Conducting Flat Drops in a Confining Potential

We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a parameter measuring the relative strength of the Coulomb interaction. As a consequence, when the potential is confining and the Coulomb repulsion strength is below a critical value, we show existence and regularity estimates for volume-constrained minimizers. We also derive the Euler-Lagrange equation satisfied by regular critical points, expressing the first variation of the Coulombic energy in terms of the normal 1/2-derivative of the capacitary potential.