Nonlinear approximation of 3D smectic liquid crystals: sharp lower bound and compactness

We consider the 3D smectic energy E-epsilon (u) = 1/2 integral(Omega) 1/epsilon (partial derivative(z)u - (partial derivative(x)u)(2) + (partial derivative(y)u)(2)/2)(2) + epsilon (partial derivative(2)(x)u + partial derivative(2)(y)u)(2) dx dy dz. The model contains as a special case the well-known 2D Aviles-Giga model. We prove a sharp lower bound on E-epsilon as epsilon -> 0 by introducing 3D analogues of the Jin-Kohn entropies Jin and Kohn (J Nonlinear Sci 10:355-390, 2000). The sharp bound corresponds to an equipartition of energy between the bending and compression strains and was previously demonstrated in the physics literature only when the approximate Gaussian curvature of each smectic layer vanishes. Also, for epsilon(n) -> 0 and an energy-bounded sequence {u(n)} with parallel to del u(n)parallel to(Lp(Omega)), parallel to del u(n)parallel to(L2(partial derivative Omega)) <= C for some p > 6, we obtain compactness of del u(n) in L-2 assuming that Delta(xy)u(n) has constant sign for each n.