  Electric double layers (EDL) with counterions only, say electrons with the
elementary charge $-e$, in thermal equilibrium at the inverse temperature
$\beta$ are considered. In particular, we study the effect of the surface
charge modulation on the particle number density profile and the pressure. The
mobile particles are constrained to the surface of a 2D cylinder and immersed
in vacuum (no dielectric image charges). An EDL corresponds to the end-circle
of the cylinder which carries a (fixed) position-dependent line charge density.
The geometries of one single EDL and two EDLs at distance $d$ are considered;
the particle density profile is studied for both geometries, the effective
interaction of two EDLs is given by the particle pressure on either of the line
walls. For any coupling constant $\Gamma\equiv \beta e^2 = 2\times {\rm
integer}$, there exists a mapping of the 2D one-component Coulomb system onto
the 1D interacting anticommuting-field theory defined on a chain of sites.
Using specific transformation symmetries of anticommuting variables, the
contact value theorem is generalized to the EDL with modulated line charge
density. For the free-fermion coupling $\Gamma=2$ it is shown that, under
certain conditions, the matrix of interaction strengths between anticommuting
variables diagonalizes itself which permits one to obtain exact formulas for
the particle density profile as well as the pressure. The obtained results
confirm the previous indications of weak-coupling and Monte Carlo observations
that the surface charge inhomogeneity implies an enhancement of the counterion
density at the contact with the charged line and a diminution of the pressure
between two parallel lines in comparison with the uniformly charged ones (with
the same mean charge densities).
