Homogenization of a Parabolic Equation in Perforated Domain with Dirichlet Boundary Condition

Proceedings Mathematical Sciences

M Rajesh¹  A K Nandakumaran² 

[1] ANLA, U.F.R. des Sciences et Techniques, Universit´e de Toulon et du Var, BP , La Garde Cedex, France$$;Department of Mathematics, Indian Institute of Science, Bangalore 0 0, India$$

关键词: Homogenization;    perforated domain;    correctors.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domainsegin{align*}𝜕_t bleft(frac{x}{d_{𝜀}},u_{𝜀}ight)-mathrm{div} a(u_{𝜀},abla u_{𝜀}) & = f(x,t)quadext{in}quad𝛺_{𝜀}×(0,T), u_{𝜀} & = 0quadext{on}quad𝜕𝛺_{𝜀}×(0,T), u_{𝜀}(x,0) & = u_0(x)quadext{in}quad𝛺_{𝜀}.end{align*}Here, $𝛺_{𝜀} = 𝛺 ackslash S_{𝜀}$ is a periodically perforated domain and $d_{𝜀}$ is a sequence of positive numbers which goes to zero. We obtain the homogenized equation. The homogenization of the equations on a fixed domain and also the case of perforated domain with Neumann boundary condition was studied by the authors. The homogenization for a fixed domain and $b(frac{x}{d_{𝜀}}, u_{𝜀}) ≡ b(u_{𝜀})$ has been done by Jian. We also obtain certain corrector results to improve the weak convergence.

【 授权许可】

Unknown