The Boundary-value Problem in Domains with Very Rapidly Oscillating Boundary

We study the asymptotic behavior of the solution to boundary-value problem for the second order elliptic equation in the bounded domain V ;Rn with a very « rapidly oscillating locally periodic boundary. We assume that the Fourier boundary condition involving a small positive parameter « is posed on the oscillating part of the boundary and that the ny1.-dimensional volume of this part goes to infinity as «ª0. Under proper normalization conditions that homogenized problem is found and the estimates of the residual are obtained. Also, we construct an additional term of the asymptotics to improve the estimates of the residual. It is shown that the limiting problem can involve Dirichlet, Fourier or Neumann boundary conditions depending on the structure of the coefficient of the original problem.