Compactness and Sharp Lower Bound for a 2D Smectics Model

We consider a 2D smectics model E_ϵ( u) =1/2∫ _1/ε( u_z-1 /2u_x^2) ^2+ε( u_xx) ^2dx dz. For ε _n→ 0 and a sequence { u_n} with bounded energies E_ε _n( u_n) , we prove compactness of {∂ _zu_n} in L^2 and {∂ _xu_n} in L^q for any 1≤ q

6 . We also prove a sharp lower bound on E_ε when ε→ 0. The sharp bound corresponds to the energy of a 1D ansatz in the transition region.