Closed curves of prescribed curvature and a pinning effect

We prove that for any H : R(2) -> R which is Z(2)-periodic, there exists H(epsilon), which is smooth, epsilon-close to H in L(1), with L(infinity)-norm controlled by the one of H, and with the same average of H, for which there exists a smooth closed curve gamma(epsilon) whose curvature is H(epsilon). A pinning phenomenon for curvature driven flow with a periodic forcing term then follows. Namely, curves in fine periodic media may be moved only by small amounts, of the order of the period.