Contact value theorem for electric double layers with modulated surface charge density

The contact value theorem was originally derived for Coulomb fluids of mobile charged particles in thermal equilibrium, in the presence of interfaces carrying a {\em uniform} surface charge density and in the absence of dielectric discontinuities. It relates the pressure (the effective force) between two parallel electric double layers to the particle number density and the surface charge density at the interface, separately for each of the two electric double layers. In this paper, we generalise the contact value theorem to electric double layers with interfaces carrying a {\em modulated} surface charge density. The derivation is based on balance of forces exerted on interfaces. The relevance of particular terms of the contact value theorem is tested on an exactly solvable two-dimensional Coulomb system with counterions only at the coupling constant $\mathrm{\Gamma }=2$.