We consider a partially overdetermined problem in a sector-like domain $\Omega$ in a cone $\Sigma$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $\Omega$ is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces $\Gamma$ with boundary which satisfy a 'gluing' condition with respect to the cone $\Sigma$. We prove that if either the cone is convex or the surface is a radial graph then $\Gamma$ must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed.

中文翻译：

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锥中的超定问题和恒定平均曲率曲面

我们考虑了在$ \ mathbb {R} ^ N $，$ N \ geq 2 $的圆锥形$ \ Sigma $中的扇形域$ \ Omega $中的部分超定问题，并证明了Serrin类型的刚性结果表示存在一个解意味着在圆锥体上的凸度假设下$ \ Omega $是一个球形扇区。我们还考虑了相关的问题，即表征具有边界的恒定平均曲率致密曲面$ \ Gamma $相对于圆锥$ \ Sigma $满足“粘合”条件。我们证明，如果圆锥体是凸的或表面是径向图，则$ \ Gamma $必须是球冠。最后，我们表明，在畴或表面的相对边界与圆锥正交相交的条件下，不需要其他假设。