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.

Conducting Flat Drops in a Confining Potential

Cyrill B. Muratov
;
Matteo Novaga;Berardo Ruffini
2022-01-01

Abstract

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.
2022
Muratov, Cyrill B.; Novaga, Matteo; Ruffini, Berardo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/1157781
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