×

On equilibrium shape of charged flat drops. (English) Zbl 1394.49046

Summary: The equilibrium shapes of two-dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while for larger charge minimizers do not exist. Analogous results hold for drops whose potential, rather than charge, is fixed.

MSC:

49S05 Variational principles of physics
76M30 Variational methods applied to problems in fluid mechanics
49J40 Variational inequalities
PDFBibTeX XMLCite
Full Text: DOI arXiv