Homogenization theory of elliptic system with lower order terms for dimension two

Wei Wang; Ting Zhang

doi:10.3934/cpaa.2023010

Article Contents

2023, Volume 22, Issue 3: 787-824. Doi: 10.3934/cpaa.2023010

This issue Previous Article On the ratio of total masses of species to resources for a logistic equation with Dirichlet boundary condition Next Article Global asymptotic stability of the rarefaction waves to the Cauchy problem for the scalar non-viscous diffusive dispersive conservation laws

Homogenization theory of elliptic system with lower order terms for dimension two

Wei Wang^1,2, and
Ting Zhang^1, ,

1.
School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, China
2.
School of Mathematical Sciences, Peking University, Beijing, 100871, China

^*Corresponding author: Ting Zhang
^*Corresponding author: Ting Zhang

Received: July 2022

Revised: July 2022

Early access: December 7, 2022

Published online: December 7, 2022

This work is partially supported by the National Natural Science Foundation of China 11931010.

Abstract / Introduction Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

In this paper, we consider the homogenization problem for generalized elliptic systems

$ \mathcal{L}_{ \varepsilon} = -\operatorname{div}[A(x/ \varepsilon)\nabla+V(x/ \varepsilon)]+B(x/ \varepsilon)\nabla+c(x/ \varepsilon)+\lambda I $

with dimension two. Precisely, we will establish the $ W^{1, p} $ estimates, Hölder estimates, Lipschitz estimates and $ L^p $ convergence results for $ \mathcal{L}_{ \varepsilon} $ with dimension two. The operator $ \mathcal{L}_{ \varepsilon} $ has been studied by Qiang Xu with dimension $ d\geq 3 $ in [22,23] and the case $ d = 2 $ is remained unsolved. As a byproduct, we will construct the Green functions for $ \mathcal{L}_{ \varepsilon} $ with $ d = 2 $ and their convergence rates.

Keywords:

Mathematics Subject Classification: Primary: 35B27; Secondary: 35B25, 35B65, 35J25.

Citation:

Full Text(HTML)

Figure 1. Balls $ B(x_i.\frac{5}{8}R) $ construct a covering of $ B(x_0, 2R)\backslash B(x_0, R) $.

Download: Full-size image PowerPoint slide

Figure 2. .

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2 edition, Elsevier/Academic Press, Amsterdam, 2003.
[2]	M. Avellaneda and F. Lin, Compactness methods in the theory of homogenization, Commun. Pure Appl. Math., 40 (1987), 803-847. doi: 10.1002/cpa.3160400607.
[3]	M. Avellaneda and F. Lin, Compactness methods in the theory of homogenization Ⅱ: Equations in non-divergence form, Commun. Pure Appl. Math., 42 (1989), 139-172. doi: 10.1002/cpa.3160420203.
[4]	D. Cioranescu and P. Donato, An Introduction to Homogenization, The Clarendon Press, Oxford University Press, New York, 1999.
[5]	S. Cho, H. Dong and S. Kim, On the Green's matrices of strongly parabolic systems of second order, Indiana Uni. Math. J., 57 (2008), 1633-1677.
[6]	H. Dong and S. Kim, Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains, Trans. Amer. Math. Soc., 361 (2009), 3303-3323. doi: 10.1090/S0002-9947-09-04805-3.
[7]	H. Dong and S. Kim, Green's function for nondivergence elliptic operators in two dimensions, SIAM J. Math. Anal., 53 (2021), 4637-4656. doi: 10.1137/20M1323618.
[8]	J. Geng and Z. Shen, Uniform regularity estimates in parabolic homogenization, Indiana Uni. Math. J., 64 (2015), 697-733. doi: 10.1512/iumj.2015.64.5503.
[9]	J. Geng and B. Shi, Green's matrices and boundary estimates in parabolic homogenization, J. Differ. Equ., 269 (2020), 3031-3066. doi: 10.1016/j.jde.2020.02.021.
[10]	M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Edizioni della Normale, Pisa, 2005.
[11]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.
[12]	C. E. Kenig, F. Lin and Z. Shen, Periodic homogenization of Green and Neumann functions, Commun. Pure Appl. Math., 67 (2014), 1219-1262. doi: 10.1002/cpa.21482.
[13]	C. E. Kenig, F. Lin and Z. Shen, Estimates of eigenvalues and eigenfunctions in periodic homogenization, J. Euro. Math. Soc., 15 (2013), 1901-1925. doi: 10.4171/JEMS/408.
[14]	Z. Shen, Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains, Amer. J. Math., 125 (2003), 1079-1115.
[15]	Z. Shen, Bounds of Riesz transforms on $ L^p $ spaces for second order elliptic operators, Ann. I. Fourier, 55 (2005), 173-197.
[16]	Z. Shen, Necessary and sufficient conditions for the solvability of the $ L^p $ Dirichlet problem on Lipschitz domains, Math. Ann., 336 (2006), 697-725. doi: 10.1007/s00208-006-0022-x.
[17]	Z. Shen, The $ L^p $ boundary value problems on Lipschitz domains, Adv. Math., 216 (2007), 212-254. doi: 10.1016/j.aim.2007.05.017.
[18]	Z. Shen, $ W^{1, p} $ estimates for elliptic homogenization problems in nonsmooth domains, Indiana Uni. Math. J., 57 (2008), 2283-2298.
[19]	Z. Shen, Periodic Homogenization of Elliptic Systems, Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-91214-1.
[20]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[21]	J. L. Taylor, S. Kim and R. M. Brown, The Green function for elliptic systems in two dimensions, Commun. Partial Differ. Equ., 38 (2013), 1574-1600. doi: 10.1080/03605302.2013.814668.
[22]	Q. Xu, Uniform regularity estimates in homogenization theory of elliptic system with lower order terms, J. Math. Anal. Appl., 438 (2016), 1066-1107. doi: 10.1016/j.jmaa.2016.02.011.
[23]	Q. Xu, Uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the Neumann boundary problem, J. Differ. Equ., 261 (2016), 4368-4423. doi: 10.1016/j.jde.2016.06.027.