Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form
- PDF / 752,198 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 10 Downloads / 200 Views
(0123456789().,-volV) (0123456789().,-volV)
ORIGINAL PAPER
Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form Xiaoqin Guo1 • Hung V. Tran1 • Yifeng Yu2 Springer Nature Switzerland AG 2020
Abstract We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either OðeÞ or Oðe2 Þ depending on the diffusion matrix A, source term f, and boundary data g. Moreover, we show that the set of diffusion matrices A that give optimal rate OðeÞ is open and dense in the set of C2 periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is OðeÞ. Keywords Homogenization Periodic setting Linear non-divergence form elliptic equations Optimal rates of convergence Mathematics Subject Classification 35B27 35B40 35D40 35J25 49L25
Dedicated to Professor Hitoshi Ishii with our admiration. The work of HT is partially supported by NSF grant DMS-1664424 and NSF CAREER award DMS1843320. This article is part of the topical collection ‘‘Viscosity solutions - Dedicated to Hitoshi Ishii on the award of the 1st Kodaira Kunihiko Prize’’ edited by Kazuhiro Ishige, Shigeaki Koike, Tohru Ozawa, and Senjo Shimizu & Hung V. Tran [email protected] Xiaoqin Guo [email protected] Yifeng Yu [email protected] 1
Department of Mathematics, University of Wisconsin Madison, Van Vleck hall, 480 Lincoln drive, Madison, WI 53706, USA
2
Department of Mathematics, University of California, Irvine, 410G Rowland Hall, Irvine, CA 92697, USA SN Partial Differential Equations and Applications
15 Page 2 of 16
SN Partial Differ. Equ. Appl. (2020)1:15
1 Introduction In this paper, we are interested in studying and characterizing the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. Let U Rn be a given bounded domain with smooth boundary. The equation of our main interest is ( x aij uexi xj ¼ f ðxÞ in U; ð1:1Þ e ue ¼ g on oU: 2
The matrix function AðyÞ ¼ ðaij Þ1 i;j n 2 C2 ðRn ; Rn Þ is always assumed to be symmetric, Zn -periodic, and positive definite for all y 2 Rn . Denote by Tn ¼ Rn =Zn the flat ndimensional torus, and S nþ the set of all real symmetric, positive definite matrices of size n n, then we can also write that A 2 C 2 Tn ; S nþ . Assume f 2 C 2 U and g 2 C 4 ðoUÞ. In this paper, we always use the Einstein summation convention. The homogenization problem (1.1) was discussed in the classical books of Bensoussan, Lions, Papanicolaou [2], Jikov, Kozlov, Oleinik [9]. It is well-known that, as e ! 0, ue ! u uniformly on U, where u solves the following effective equation aij uxi xj ¼ f ðxÞ in U; ð1:2Þ u¼g on oU: Here, A ¼ faij g1 i;j n is the effective matrix with constant entries, which is determined as follows. For each fixed ðk; lÞ 2 f1; . . .; ng2 , consider the solution vkl of the (k, l)-th cell problem aij ðyÞvkl y
Data Loading...
