We consider the following semilinear overdetermined problem on a two dimensional bounded or unbounded domain Ω with analytic boundary ∂Ω having at least one bounded connected component

$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u = g(u)}\;\;\;\;\;\;\;&{\text{in}\;\Omega ,} \\ {\tfrac{{\partial u}}{{\partial v}} = 0\;\text{and}\;u = c}&{\text{on}\;\partial \Omega ,} \end{array}} \right.$$

where c is a constant. When g(c) = 0 the constant solution u ≡ c is the unique solution. For g(c) ≠ 0, we show that the boundary is a circle if and only if the problem admits a solution that has a constant third or fourth normal derivative along the boundary. A similar result involving the fifth normal derivative is proved.

中文翻译：

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与希弗问题有关的一些结果

我们考虑解析边界∂Ω具有至少一个有界连接分量的二维有界或无界域Ω上的以下半线性超定问题

$$ \ left \ {{\\ begin {array} {* {20} {c}}} {-\ Delta u = g（u）} \; \; \; \; \; \; \;＆{\ text {in} \; \ Omega，} \\ {\ tfrac {{\ partial u}} {{\ partial v}} = 0 \; \ text {and} \; u = c}＆{\ text {on} \; \ partial \ Omega，} \ end {array}} \ right。$$

其中c是常数。当克（ç）= 0的恒定溶液û ≡ Ç是独特的解决方案。对于g（c）≠0，当且仅当问题允许沿着边界具有恒定的三阶或四阶导数的解时，我们证明边界为圆。证明了涉及第五正态导数的相似结果。