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

We consider the 3D smectic energy $$\begin{aligned} {\mathcal {E}}_{\epsilon }\left( u\right) =\frac{1}{2}\int _{\Omega }\frac{1}{\varepsilon } \left( \partial _{z}u-\frac{(\partial _{x}u)^{2}+(\partial _{y}u)^{2}}{2}\right) ^{2} +\varepsilon _{x}^{2}u+\partial _{y}^{2}u\right) ^{2}dx\,dy\,dz. \end{aligned}$$ The model contains as a special case well-known 2D Aviles-Giga model. prove sharp lower bound on $${\mathcal {E}}_{\varepsilon }$$ $$\varepsilon \rightarrow 0$$ by introducing analogues of Jin–Kohn entropies Jin and Kohn (J Nonlinear Sci 10:355–390, 2000). corresponds to an equipartition between bending compression strains was previously demonstrated in physics literature only when approximate Gaussian curvature each layer vanishes. Also, for _{n}\rightarrow energy-bounded sequence $$\{u_n \}$$ with $$\Vert \nabla u_n\Vert _{L^{p}(\Omega )},\, \Vert _{L^2(\partial \Omega )}\le C$$ some $$p>6$$ , we obtain compactness $$\nabla u_{n}$$ $$L^{2}$$ assuming that $$\Delta _{xy}u_{n}$$ has constant sign n.