Correctors for Some Nonlinear Monotone Operators
Tóm tắt
In this paper we study homogenization of quasi-linear partial differential equations of the form — div(a(x, x/εh, Duh)) = fh on Ω with Dirichlet boundary conditions. Here the sequence (εh) tends to 0 as h → ∞ and the map a(x, y, ξ) is periodic in y, monotone in ξ and satisfies suitable continuity conditions. We prove that uh → u weakly in
$$W_0^{1,p}(\Omega )$$
as h → ∞, where u is the solution of a homogenized problem of the form — div (b (x, Du)) = f on Ω. We also derive an explicit expression for the homogenized operator b and prove some corrector results, i.e. we find (Ph) such that Duh — Ph (Du) → 0 in Lp (Ω, Rn).
Tài liệu tham khảo
Allaire G, Homogenization and Two-Scale Convergence, SIAM J. Math. Anal., 1992, V.23, N 6, 1482–1518.
Bensoussan A, Lions J L and Papanicolaou G, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam, 1978.
Braides A, Correctors for the Homogenization of Almost Periodic Monotone Operators, Asymptotic Analysis, 1991, V.5, 47–74.
Braides A, Chiado Piat V and Defransceschi A, Homogenization of Almost Periodic Monotone Operators, Ann. Inst. Henri Poincare, Anal. Non Lineaire, 1992, V.9, N 4, 399–432.
Byström J, Correctors for Some Nonlinear Monotone Operators, Research Report, N 11, ISSN: 1400–4003, Department of Mathematics, Luleå University of Technology, 1999.
Chiado Piat V and Defransceschi A, Homogenization of Monotone Operators, Nonlinear Analysis, Theory, Methods and Applications, 1990, V.14, N 9, 717–732.
Dal Maso G and Defransceschi A, Correctors for the Homogenization of Monotone Operators, Differential and Integral Equations, 1990, V.3, N 6, 1151–1166.
Defransceschi A, An Introduction to Homogenization and G-Convergence, Lecture Notes, School on Homogenization, ICTP, Trieste, 1993.
Fusco N and Moscariello G, On the Homogenization of Quasilinear Divergence Structure Operators, Annali Mat. Pura Appl., 1987, V.146, 1–13.
Jikov V, Kozlov S and Oleinik O, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin – Heidelberg – New York, 1994.
Meyers N and Elcrat A, On Non-Linear Elliptic Systems and Quasi Regular Functions, Duke Math. J., 1975, V.42, 121–136.
Murat F, Compacité par Compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1978, V.5, N 4, 489–507.
Persson L, Persson L-E, Svanstedt N and Wyller J, The Homogenization Method, An Introduction, Studentlitteratur, Lund, 1993.
Royden H, Real Analysis, Macmillan, New York, Third Edition, 1988.
Wall P, Some Homogenization and Corrector Results for Nonlinear Monotone Operators, J. Nonlin. Math. Phys., 1998, V.5, N 3, 331–348.
Zeidler E, Nonlinear Functional Analysis and its Applications, Vol.4, Springer Verlag, New York, 1990.
Zeidler E, Nonlinear Functional Analysis and its Applications, Vol.2b, Springer Verlag, New York, 1990.