Dirichlet’s Problem for an Equation with Periodic Coefficients Depending on a Small Parameter

This paper studies the limiting behavior of the solution u^ε (x) of Dirichlet’s problem for \[ L^ε u^ε = 1/2∑ {aᵢⱼ } ( {x/ε } ){{∂ ^2 u^ε }}{{∂ x^i ∂ x^j }} + ∑ {b_i } ( {x/ε } ){{∂ u^ε }}{{∂ x^i }} - c( {x/ε } )u^ε = 0, \] when ε → 0. The coefficients of the operator L¹ are assumed to be periodic. It is proved that lim ε → 0 u^ε (x) = u(x) exists. The function $u(x)$ is a solution of Dirichlet’s problem for the equation Lu = 0, where the coefficients of the operator L are obtained by averaging the coefficients of the operator L^ε.