This paper provides expressions for the boundary potential that provides the best electrostatic potential approximation of a given $$L^2$$ vector field on a nice bounded region in $${\mathbb R}^N$$. The permittivity of the region is assumed to be known and the potential is required to be zero on the conducting part of the boundary. The boundary potential is found by solving the minimization conditions and using a special basis of the trace space for the space of allowable potentials. The trace space is identified by its representation with respect to a basis of $$\varSigma $$-Steklov eigenfunctions.