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PreprintCC BY 4.0arXiv

An exact Delaunay flux identity for compact equilibria in the liquid drop model

  • Averkios Averkiou
  • Manuel del Pino
  • Monica Musso
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Abstract

We revisit the construction of compact liquid drop equilibria from Delaunay unduloids in [Arch. Ration. Mech. Anal. 249 (2025), Paper No. 74]. We evaluate exactly the coefficient arising in the final solvability condition of the Lyapunov-Schmidt reduction, identifying it as the product of the axial period and the normalised Delaunay flux. Using this identity and the monotonicity of the period, we establish the conjectured positivity of the coefficient and prove that its product with the period strictly increases with the neck radius. We also derive an exact relation between the volume and the surface area of one period and prove that both strictly decrease with the neck radius. These results extend the construction throughout the embedded unduloid family and yield compact equilibria of every sufficiently large mass, with reference neck radii converging to any prescribed interior value as the mass tends to infinity.

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Cite

BibTeXRIS
BibTeX
@misc{averkiou2026exact,
  title = {{An exact Delaunay flux identity for compact equilibria in the liquid drop model}},
  author = {Averkios Averkiou and Manuel del Pino and Monica Musso},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12689},
  eprint = {2610.12689},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12689},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An exact Delaunay flux identity for compact equilibria in the liquid drop model
AU  - Averkios Averkiou
AU  - Manuel del Pino
AU  - Monica Musso
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12689
UR  - https://arxiv.org/abs/2610.12689
AB  - We revisit the construction of compact liquid drop equilibria from Delaunay unduloids in [Arch. Ration. Mech. Anal. 249 (2025), Paper No. 74]. We evaluate exactly the coefficient arising in the final solvability condition of the Lyapunov-Schmidt reduction, identifying it as the product of the axial period and the normalised Delaunay flux. Using this identity and the monotonicity of the period, we establish the conjectured positivity of the coefficient and prove that its product with the period strictly increases with the neck radius. We also derive an exact relation between the volume and the surface area of one period and prove that both strictly decrease with the neck radius. These results extend the construction throughout the embedded unduloid family and yield compact equilibria of every sufficiently large mass, with reference neck radii converging to any prescribed interior value as the mass tends to infinity.
ER  -