We consider a 2-dimensional thin domain with order of thickness {epsilon} which presents oscillations of amplitude also {epsilon} on both boundaries, top and bottom, but the period of the oscillations are of different order at the top and at the bott
om. We study the behavior of the Laplace operator with Neumann boundary condition and obtain its asymptotic homogenized limit as the parameter {epsilon} goes to 0. We are interested in understanding how this different oscillatory behavior at the boundary, influences the limit problem.
We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $Omega$ and on the domain $phi(Omega)$ resulting from $Omega$ by means of a bi-Lipschitz map $phi$. We c
onsider the solutions $u$ and $tilde u$ of the corresponding elliptic equations with the same right-hand side $fin L^2(Omegacupphi(Omega))$. Under certain assumptions we estimate the difference $| ablatilde u- abla u|_{L^2(Omegacupphi(Omega))}$ in terms of certain measure of vicinity of $phi$ to the identity map. For domains within a certain class this provides estimates in terms of the Lebesgue measure of the symmetric difference of $phi(Omega)$ and $Omega$, that is $|phi(Omega)triangle Omega|$. We provide an example which shows that the estimates obtained are in a certain sense sharp.
