Non-spherical equilibrium shapes in the liquid drop model

Abstract

We prove the existence of a family of volume-constrained critical points of the liquid drop functional, which are cylindrically but not spherically symmetric. This family bifurcates from the ball and exchanges stability with it. We justify a formula of Bohr and Wheeler for the energy of these sets.

Existence and nonexistence in the liquid drop model

“…Then the desired conclusion follows from (10) and the fact that 2 10) is trivial when α = 1, so we will distinguish two cases α ∈ (0, 1) and α ∈ (1, 2).…”

“…In the last decade, this model (for general λ and N ) has gained renewed interest in the mathematics literature. We refer to [6] for a review and, for instance, to [18,22,19,16,2,12,9,11,17,10] and references therein; see also [14,24]. A variant of the problem with a constant background has also been intensely studied, see, for instance, [1,4,5,3,20,8,13] and references therein.…”

Some minimization problems for mean field models with competing forces

“…Finally, returning to the standard liquid drop model with λ = 1 and N = 3, we mention the open problem to make the global bifurcation picture of Bohr and Wheeler [2] rigorous. For an initial local bifurcation result, see [24].…”

On the Ternary Ohta-Kawasaki Free Energy and Its One-dimensional Global Minimizers

“…For a small volume (i.e., Vol {𝑢 = 1} 1), the global minimizer is a perfect ball; for a large volume, the global minimizer does not exist, indicating nuclear fission. Recent works on the liquid drop model include [28]. Another example is the case when 𝐺 is replaced by the screened Coulomb kernel with 𝛾 1 and 1 + 𝜔 1 but without the volume constraint.…”