The thermodynamic potential for fixed charges in dielectric continua is given for arbitrary values of the polarization density. Minimization of the functional gives the equilibrium polarization density (equivalently image charges) and the electrostatic potential througnout the media. The functional, which in general involves volume integrals, for the case of piecewise uniform dielectrica is reduced to surface integrals of the constrained surface charge density at the dielectric discontinuities. A linear integral equation for the liduced surface charge, which can be solved by iteration, is given. The conjugate thermodynamic force for the constrained surface charge is also given. The latter formulation is suitable for Car-Parrinello or Lagrangian molecular dynamics in complex geometries.
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