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We study the statistical mechanics of classical Coulomb systems in a lowcoupling regime (Debye-Huckel regime) in a confined geometry with Dirichletboundary conditions. We use a method recently developed by the authors whichrelates the grand partition function of a Coulomb system in a confined geometrywith a certain regularization of the determinant of the Laplacian on thatgeometry with Dirichlet boundary conditions. We study several examples ofconfining geometry in two and three dimensions and semi-confined geometrieswhere the system is confined only in one or two directions of the space. Wealso generalize the method to study systems confined in arbitrary geometrieswith smooth boundary. We find a relation between the expansion for smallargument of the heat kernel of the Laplacian and the large-size expansion ofthe grand potential of the Coulomb system. This allow us to find thefinite-size expansion of the grand potential of the system in general. Werecover known results for the bulk grand potential (in two and threedimensions) and the surface tension (for two dimensional systems). We find thesurface tension for three dimensional systems. For two dimensional systems ourgeneral calculation of the finite-size expansion gives a proof of the existencea universal logarithmic finite-size correction predicted some time ago, atleast in the low coupling regime. For three dimensional systems we obtain aprediction for the perimeter correction to the grand potential of a confinedsystem.Comment: 26 page

General Considerations on the Finite-Size Corrections for Coulomb Systems in the Debye--H�ckel Regime