我們在這篇論文中討論了一些特殊退化二次橢圓算子的勻質化問題, 並利用有限元素法做了幾個數值上的計算

Abstract

Homogenization of some special degenerate second order linear elliptic operators and its numerical computation
Lin-Hong Shen, Avisor:Assistant Professor Chia-Chieh Chu
Department of Mathematics
National Tsing Hua University, Hsin-Chu City,Taiwan

In many area, homogenization is an alternative way to find out the asymptotic behaviour of partial differential equation. This arti- cle is about homogenization process of degenerate second order linear elliptic operators. In this article, we give both theoretical and com- putational analysis to the asymptotic behaviour of the solution of the equation.
−div(a( x )Duh) = f on Ω ,
uh |∂Ω= 0 on ∂Ω ,

when Eh tends to zero, where aij (x) is Y -periodic, nonnegative defi- nite for almost every x in domain Ω and vanishes at some points in Ω. We find out that the homogenization process of degenerate ellip- tic equation in rectangle domain is still available for some particular coefficient functions with its inverse is integrable
Key words: homogenization, degenerate elliptic equation, asymp- totic behaviour, numerical analysis

Contents
1 Introduction ............................................1
2 Homogenization of some special degenerate second order linear elliptic operators .................................3
2.1 Setting of the problem ................................3 2.2 One dimensional cases .................................6 2.3 Asymptotic expansions using multiple scales ..........10 2.4 Homogenization .......................................14
3 Numerical Computation ..................................18 3.1 Examples in R ........................................18 3.2 Examples in R2 .......................................20
4 Conclusion .............................................23
A Special case in layered material .......................23
B Special case in coefficient function with separation variable .................................................24

References
[AF03] Robert A. Adams and John J.F. Fournier. Sobolev spaces. Academic Press, second edition, 2003.
[BLP78] A. Bensoussan, J.-L. Lions, and G. Papanicolaou. Asymptotic anal- ysis for periodic structures. AMS Chelsea Publishing, Providence, RI, 1978.
[BS08] Susanne C. Brenner and L.Ridgway Scott. The mathematical theory of the finite element methods. Springer, 2008.
[Cav02] Albo Carlos Cavalheiro. An approximation theorem for solutions of degenerate elliptic equations. Proc. Edinb. Math. Soc. (2), 45:363– 389, 2002.
[Def93] Anneliese. Defranceschi. An introduction to homogenization and g-convergence. Technical report, ICTP, September 1993.
[FKS82] E. Fabes, C. Kenig, and R. Serapioni. The local regularity of so- lutions of degenerate elliptic equations. Communication in PDEs, 7(1):77–116, 1982.
[GT77] D. Gilbarg and N. Trudinger. Elliptic partial differential equations of second order. Springer-Verlag, Berlin, 1977.
[GU09] V. Gol’dshtein and A. Ukhlov. Weighted sobolev space and embed- ding theorems. Transactions of the American Mathematical Society, 361(7):3829–3850, 2009.
[Kuf85] Alois. Kufner. weighted Sobolev Spaces. A Wiley-Interscience Pub- lication. John Wiley & Sons, Inc., New York, 1985.
26
[PS08] Grigorios A. Pavliotis and Andrew M. Stuart. Multiscale Methods, volume 53 of Texts in Applied Mathematics. Springer, New York, 2008.
[To¨l12] Jonas M. To¨lle. Uniqueness of weighted sobolev spaces with weakly differentiable weights. Journal of Functional Analysis, 263(10):3195–3223, 2012.