Calculus of Variations and Geometric Measure Theory

C. Muratov - M. Novaga - B. Ruffini

Conducting flat drops in a confining potential

created by novaga on 03 Jun 2020
modified on 16 Feb 2022


Published Paper

Inserted: 3 jun 2020
Last Updated: 16 feb 2022

Journal: Arch. Rat. Mech. Anal.
Volume: 243
Number: 3
Pages: 1773-1810
Year: 2022

ArXiv: 2006.02839 PDF


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 partial regularity of 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 $\frac12$-derivative of the capacitary potential.