Existence and nonexistence in the liquid drop model

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.

The perimeter Per Ω is taken in the sense of De Giorgi, namely Per Ω = sup which is simply the surface area of Ω when the boundary is smooth. We consider the minimization problem E(m) = inf |Ω|=m E(Ω).
The most important case is λ = 1 in dimension N = 3, which goes back to Gamow's liquid drop model for atomic nuclei [15]. In this case, a nucleus is thought of consisting of nucleons (protons and neutrons) in a set Ω ⊂ R N . The nucleons are assumed to be concentrated with constant density, which implies that the number of nucleons is proportional to |Ω|. The perimeter term in the energy functional corresponds to a surface tension, which holds the nuclei together. The second term in the energy functional corresponds to a Coulomb repulsion among the protons. Here for simplicity we have scaled all physical constants to be unity.
In principle, the two terms in E(Ω) are competing against each other: balls minimize the first term (by the isoperimetric inequality [7], see also [23,Theorem 14.1]) and maximize the second term (by the Riesz rearrangement inequality [25], see also [21,Theorem 3.7]). Thus the question about the existence of a minimizer for E(m) is nontrivial.
Clearly, the existence will depend on the parameter m > 0. By scaling Ω → m 1/N U with |U | = 1, we see that Note that (N +1−λ)/N > 0. This suggests that for small m the short range attraction due to the perimeter term is dominant, whereas for large m the long range repulsion due to the Riesz potential is dominant. Correspondingly, we expect that there is a minimizer for small m and there is no minimizer for large m.
In the case λ = 1, N = 3, the physics literature suggests that there is a critical volume m * > 0 such that balls are unique minimizers for E(m) when m ≤ m * and there is no minimizer when m > m * . The value m * corresponds to the threshold where the energy of a ball of volume m is equal to that of two balls of mass m/2 each spaced infinitely far apart. It can be computed explicitly to be (see [5,12]) with B 1 the unit ball in R 3 . A mathematical proof of this remains unknown.
In the present paper, we consider the general case N ≥ 2 and λ ∈ (0, N ). We define the critical volume m * to be the unique value such that namely, Here B 1 is the unit ball in R N (hence, (m/|B 1 |) 1/N B 1 is a ball of measure m). Thus, just like in the special case λ = 1, N = 3, this is the critical value where the energy of a ball of volume m * is equal to that of two balls of mass m * /2, each spaced infinitely far apart, and it is natural to conjecture that m * divides the regime where minimizers are balls from the regime where there are no minimizers.
Our first new result concerns the existence in (a). Except when λ > 0 is small, the existence of minimizers for E(m) is known only for small m. In this paper, we extend the existence to what is conjectured to be the optimal range of parameters. Theorem 1. Let N ≥ 2 and λ ∈ (0, N ). Then the variational problem E(m) has a minimizer for every 0 < m ≤ m * , where m * is defined in (1).
We will prove Theorem 1 by establishing the strict binding inequality [12] for all m < m * . As a by product of our proof, we obtain the following conditional uniqueness of minimizers.
has no minimizer when m > m * , then balls are minimizers for E(m) when m ≤ m * and they are unique minimizers when m < m * .
So far, the non-existence result in the sharp range m > m * is only available for λ > 0 small [18,2]. For larger λ and a nonexplicit range of m, we have This result is due to [18,19,22] for λ < 2 and seems to be unpublished for λ = 2. We will combine the methods in [11] and [18,19]. It is an open problem whether the nonexistence result also holds for 2 < λ < N when N ≥ 3.

Existence
In this section we prove Theorem 1. We will deduce Theorem 1 from the following strict binding inequality.
Thanks to [12, Theorem 3.1], the strict binding inequality (3) is a sufficient condition for the existence of minimizers of E(m). Moreover, by [12,Theorem 3.4], the set {m > 0 : E(m) has a minimizer} is closed in (0, ∞). Hence, Theorem 4 implies the existence of minimizers of E(m) for all 0 < m ≤ m * . Note that the proofs of Theorems 3.1 and 3.4 in [12] extend, without modifications, to the case λ = 1; see Remark 3.7 in that paper.
We will prove the strict binding inequality using a scaling argument, based on the following key observation which uses only the isoperimetric inequality.
Lemma 5. If 0 < m 1 < m, then we have, with s = m 1 /m ∈ (0, 1) and B 1 the unit ball in R N , Proof. Take Ω ⊂ R N such that |Ω| = m 1 . Then |s −1/N Ω| = m, and hence By the isoperimetric inequality Thus Optimizing over all Ω satisfying |Ω| = m 1 we get which is equivalent to the desired inequality.
The rest is exactly the same as in Case 1. This completes the proof of Lemma 6.

Uniqueness
Proof of Theorem 2.

Thus the variational problem E(m) has a minimizer for all
This is a contradiction to the assumption that E(m) has no minimizer if m > m * . Thus we conclude that balls are minimizers for E(m * ).
Step 2. Now we prove that if m < m * , then balls are unique minimizers for E(m). This fact follows from [2, Theorem 2.10] which states that the set where balls are minimizers is an interval and that one has uniqueness away from the endpoint (note that this part does not require the assumption λ < N − 1 which is imposed in the rest of [2]). For the reader's convenience, we provide a direct proof below.
Consider an arbitrary measurable set Ω ⊂ R N with |Ω| = m < m * . Then proceeding as in the proof of Lemma 5, we find that with s = m/m * ∈ (0, 1) and the equality occurs if and only if Ω is a ball. On the other hand, we know that balls are minimizers for E(m * ), namely Inserting the latter equality in (12), we obtain Thus balls are minimizers for E(m); moreover, if Ω is a minimizer for E(m), then the equality occurs in (12) and Ω is a ball.
On the other hand, for any ρ ≥ 0, The last double integral tends to zero as ρ → ∞ Inserting these facts into (13) and letting ρ → ∞, we infer Per Ω + 2σ(Ω ∩ {ν · x = t}) Note that the double integral here can be written as Ω×Ω |x − y| −λ 1 {ν·x>t>ν·u} dx dy. Thus, integrating the inequality with respect to t ∈ R gives, by Fubini's theorem, Finally, we average this inequality with respect to ν ∈ S N −1 and use the fact that to obtain the bound in the lemma.
It remains to deal with the case 1 < λ ≤ 2. The key is the following bound, which, in the special case N = 3 and λ = 1 appears in [22,Eq. (2.12)]. The proof there extends immediately to the general case, since the analogues of [22, Lemma 3 (ii) and Lemma 4] hold according to [19,  Here diam Ω in the lemma is understood as the diameter of the set {x ∈ R N : |Ω ∩ B r (x)| > 0 for all r > 0}.