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PreprintCC BY 4.0arXiv

Overdetermined boundary problems for maximal graphs

  • Mijia Lai
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Abstract

We study overdetermined boundary problems for the maximal surface equation with zero Dirichlet data and constant nonzero Neumann data . For the exterior of a bounded domain with connected boundary in ℝn, n≥2, we prove that such domain must be the complement of a ball and the solution is radial. In dimension two, we also show that a positive solution on a domain bounded by a properly embedded noncompact connected curve forces the domain to be a half-plane and the solution to be affine. The proofs combine the divergence structure of the equation with the induced geometry of maximal graphs.

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Cite

BibTeXRIS
BibTeX
@misc{lai2026overdetermined,
  title = {{Overdetermined boundary problems for maximal graphs}},
  author = {Mijia Lai},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04132},
  eprint = {2611.04132},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04132},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Overdetermined boundary problems for maximal graphs
AU  - Mijia Lai
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04132
UR  - https://arxiv.org/abs/2611.04132
AB  - We study overdetermined boundary problems for the maximal surface equation with zero Dirichlet data and constant nonzero Neumann data . For the exterior of a bounded domain with connected boundary in ℝ^n, n≥2, we prove that such domain must be the complement of a ball and the solution is radial. In dimension two, we also show that a positive solution on a domain bounded by a properly embedded noncompact connected curve forces the domain to be a half-plane and the solution to be affine. The proofs combine the divergence structure of the equation with the induced geometry of maximal graphs.
ER  -