  It has been argued that for a finite two-dimensional classical Coulomb system
of characteristic size $R$, in its conducting phase, as $R\to \infty $ the
total free energy (times the inverse temperature $\beta$) admits an expansion
of the form: $\beta F=AR^{2}+BR+{1/6}\chi \ln R,$ where $\chi $ is the Euler
characteristic of the manifold where the system lives. The first two terms
represent the bulk free energy and the surface free energy respectively. The
coefficients $A$ and $B$ are non-universal but the coefficient of $\ln R$ is
universal: it does not depend on the detail of the microscopic constitution of
the system (particle densities, temperature, etc...). By doing the usual
Legendre transform this universal finite-size correction is also present in the
grand potential. The explicit form of the expansion has been checked for some
exactly solvable models for a special value of the coulombic coupling. In this
paper we present a method to obtain these finite-size corrections at the
Debye--H\"uckel regime. It is based on the sine-Gordon field theory to find an
expression for the grand canonical partition function in terms of the spectrum
of the Laplace operator. As examples we find explicitly the grand potential
expansion for a Coulomb system confined in a disk and in an annulus with ideal
conductor walls.
