  We consider the 3D smectic energy $$\mathcal{E}_{\epsilon }\left( u\right)
=\frac{1}{2}\int_{\Omega }\frac{1}{\varepsilon } \left( u_z-\frac{( u_x)^{2}+(
u_y)^{2}}{2}\right) ^{2}+\varepsilon \left(
u_{xx}+u_{yy}\right)^{2}\,dx\,dy\,dz. $$ The model contains as a special case
the well-known 2D Aviles-Giga model. We prove a sharp lower bound on
$\mathcal{E}_{\varepsilon}$ as $\varepsilon \to 0$ by introducing 3D analogues
of the Jin-Kohn entropies. 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 $\varepsilon_{n}\rightarrow
0$ and an energy-bounded sequence $\{u_n \}$ with $\|\nabla
u_n\|_{L^{p}(\Omega)},\,\|\nabla u_n\|_{L^2(\partial \Omega)}\leq C$ for some
$p>6$, we obtain compactness of $\nabla u_{n}$ in $L^{2}$ assuming that
$\Delta_{xy}u_{n}$ has constant sign for each $n$.
