Convergence Rates of Solutions for Elliptic Reiterated Homogenization Problems
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DOI: 10.1007/s13226-020-0435-3
CONVERGENCE RATES OF SOLUTIONS FOR ELLIPTIC REITERATED HOMOGENIZATION PROBLEMS Juan Wang and Jie Zhao College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China e-mails: [email protected]; [email protected] (Received 21 November 2018; accepted 26 April 2019) In this paper, we study reiterated homogenization problems for equations −div(A(x/ε, x/ε2 )∇uε ) = f (x). We introduce auxiliary functions and obtain the representation formula satisfied by uε and homogenized solution u0 . Then we utilize this formula in combination with the asymptotic estimates of Neumann functions for operators and uniform regularity estimates of solutions to obtain convergence rates in Lp for solutions as well as gradient error estimates for Neumann problems. Key words : Reiterated homogenization; convergence rates; Neumann functions; regularity estimate. 2010 Mathematics Subject Classification : 35B27, 35J15.
1. I NTRODUCTION This paper concerns with the asymptotic behavior of solutions to reiterated homogenization equations with Neumann boundary conditions. More precisely, given a bounded C 1,1 domain Ω ⊂ Rn , we consider
µ ¶ ∂uε ∂ 2 ∂uε aij (x/ε, x/ε ) = f in Ω and = 0 on ∂Ω, (1) Lε uε = − ∂xi ∂xj ∂νε ∂uε ∂uε = ni aij denotes the conormal derivative with Lε and n(x) is the outward unit normal where ∂νε ∂xj to ∂Ω at the point x. Throughout this paper, the summation convention is used. We assume that the coefficient matrix A(y, z) = (aij (y, z)) with 1 ≤ i, j ≤ n is real symmetric and satisfies the ellipticity condition λ | ξ |2 ≤ aij (y, z)ξi ξj ≤
1 | ξ |2 , for y, z ∈ Rn and ξ = (ξi ) ∈ Rn , λ
(2)
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JUAN WANG AND JIE ZHAO
where λ > 0, and the periodicity condition A(y + l, z + h) = A(y, z), for y, z ∈ Rn and l, h ∈ Zn .
(3)
We impose the smoothness condition k A(y, z) kC 1,α (Rn ×Rn ) ≤ Λ, for some α ∈ (0, 1) and Λ > 0. Without loss of generality, we also assume the compatibility condition Z Z uε (x)dσ(x) = f (x)dx = 0. ∂Ω
(4)
(5)
Ω
Error estimates of solutions is one of the main questions in homogenization theory. There are many papers about convergence of solutions for elliptic homogenization problems. Assume that all of functions are smooth enough, the O(ε) error estimate in L∞ was presented by Bensoussan, Lions and Papanicolaou [5]. In 1987, Avelcaneda and Lin [3] proved Lp convergence by the method of maximum principle. At the same year, they [4] obtained L∞ error estimate when f is less regular than Bensoussan, Lions and Papanicolaou’s. After that, Griso [11, 12] studied interior error estimates by using the periodic unfolding method. In 2012, Kenig, Lin and Shen [16] obtained convergence 1
of solutions in L2 and H 2 in Lipschitz domains with Dirichlet or Neumann boundary conditions. In 2014, they [17] have also studied the asymptotic behavior of the Green and Neumann functions obtaining some error estimates of solutions. One may consult several outstanding sources [1, 5, 7, 9, 10, 13, 20] for background and overview of the homogenization theory. In this w
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