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arXiv · 2611.04132

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 $\mathbb{R}^n$, $n\geq2$, 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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Mijia Lai. 2026-11-03. Overdetermined boundary problems for maximal graphs. https://arxiv.org/abs/2611.04132

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