Minimizers and critical volume for the three-dimensional liquid-drop energy

For the three-dimensional liquid-drop energy—De Giorgi perimeter plus half the Coulomb self-interaction—determine which sets minimize energy at fixed volume. The stated target is unique ball minimization, up to translation and null sets, for 0<V≤5(2−2^(2/3))/(2^(2/3)−1)≈3.51, and no minimizer above that threshold.

Why it matters and what counts as progress

Why it matters: It asks for the exact balance between surface energy and Coulomb repulsion, including both optimal shape and the loss of minimizers above threshold.

What counts as progress: Identify the exact volume threshold, prove ball optimality and uniqueness below it, and prove nonexistence above it in the same energy normalization.

How the problem developed

12 Aug 2026 · Minimization theorem [1]
Otis Chodosh and Matilde Gianocca, with ChatGPT 5.6 Pro credited for the results Proved that balls uniquely minimize the three-dimensional Gamow energy up to the critical volume V*, and that no minimizer exists above V*.

AI contribution

No compromise in the liquid drop model

Otis Chodosh, Matilde Gianocca

For the three-dimensional liquid-drop energy given by De Giorgi perimeter plus half the Coulomb self-interaction, balls uniquely minimize at every fixed volume 0<V≤5(2−2^(2/3))/(2^(2/3)−1)≈3.51, up to translation and null sets. Above that threshold, no minimizer exists.

AI involvementAI-autonomous

ImportanceMajor

Technical verificationFormal (human)

Human understandingFull