Dirichlet's Problem for an Equation with Periodical Coefficients Depending on a~Small Parameter

M. I. Freǐdlin

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 1, pp. 133-139

Résumé

This paper studies the limiting behavior of the solution $u^\varepsilon(x)$ of Dirichlet's problem for $$ L^\varepsilon u^\varepsilon=\frac12\sum a_{ij}\left(\frac x\varepsilon\right)\frac{\partial^2 u^\varepsilon}{\partial x^i\partial x^j}+\sum b_i\left(\frac x\varepsilon\right)\frac{\partial u^\varepsilon}{\partial x^i}-c\left(\frac x\varepsilon\right)u^\varepsilon=0, $$ when $\varepsilon\to 0$. The coefficients of the operator $L^1$ are assumed to be periodic. It is proved that $\lim\limits_{\varepsilon\to 0}u^\varepsilon(x)=u(x)$ exists. The function $u(x)$ is a solution of Dirichlet's problem for the equation $\bar Lu=0$, where the coefficients of the operator $\bar L$ are obtained by averaging the coefficients of the operator $L^\varepsilon$.