Operator estimates for elliptic problems in domains with fast locally periodically oscillating boundary and Robin boundary condition: homogenized Dirichlet condition

Abstract

We consider a general elliptic operator in a multidimensional domain with a fast locally periodically oscillating boundary. On this oscillating boundary we impose the Robin condition with a strictly positive coefficient. It is supposed that the amplitude of the boundary oscillations is much larger than the linear size of the periodicity cell. Under such assumptions, we show that the homogenization leads to the Dirichlet condition on the limiting boundary. Our main result are $W_2^1$– and $L_2$–operator estimates.