No compromise in the liquid drop model

Abstract: We characterize minimizers in Gamow's liquid drop problem.

• The paper proves that balls are the unique volume-constrained minimizers of Gamow’s liquid drop energy for volumes V≤V*≈3.51, while no minimizer exists above this threshold.
• The authors combine Euler–Lagrange and dilation identities with capacitary potential estimates, Bochner inequalities, and Gauss–Bonnet to rule out globally minimizing nonspherical shapes.
• The result establishes a sharp transition from one spherical drop to nonattainment through fission into separated components, while distinguishing global minimizers from non-spherical stationary configurations.

No compromise in the liquid drop model

Problem formulation and principal result

Chodosh and Gianocca study Gamow’s liquid drop functional in $\mathbb{R}^3$,

$E(\Omega)=P(\Omega)+D(\Omega),$

where $P(\Omega)$ is the perimeter and

$$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$

is the Newtonian Coulomb self-energy. The variational problem fixes the volume $|\Omega|=V$ and minimizes $E$ over measurable sets. The perimeter term favors compact, spherical configurations, whereas the repulsive Coulomb term favors spatial separation. The central question is whether these competing effects can produce a nonspherical connected minimizer, or whether the global problem exhibits a sharp transition directly from one ball to nonattainment through fission into widely separated components.

The main theorem gives a complete answer. Define

$V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$

Then:

• for $V\leq V_*$, the unique minimizers, modulo translations, are balls of volume $V$;
• for $V>V_*$, no volume-constrained minimizer exists.

Thus, there is no intermediate regime in which a non-spherical or connected nonspherical drop globally minimizes the functional. The threshold $E(\Omega)=P(\Omega)+D(\Omega),$0 is exactly the point at which the energy of one ball coincides with the limiting energy of two equal balls sent infinitely far apart. The result resolves the conjectured sharp transition between spherical minimizers and loss of compactness (2608.11517).

This conclusion is stronger than the previously available small-volume minimality and large-volume nonexistence results. In particular, it excludes global minimizers associated with the non-spherical stationary configurations studied in related work, even though such equilibria may exist as critical points.

Scaling and the competing-ball mechanism

For a ball $E(\Omega)=P(\Omega)+D(\Omega),$1 of volume $E(\Omega)=P(\Omega)+D(\Omega),$2, with radius $E(\Omega)=P(\Omega)+D(\Omega),$3, the perimeter and Coulomb energy satisfy

$E(\Omega)=P(\Omega)+D(\Omega),$4

Consequently,

$E(\Omega)=P(\Omega)+D(\Omega),$5

The perimeter contribution scales like $E(\Omega)=P(\Omega)+D(\Omega),$6, while the Coulomb contribution scales like $E(\Omega)=P(\Omega)+D(\Omega),$7. For sufficiently large volume, it is therefore energetically preferable to divide the mass into multiple pieces. Two equal balls of volume $E(\Omega)=P(\Omega)+D(\Omega),$8 placed at arbitrarily large separation provide a minimizing sequence whose limiting energy is

$E(\Omega)=P(\Omega)+D(\Omega),$9

Equating this quantity with $P(\Omega)$0 produces the explicit threshold $P(\Omega)$1. The comparison is not merely asymptotic: the theorem proves that the two-ball competitor determines the exact global transition. For $P(\Omega)$2, a hypothetical minimizer would have to beat the separated two-ball energy, but the paper derives a contradiction from the stationarity equation and a capacitary inequality.

The same scaling calculation yields the minimal binding energy

$P(\Omega)$3

and identifies the optimizing volume as $P(\Omega)$4. The optimizer is a ball. This gives an exact value for the least energy per unit volume, not merely a qualitative characterization.

Stationarity and the capacitary potential

The proof is based on a refined treatment of the Euler–Lagrange equation. A smooth stationary domain satisfies

$P(\Omega)$5

where $P(\Omega)$6 is the sum of the principal curvatures with respect to the outward normal and

$P(\Omega)$7

is the Coulomb potential. Here $P(\Omega)$8 is the Lagrange multiplier associated with the volume constraint.

The central methodological step is to introduce the capacitary potential $P(\Omega)$9 of the filled hull $$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$0 of $$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$1. It solves

$$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$2

The capacity is normalized by

$$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$3

Rather than integrating the Euler–Lagrange equation against the constant function on the boundary, the authors weight it by $$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$4. This produces an exact identity linking the mean-curvature term, the Coulomb term, the volume, the Lagrange multiplier, and the capacity. Specifically, an integration-by-parts argument gives

$$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$5

The choice of weight is structurally adapted to the Coulomb interaction: the Newtonian potential satisfies $$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$6, while $$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$7 is harmonic outside $$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$8 and constant inside it. This converts the nonlocal term into the volume exactly.

The capacitary inequality

The main geometric estimate is derived from the level sets of $$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$9. Define

$|\Omega|=V$0

For balls, $|\Omega|=V$1 is identically $|\Omega|=V$2. General capacitary monotonicity results imply that $|\Omega|=V$3 is nonincreasing and convex, with $|\Omega|=V$4 as $|\Omega|=V$5.

The authors combine a Bochner inequality for $|\Omega|=V$6, the geometry of the level sets, and Gauss–Bonnet. The resulting estimate is

$|\Omega|=V$7

At the boundary level $|\Omega|=V$8, this implies

$|\Omega|=V$9

Using both the capacity identity and Cauchy–Schwarz, the authors obtain

$E$0

After eliminating the capacity, this yields the decisive lower bound

$E$1

The estimate is dimension-specific and uses the topology of connected level surfaces in $E$2. The Gauss–Bonnet contribution supplies the sharp $E$3 term; the authors emphasize that the improvement over earlier capacitary estimates depends on this precise geometric structure. Equality is rigidly associated with the spherical case.

Proof of spherical minimality

Existence of a minimizer is known throughout the range $E$4. Regularity theory gives a bounded representative with $E$5 boundary, while connectedness follows because translating one connected component to infinity strictly reduces the positive Coulomb interaction.

A dilation variation gives the identity

$E$6

Let $E$7 denote the perimeter of the ball of volume $E$8, and write

$E$9

The isoperimetric inequality guarantees $V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$0, with equality only for a ball. Comparison with the ball bounds $V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$1 from above. The dilation identity then gives an upper bound on $V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$2, while the capacitary estimate gives the lower bound $V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$3 because $V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$4.

The two estimates are incompatible whenever $V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$5. More explicitly, their difference contains the strictly positive factor

$V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$6

Hence $V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$7, and the equality case of the isoperimetric inequality forces $V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$8 to be a translate of the ball. This establishes both minimality and uniqueness.

The argument is notable because it does not require a priori convexity, symmetry, or perturbative closeness to a sphere. The conclusion follows from the interaction of the exact dilation identity, the global ball comparison, and the capacitary lower bound.

Proof of nonexistence beyond the threshold

For $V_*= 5\frac{2-2^{2/3}}{2^{2/3}-1} \approx 3.51.$9, the authors assume that a minimizer exists and compare it with two equal balls at infinite separation. This gives

$V\leq V_*$0

The same perimeter excess parameter $V\leq V_*$1 is used to convert the energy comparison into an upper bound for the Lagrange multiplier. The capacitary estimate then provides a lower bound. The contradiction is handled in two volume ranges.

For $V\leq V_*$2, the discrepancy between the lower and upper multiplier bounds decomposes into a term proportional to $V\leq V_*$3 and a term proportional to $V\leq V_*$4. Both are positive under the relevant perimeter constraint. Thus no stationary minimizer can exist.

For $V\leq V_*$5, the authors use the second branch of the capacitary estimate and optimize a cubic expression. The resulting upper bound is strictly smaller than $V\leq V_*$6, contradicting the lower bound. Volumes beyond this interval were already covered by prior nonexistence results, including the bound $V\leq V_*$7 and its subsequent improvement to $V\leq V_*$8 (Schulz, 10 Aug 2026).

The proof therefore closes the previously unresolved interval between the exact fission threshold and the established large-volume nonexistence regime. The claim is particularly strong at the transition: at $V\leq V_*$9 the ball remains the unique minimizer, whereas immediately above $V$0 the infimum is not attained. There is no minimizer at the threshold corresponding to two finite, separated components; the optimal separation diverges.

Relation to stationary non-spherical configurations

The theorem concerns global minimizers, not all stationary domains. The liquid drop Euler–Lagrange equation can admit non-spherical equilibria, including bifurcating or otherwise geometrically structured solutions. Such configurations are not contradicted by the result. Instead, the theorem shows that they cannot attain the global constrained minimum in the parameter range where the ball is minimizing or where the infimum is realized only through splitting.

This distinction is important for both variational analysis and physical interpretation. The energy landscape may contain metastable or unstable stationary shapes even though the global variational problem has only spherical minimizers below $V$1 and no minimizers above it. The paper consequently separates the classification of critical points from the classification of global ground states.

Implications and future directions

The result gives a complete global phase diagram for the three-dimensional, isotropic, Newtonian liquid drop model. The transition is governed exactly by comparison between a single ball and two equal balls, rather than by a nonspherical compromise between perimeter and Coulomb energy. This supports a sharp fission interpretation: once the repulsive interaction dominates sufficiently, the variational problem loses compactness instead of selecting a finite-size deformed minimizer.

Several extensions remain open. The capacitary argument is tightly tied to three-dimensional Newtonian potential theory, Gauss–Bonnet, and connectedness properties of harmonic level sets. Generalizations to other dimensions, Riesz kernels, anisotropic perimeters, or screened interactions would require replacement inequalities with comparable rigidity. In anisotropic models, Wulff shapes may replace balls, but the exact threshold and the structure of the nonattainment mechanism are not automatic.

The paper also illustrates a technically nontrivial use of AI-assisted mathematical discovery. The manuscript reports that ChatGPT generated the fundamental proof strategy, which the authors subsequently checked and reworked. This does not alter the mathematical content, but it raises methodological questions about formal verification, proof provenance, and the role of LLMs in discovering combinations of existing geometric identities. For future AI systems, the relevant advance would not be fluent exposition alone, but reliable generation of verifiable arguments involving regularity assumptions, distributional inequalities, equality cases, and sharp constants. Formal proof assistants could be particularly useful for auditing the capacitary estimate and the transition-case algebra.

Conclusion

“No compromise in the liquid drop model” (2608.11517) establishes the exact global behavior of Gamow’s liquid drop functional in $V$2. Balls uniquely minimize the energy up to the sharp threshold $V$3, while no minimizer exists above it. The proof combines first variation, scaling, Newtonian potential theory, capacitary monotonicity, Bochner identities, and Gauss–Bonnet. Its central conclusion is that the competition between surface tension and Coulomb repulsion produces no globally optimal intermediate deformation: the system remains spherical until the variational problem resolves the competition by splitting and loss of compactness.