Delaunay-Like Compact Equilibria in the Liquid Drop Model

The liquid drop model was introduced by Gamow in 1928 and Bohr–Wheeler in 1938 to model atomic nuclei. The model describes the competition between the surface tension, which keeps the nuclei together, and the Coulomb force, corresponding to repulsion among protons. More precisely, the problem consists of finding a surface Σ =∂ Ω Σ = ∂ Ω in R³ R 3 that is critical for the energy aligned E (Ω ) = Per\, (Ω ) + 1/2 ∫ _Ω ∫ _Ω dxdy|x-y| aligned E ( Ω ) = Per ( Ω ) + 1 2 ∫ Ω ∫ Ω d x d y | x - y | under the volume constraint |Ω | = m | Ω | = m . The term Per\, (Ω ) Per ( Ω ) corresponds to the surface area of Σ Σ . The associated Euler–Lagrange equation is aligned H_Σ (x) + ∫ Ω dy|x-y| = λ for all x∈ Σ , aligned H Σ ( x ) + ∫ Ω d y | x - y | = λ for all x ∈ Σ ,