Variational principle for Poisson's equation for an impurity ion in a medium with spatially variable dielectric constant
Kenneth R. Brownstein · Physical review. B, Solid state · 1977
A recent article by Csavinszky deals with a variational principle used to solve a Poisson's equation containing a spatially variable dielectric constant $K(r)$. However, a term involving $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ abla}}K$ is neglected as an approximation. Here it is shown how to construct a functional leading to the correct Poisson's equation including the neglected $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ abla}}K$ term. This functional is further modified to eliminate a self-energy infinity associated with the point-charge type boundary condition.