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 int...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis Vol. 243; no. 3; pp. 1773 - 1810
Main Authors: Muratov, Cyrill B., Novaga, Matteo, Ruffini, Berardo
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2022
Springer Nature B.V
Subjects:
Classical Mechanics
Complex Systems
Confining
Critical point
Digital Object Identifier
Energy
Euler-Lagrange equation
Fluid- and Aerodynamics
Mathematical and Computational Physics
Physics
Physics and Astronomy
Theoretical
Two dimensional models
Variational geometry
ISSN:0003-9527, 1432-0673
Online Access:Get full text
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Summary: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.
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ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-021-01738-0