On Equilibrium Shape of Charged Flat Drops

Abstract

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.

A spectral shape optimization problem with a nonlocal competing term

“…In turn this led to a renewed interest in several variational models where a competition between a cohesive term is balanced by a repulsive term. A non-exhaustive list of papers in this field is [2,6,11,12,[14][15][16]19,20,23,26,29,33].…”

“…, but this does not affect our argument. Keeping in mind (33), we know that t → m(t) is an increasing function, and that for a.e. t ∈ R one has m (t) = ε(t).…”

On minimizers of an isoperimetric problem with long-range interactions under a convexity constraint

“…Let us point out that in [23], still another regularizing mechanism is proposed. There, wellposedness is obtained by adding an entropic term which prevents charges to concentrate too much on the boundary of E. We point out that it has been recently shown in [24] that in the borderline case α = 1, N = 2 such a regularization is not needed for the model to be well-posed. For a more exhaustive discussion about the physical motivations and the literature on related problems we refer to the papers [23,11].…”

Reifenberg flatness for almost-minimizers of the perimeter under minimal assumptions

“…Motivated by applications to a variational model for charged liquid drops (see [10,11,7]), the aim of this note is to study the regularity of quasi-minimizers of the (anisotropic) perimeter under minimal assumptions. For a uniformly elliptic and regular anisotropy Φ (see Definition 2.1) and a set of finite perimeter E, we define P Φ (E) as…”