An Improved Nonexistence Bound for the Liquid Drop Model

Abstract: For Gamow's liquid drop model, we improve the nonexistence bound of Frank-Killip-Nam from 8 to 7.5. The key additional input is a geometric perimeter inequality arising from the capillarity problem. This yields a quantitative gain in the averaged slicing argument and shows that the variational problem admits no minimizer for $A\geq 7.5$.

• The paper improves the Frank–Killip–Nam nonexistence threshold from A ≤ 8 to A < 7.5 by proving that any attained minimizer must have volume below 7.5.
• It strengthens the averaged slicing method by applying a sharp half-space capillarity isoperimetric inequality to show that the cutting deficit is quantitatively positive.
• The resulting bootstrap estimate combines geometric bounds, Coulomb-energy inequalities, and explicit numerical optimization, while leaving the conjectured splitting threshold near A ≈ 3.512 unresolved.

The paper under review sharpens a nonexistence result for Gamow's liquid drop model, proving that if the variational problem at fixed volume $A$ is attained by a minimizer, then necessarily $A < 7.5$. This improves the classical bound $A \leq 8$ of Frank–Killip–Nam [FKN:2016]. The mechanism is a quantitative strengthening of the averaged slicing argument: where Frank–Killip–Nam used only nonnegativity of a cutting deficit, the author extracts a strictly positive lower bound on that deficit using a capillarity isoperimetric inequality for half-space cuts.

Background and main theorem

For a measurable set $\Omega \subset \mathbb{R}^3$ of finite perimeter, the liquid drop energy is

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

and one studies $E(A) = \inf\{E(\Omega): |\Omega| = A\}$. Perimeter favors a single ball while the repulsive Coulomb term favors fragmentation. The conjectured picture is that balls minimize up to the splitting threshold $A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$, beyond which two distant balls are preferable; Chodosh and Ruohoniemi proved ball minimality for $A \leq 1$. The main theorem here states:

Theorem. If $E(A)$ is attained, then $A < 7.5$.

Since existence of minimizers is not known in general (and is expected to fail for large $A < 7.5$0), this is a nonexistence statement: no minimizer can exist above volume $A < 7.5$1, closing part of the gap between the known threshold behavior near $A < 7.5$2 and the previous upper bound of $A < 7.5$3.

The averaged slicing argument and its quantitative gap

Fixing a hypothetical minimizer $A < 7.5$4 of volume $A < 7.5$5, the Frank–Killip–Nam strategy slices $A < 7.5$6 by planes $A < 7.5$7 with normal $A < 7.5$8. Writing $A < 7.5$9 for the two pieces, $A \leq 8$0 for the cross-sectional area, and

$A \leq 8$1

for the cutting deficit, cutting formulas give $A \leq 8$2, where $A \leq 8$3 is the cross Coulomb interaction between the two halves. Minimality forces $A \leq 8$4, and averaging over $A \leq 8$5 and $A \leq 8$6 via Cavalieri's principle yields

$A \leq 8$7

which is nonnegative only if $A \leq 8$8. The improvement requires showing that $A \leq 8$9 cannot be identically zero — that the three nonnegative terms in its decomposition,

$\Omega \subset \mathbb{R}^3$0

with $\Omega \subset \mathbb{R}^3$1, cannot all vanish simultaneously.

Capillarity inequalities as the new input

The key geometric observation is that each piece $\Omega \subset \mathbb{R}^3$2 lies in a half-space and touches the cutting plane along area $\Omega \subset \mathbb{R}^3$3. Applying the sharp relative isoperimetric inequality for the capillarity functional $\Omega \subset \mathbb{R}^3$4 — whose minimizers are spherical caps, per Pascale–Pozzetta — yields, for every $\Omega \subset \mathbb{R}^3$5,

$\Omega \subset \mathbb{R}^3$6

where $\Omega \subset \mathbb{R}^3$7. Combined with the lower bound $\Omega \subset \mathbb{R}^3$8 on the Coulomb term (from Chodosh–Ruohoniemi), this gives an explicit lower bound on $\Omega \subset \mathbb{R}^3$9 in terms of the dimensionless ratio $$E(\Omega) = P(\Omega) + D(\Omega), \qquad D(\Omega) = \frac12 \iint_{\Omega\times\Omega} \frac{dx\,dy}{|x-y|},$$0 through the function

$$E(\Omega) = P(\Omega) + D(\Omega), \qquad D(\Omega) = \frac12 \iint_{\Omega\times\Omega} \frac{dx\,dy}{|x-y|},$$1

which admits the closed form $$E(\Omega) = P(\Omega) + D(\Omega), \qquad D(\Omega) = \frac12 \iint_{\Omega\times\Omega} \frac{dx\,dy}{|x-y|},$$2 and satisfies $$E(\Omega) = P(\Omega) + D(\Omega), \qquad D(\Omega) = \frac12 \iint_{\Omega\times\Omega} \frac{dx\,dy}{|x-y|},$$3.

Lower bound on the deficit and the bootstrap

Two further estimates complete the quantitative control. First, the third decomposition term satisfies $$E(\Omega) = P(\Omega) + D(\Omega), \qquad D(\Omega) = \frac12 \iint_{\Omega\times\Omega} \frac{dx\,dy}{|x-y|},$$4, where $$E(\Omega) = P(\Omega) + D(\Omega), \qquad D(\Omega) = \frac12 \iint_{\Omega\times\Omega} \frac{dx\,dy}{|x-y|},$$5 combines the exact value $$E(\Omega) = P(\Omega) + D(\Omega), \qquad D(\Omega) = \frac12 \iint_{\Omega\times\Omega} \frac{dx\,dy}{|x-y|},$$6 valid for $$E(\Omega) = P(\Omega) + D(\Omega), \qquad D(\Omega) = \frac12 \iint_{\Omega\times\Omega} \frac{dx\,dy}{|x-y|},$$7 (Chodosh–Ruohoniemi), the Frank–Nam interpolation inequality $$E(\Omega) = P(\Omega) + D(\Omega), \qquad D(\Omega) = \frac12 \iint_{\Omega\times\Omega} \frac{dx\,dy}{|x-y|},$$8, and the trial upper bound $$E(\Omega) = P(\Omega) + D(\Omega), \qquad D(\Omega) = \frac12 \iint_{\Omega\times\Omega} \frac{dx\,dy}{|x-y|},$$9 from three equal balls. Second, these are assembled into

$E(A) = \inf\{E(\Omega): |\Omega| = A\}$0

with $E(A) = \inf\{E(\Omega): |\Omega| = A\}$1.

Integrating over the region where $E(A) = \inf\{E(\Omega): |\Omega| = A\}$2 (i.e., slices cutting off volume at most $E(A) = \inf\{E(\Omega): |\Omega| = A\}$3 from either side) and combining with the FKN identity gives the bootstrap inequality

$E(A) = \inf\{E(\Omega): |\Omega| = A\}$4

Two properties of $E(A) = \inf\{E(\Omega): |\Omega| = A\}$5 drive the argument: a Lipschitz estimate $E(A) = \inf\{E(\Omega): |\Omega| = A\}$6, which makes $E(A) = \inf\{E(\Omega): |\Omega| = A\}$7 with $E(A) = \inf\{E(\Omega): |\Omega| = A\}$8 strictly decreasing on $E(A) = \inf\{E(\Omega): |\Omega| = A\}$9 since $A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$0; and the explicit numerical bound

$A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$1

The latter is proved by constructing an admissible choice $A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$2 for $A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$3, deriving the polynomial lower bound $A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$4 with $A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$5, and evaluating the resulting integrals explicitly to obtain $A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$6. Since $A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$7, monotonicity forces $A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$8.

Limitations and open questions

Several restrictions should be noted plainly. The quantitative gain relies on the Chodosh–Ruohoniemi classification of minimizers for volumes $A_c = 5(2-2^{2/3})/(2^{2/3}-1) \approx 3.512$9, so the argument inherits that result's validity; it also uses the unproven conjectural structure only implicitly through trial configurations, but the bootstrap is confined to $A \leq 1$0, and the numerical margin over $A \leq 1$1 ($A \leq 1$2) is small, so pushing below $A \leq 1$3 would require sharper estimates on $A \leq 1$4. The paper does not address whether minimizers exist for any $A \leq 1$5, nor does it narrow the gap between the conjectured splitting threshold $A \leq 1$6 and the new nonexistence bound. Whether the capillarity-based deficit estimate can be extended to slices cutting off volumes larger than $A \leq 1$7 — which would remove the restriction $A \leq 1$8 and potentially improve the constant substantially — remains open.

Conclusion

The paper improves the nonexistence threshold for Gamow's liquid drop model from $A \leq 1$9 to $E(A)$0 by supplying a positive lower bound, proportional to the cross-sectional area, on the cutting deficit in the averaged slicing argument. The essential new ingredient is the capillarity isoperimetric inequality applied to half-space sections, combined with explicit optimization and a contraction/bootstrap step whose numerical verification is carried out in full detail. The result narrows, but does not close, the interval of volumes in which the variational problem may still admit minimizers.