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G. Chechkin
(Russia)

 

Homogenization of the Lavrentiev-Bitsadze equation in partially perforated domain

Gregory A.Chechkin

Department of Differential Equations, Faculty of Mechanics and Mathematics
Lomonosov Moscow State University, Moscow 119899 Russia
chechkin@mech.math.msu.su

We consider a model homogenization problem for the Lavrentiev-Bitsadze equation

\begin{displaymath} u^\varepsilon_{tt}+\ (\hbox{sign}\, t)\, u^\varepsilon_{xx}=f(x,t) \end{displaymath}

in a partially perforated domain $\Omega_\varepsilon\in \hbox{\msbm\symbol{'122}}^2$ with a Neumann boundary condition on the boundary of holes. The perforated part of the domain lies in the upper half-plane $t>0$ and has a locally periodic structure. The parameter $\varepsilon$ characterizes the cell of periodicity. We study the asymptotic behavior of the solution of such a problem as $\varepsilon\to 0$. We prove the existence and the uniqueness theorem for the posed problem. Then, we construct the homogenized problem and on the base of the a priori estimates of the solution we prove the weak convergence of the solutions of this problem to the solution of the homogenized problem.

This is a joint work with Vladimir A. Kondratiev.


 

 

 

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Last modified: December 20, 1999