amer. Math. Soc. 6, 310–313 (1955) · Zbl 0065.01604 · doi:10.1090/S0002-9939-1955-0068030-2
[18] He J.-H.: Variational principle for non-Newtonian lubrication: Rabinowitsch fluid model. Appl. Math. Comp. 157, 281–286 (2004) · Zbl 1095.76046 · doi:10.1016/j.amc.2003.07.028
[19] Kwame Essel, E.: Homogenization of Reynolds equations and of some parabolic problems via Rothes method. PhD thesis, Luleå University of Technology, Sweden, 2008
[20] Larsen R.: Banach algebras. Marcel Dekker, New York (1973)
[21] Lindenstrauss J.: On non-separable reflexive Banach spaces. Bull. Amer. Math. Soc. 72, 967–970 (1966) · Zbl 0156.36403 · doi:10.1090/S0002-9904-1966-11606-3
[22] Lions J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969)
[23] Lions J.L., Magenes E.: Problèmes aux limites non homog ènes et applications Vol. 1. Dunod, Paris (1968) · Zbl 0165.10801
[24] Lukkassen D., Nguetseng G., Wall P.: Two scale convergence. Int. J. Pure Appl. Math. 1, 35–86 (2002) · Zbl 1061.35015
[25] Lukkassen D., Nguetseng G., Nnang H., Wall P.: Reiterated homogenization of nonlinear elliptic operators in a general deterministic setting. J. Funct. Spaces Appl. 7, 121–152 (2009) · Zbl 1178.35048
[26] Nguetseng G.: Homogenization structures and applications I. Z. Anal. Anwen. 22, 73–107 (2003) · Zbl 1045.46031 · doi:10.4171/ZAA/1133
[27] Nguetseng G.: Mean value on locally compact abelian groups. Acta Sci. Math. 69, 203–221 (2003) · Zbl 1047.43003
[28] Nguetseng G.: Deterministic homogenization of a semilinear elliptic equation. Math Reports 8, 167–195 (2006) · Zbl 1150.35010
[29] Nguetseng, G.: Almost periodic homogenization: asymptotic analysis of a second order elliptic equation. Preprint · Zbl 1150.35010
[30] Rakotoson J.M.: Uniqueness of renormalized solutions in a T -set for the L 1-data problem and the link between various formulations. Indiana Univ. Math. J. 43, 685–702 (1994) · Zbl 0805.35035 · doi:10.1512/iumj.1994.43.43029
[31] Woukeng J.L.: Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales. Ann. Mat. Pura Appl. 189(3), 357–379 (2010) · Zbl 1213.35067 · doi:10.1007/s10231-009-0112-y
[32] Woukeng, J.L.: Homogenization of nonlinear degenerate non-monotone elliptic operators in domains perforated with tiny holes. Acta Appl. Math. (2009) doi: 10.1007/s10440-009-9552-z , 2009 · Zbl 1203.35028
[33] Zhikov, V.V., Krivenko, E.V.: Homogenization of singularly perturbed elliptic operators. Matem. Zametki. 33, 571–582 (1983) (English transl.: Math. Notes 33, 294–300 (1983)) · Zbl 0556.35003
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