  In a continuum model of the solvation of charged molecules in an aqueous
solvent, the classical Poisson-Boltzmann (PB) theory is generalized to include
the solute point charges and the dielectric boundary that separates the
high-dielectric solvent from the low-dielectric solutes. With such a setting,
we construct an effective electrostatic free-energy functional of ionic
concentrations, where the solute point charges are regularized by a reaction
field. We prove that such a functional admits a unique minimizer in a class of
admissible ionic concentrations and that the corresponding electrostatic
potential is the unique solution to the boundary-value problem of the
dielectric-boundary PB equation. The negative first variation of this minimum
free energy with respect to variations of the dielectric boundary defines the
normal component of the dielectric boundary force. Together with the
solute-solvent interfacial tension and van der Waals interaction forces, such
boundary force drives an underlying charged molecular system to a stable
equilibrium, as described by a variational implicit-solvent model. We develop
an $L^2$-theory for the continuity and differentiability of solutions to
elliptic interface problems with respect to boundary variations, and derive an
explicit formula of the dielectric boundary force. With a continuum
description, our result of the dielectric boundary force confirms a
molecular-level prediction that the electrostatic force points from the
high-dielectric and polarizable aqueous solvent to the charged molecules. Our
method of analysis is general as it does not rely on any variational
principles.
