Conducting Flat Drops in a Confining Potential

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