Compactness and Sharp Lower Bound for a 2D Smectics Model

We consider a 2D smectics model E-is an element of (u) = 1/2 integral(Omega) 1/epsilon (u(z) - 1/2 u(x)(2)) (2) + epsilon (u(xx))(2) dx dz. For epsilon(n) -> 0 and a sequence {u(n)} with bounded energies E epsilon(n) (u(n)), we prove compactness of {partial derivative(z)u(n)} in L-2 and {partial derivative(x)u(n)} in Lq for any 1 <= q < p under the additional assumption parallel to partial derivative(x)u(n)parallel to L p <= C for some p > 6. We also prove a sharp lower bound on Ee when epsilon -> 0. The sharp bound corresponds to the energy of a 1D ansatz in the transition region.