Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 41 min ago
PreprintarXiv licencearXiv

The stability conjecture for the Bernoulli problem in dimension four

  • Xavier Fernández-Real
  • Joaquim Serra
Open at sourcePDFDOICite

Abstract

We prove that every global classical stable solution to the one-phase Bernoulli problem in ℝ4 is one-dimensional. In particular, if nonempty, its free boundary consists of one or two parallel hyperplanes. This settles the stability conjecture for the Bernoulli problem in dimension four. As consequences, we obtain universal local Hessian and free-boundary curvature estimates for classical stable solutions in ℝ4, as well as axial symmetry of global classical solutions with finite Morse index in ℝ4.

Related

  1. The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture
    Deep Bhattacharjee, Ushashi Bhattacharya, Priyabrata Mandal +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. On the Hodge conjecture for hyper-Kähler varieties of generalized Kummer type of dimension divisible by four
    Salvatore Floccari, Lie Fu
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Quillen's local parameter conjecture in dimension four
    Yunpeng Li
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Uniform Stability of Scott-Vogelius Elements on Three-Dimensional Freudenthal Meshes in Degrees Four and Five: Resolving the Farrell-Mitchell-Scott Conjecture
    Hanbing Liang, Fujun Liu
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingComputational physicsarXiv
  5. Linear Stability of Double-Double Orbits in the Equal-Mass Four-Body Problem via Index Theory
    Zijian Meng
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv

Cite

BibTeXRIS
BibTeX
@misc{fernandezreal2026stability,
  title = {{The stability conjecture for the Bernoulli problem in dimension four}},
  author = {Xavier Fernández-Real and Joaquim Serra},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05669},
  eprint = {2609.05669},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05669},
}
RIS
TY  - UNPB
TI  - The stability conjecture for the Bernoulli problem in dimension four
AU  - Xavier Fernández-Real
AU  - Joaquim Serra
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05669
UR  - https://arxiv.org/abs/2609.05669
AB  - We prove that every global classical stable solution to the one-phase Bernoulli problem in ℝ^4 is one-dimensional. In particular, if nonempty, its free boundary consists of one or two parallel hyperplanes. This settles the stability conjecture for the Bernoulli problem in dimension four. As consequences, we obtain universal local Hessian and free-boundary curvature estimates for classical stable solutions in ℝ^4, as well as axial symmetry of global classical solutions with finite Morse index in ℝ^4.
ER  -