Formal compactness and lim-inf conjecture for the three-dimensional bilayer membrane model

Let $(u_\varepsilon,v_\varepsilon)_{\varepsilon>0}$ be a sequence in $\mathcal K_\varepsilon$ satisfying

$$\mathcal G_\varepsilon(u_\varepsilon,v_\varepsilon)\leq\Lambda.$$

A finite union of generalized surfaces $S_j$, with associated even density functions $\vartheta_j$ and generalized second fundamental form $\Pi$, is obtained along a subsequence, with each $S_j$ having empty boundary. If $H$ and $K$ denote the trace and determinant of $\Pi$, respectively, then

Formal compactness and lim-inf conjecture. There exists a subsequence, not relabelled, such that

u_\u_\varepsilon\to\sum_{j=1}^L\vartheta_j\mathcal H^2\lfloor S_j\quad\text{as }\varepsilon\to0,

and

$$\sum_{j=1}^L\int_{S_j}\left(\frac14H^2-\frac16K\right)\vartheta_j\,d\mathcal H^2\leq\liminf_{\varepsilon\to0}\mathcal G_\varepsilon(u_\varepsilon,v_\varepsilon).$$

This is presented in the source as a formal three-dimensional analogue of the established two-dimensional compactness and Gamma-convergence theory. The generalized surfaces and curvature quantities indicate the expected limiting geometric structure, while a full Gamma-convergence result in three dimensions is stated to be substantially more difficult and is not established here.