Let $u$ denote a solution to a rotationally invariant Hessian equation $F\left(D^{2}u\right)=0$ on a bounded simply connected domain $\Omega \subset {ℝ}^2$, with constant Dirichlet and Neumann data on $\partial \Omega$. We prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $\Omega$ is a disk. The fully nonlinear operator $F\not\equiv 0$ is of general type and, in particular, not assumed to be elliptic. We also show that the result is sharp, in the sense that it is not true if $\Omega$ is not simply connected, or if $u$ is $C^{\infty }$ but not real-analytic.

中文翻译：

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完全非线性非椭圆方程的塞林超定问题

让 $你$ 表示旋转不变的 Hessian 方程的解 $F\left(D^2你\right)=0$ 在有界单连通域上 $\Omega \subset {ℝ}^2$, 用常数 Dirichlet 和 Neumann 数据 $\partial \Omega$. 我们证明如果$你$ 是实解析的并且不完全为零，那么 $你$ 是径向的并且 $\Omega$是一个磁盘。完全非线性算子$F\not\equiv 0$是一般类型，特别是不假定为椭圆形。我们还表明结果是尖锐的，因为如果$\Omega$ 不是简单地连接，或者如果 $你$ 是 ${丙}^{\infty }$ 但不是实解析的。