×

zbMATH — the first resource for mathematics

On the homogenization of quasilinear divergence structure operators. (English) Zbl 0636.35027
A family of boundary value problems of the type \[ (1)\quad -div a(x/\epsilon,u,Du)=f,\quad u\in H_ 0^{1,p}(\Omega) \] is considered, where a(x,u,\(\epsilon)\) is periodic in x and satisfies suitable growth conditions, \(\epsilon >0\), \(p>1\). It is proved that the solutions \(u_{\epsilon}\) of (1) converge weakly to \(u_ 0\) satisfying \[ -div b(u,Du)=f,\quad u\in H_ 0^{1,r}(\Omega), \] and \(a(x/\epsilon,u_{\epsilon},Du_{\epsilon})\) converge to \(b(u_ 0,Du_ 0)\) weakly in \(L^{p'},p'=p/(p-1)\), where b(u,\(\xi)\) is given by an explicit formula. Moreover, it is shown that certain structure conditions are preserved for the limit operator.
Reviewer: M.Kucera

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35A35 Theoretical approximation in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Babuska, I., Solutions of interface problems by homogenization, SIAM J. Math. Anal., 7, 603-645 (1976) · Zbl 0343.35023
[2] Babuska, I., Solutions of interface problems by homogenization, SIAM J. Math. Anal., 8, 923-937 (1977) · Zbl 0402.35046
[3] A.Bensoussan - J. L.Lions, - G.Papanicolau,Asymptotic methods in periodic structures, North Holland, 1978.
[4] L.Boccardo - F.Murat,Homogéneization de problèmes quasilinéaires, Proceedings of the Meeting «Studio di problemi limite dell’Analisi Funzionale», Bressanone, 1981; Ed. Pitagora, 1982.
[5] Carbone, L.; Sbordone, C., Some properties of Γ-limits of integral functionals, Ann. Mat. Pura Appl., 122, 1-60 (1979) · Zbl 0474.49016
[6] De Giorgi, E.; Spagnolo, S., Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital., 8, 391-411 (1973) · Zbl 0274.35002
[7] Fusco, N.; Moscariello, G., An application of duality to homogenization of integral functionals, Memorie dell’Acc. dei Lincei, 17, I, 361-372 (1984) · Zbl 0672.49007
[8] D.Gilbarg- N. S.TrudingerElliptic Partial Differential Equations of Second Order; Springer, 1977, second ed., 1983.
[9] Marcellini, P., Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl., (4), 117, 139-152 (1978) · Zbl 0395.49007
[10] Marcellini, P.; Sbordone, C., Homogenization of non uniformly elliptic operators, Applicable Analysis, 68, 101-114 (1978) · Zbl 0406.35014
[11] Murat, F., Compacité par compensation, Ann. Scuola Norm. Sup. Pisa, 5, 489-507 (1978) · Zbl 0399.46022
[12] Raitum, U. E., On the G-convergence of quasilinear elliptic operators with unbounded coefficients, Sov. Math. Dokl., 24, 472-475 (1981)
[13] Palencia, E. Sanchez, Nonhomogeneous media and vibration theory, Lecture Notes in Physics 127 (1980), Berlin: Springer, Berlin
[14] P.Suquet,Plasticité et homogéneization, Thèse, Paris VI, 1982.
[15] L.Tartar,Topics in nonlinear Analysis, Publ. Math. d’Orsay,13 (1978). · Zbl 0395.00008
[16] L.Tartar,Homogéneization et compacité par compensation, Séminaire Schwartz Exposé, n. 9 (1978). · Zbl 0406.35055
[17] L.Tartar,Convergence d’operateurs différentiels, Proceedings of the Meeting «Analisi Convessa e Applicazioni»>, Roma, 1974.
[18] L.Tartar,Cours Pecot, Collège de France, partiellement rédigé par F.Murat,H-convergence, Séminaire d’Analise Fonctionnelle et Numérique de l’Université d’Alger, 1977-78.
[19] Trudinger, N. S., On the Comparison Principle for Quasilinear Divergence Structure Equations, Arch. for Rat. Mech. and Anal., 57, 128-133 (1974) · Zbl 0311.35046
[20] Ngoan, Zhikov-Kozlov-Oleinik-Khat’En, Averaging and G-convergence of differential operators, Russian Math. Surveys, 34, 69-147 (1979) · Zbl 0445.35096
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.