New variational analysis on the sharp interface of multiscale implicit solvation: General expressions and applications

The interface definition between regions of different scales becomes a key component of a multiscale model in mathematical biology and other fields. Differential geometry based surface models have been proposed to apply the theory of differential geometry as a natural means to couple polar-nonpolar and solute-solvent interactions. As a consequence, the variational analysis of such models heavily relies on the variation of the interface. In this work, we provide a new variational approach to conduct the variational analysis on the sharp interface of multiscale implicit solvation models. It largely simplifies the computations of variations of the area and volume functionals. Moreover, general expressions of the second variation formulas of the solvation energy functional are obtained and used for the stability analysis of the equilibrium interface. Finally, we establish a reasonable concept of stability which generalizes the well-known results in minimal surfaces with constant volume and then the necessary and sufficient condition for stability. Our work paves the way to conducting stability analysis for a general energy functional especially with constant volume.