Existence and nonexistence in the liquid drop model

Abstract

We revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei.

Some minimization problems for mean field models with competing forces

“…As a first step towards this conjecture, before asking whether minimizers for E gld λ (m) are balls for all m ≤ m * , it is natural to ask whether minimizers exist for all m ≤ m * . This is indeed the case, as shown in [32]. Moreover, it is shown there as well that if there are no minimizers for m > m * , then balls are minimizers for m ≤ m * .…”

“…Then, if λ ≤ 2 (and λ < N, as always), one can show that m n.e. c < ∞, that is, there is no minimizer for large m. This is due to [38,39,43,32]. It seems to be unknown whether m n.e.…”