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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| Veröffentlicht in: | Archive for rational mechanics and analysis Jg. 243; H. 3; S. 1773 - 1810 |
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01.03.2022
Springer Nature B.V |
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| 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. |
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| AbstractList | 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. 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 12-derivative of the capacitary potential. |
| Author | Muratov, Cyrill B. Ruffini, Berardo Novaga, Matteo |
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| Cites_doi | 10.1007/s00205-014-0827-9 10.1098/rspa.1964.0151 10.1103/PhysRevLett.106.144501 10.1080/14786448208628425 10.1063/1.857551 10.1007/978-3-642-65183-0 10.1016/j.bulsci.2011.12.004 10.1103/PhysRevE.89.033011 10.4007/annals.2008.168.941 10.1007/s00205-010-0383-x 10.1007/PL00011302 10.1007/s00332-015-9259-4 10.1017/CBO9781139108133 10.1002/(SICI)1096-9888(199707)32:7<677::AID-JMS536>3.0.CO;2-G 10.1051/cocv/2012041 10.5565/PUBLMAT_60116_01 10.1007/s00205-003-0298-x 10.1007/s10404-015-1603-3 10.1007/s10231-016-0632-1 10.1090/S0002-9939-2014-12381-1 10.1146/annurev.fluid.39.050905.110245 10.4171/RMI/902 10.1146/annurev.fluid.39.050905.110159 10.2140/apde.2018.11.1113 10.1007/BF02937431 10.1137/20M1346584 10.1002/cpa.21739 10.1155/2019/8283496 |
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