Given a bounded domain of \({{\mathbb {R}}}^{n}\) of class \( C^{2}\), we prove the symmetry of solutions to overdetermined problems obtained by adding both zero Dirichlet and constant Neumann boundary conditions to a class of fully nonlinear equations \(\sigma _{k}(\lambda )=C_{n}^{k}\), where \(\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _n)\) are the principal curvatures of a graph. Our method of proof relies on the maximum principle for a suitable P-function and associated Pohozaev type identities.

中文翻译：

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Weingarten超曲面的超定问题

给定类\（C ^ {2} \）的\（{{{mathbb {R}}} ^ {n} \）的有界域，我们证明了通过将零Dirichlet和D都加来的超定问题的解的对称性。一类完全非线性方程\（\ sigma _ {k}（\ lambda）= C_ {n} ^ {k} \）的常数Neumann边界条件，其中\（\ lambda =（\ lambda _1，\ lambda _2， \ ldots，\ lambda _n）\）是图的主曲率。我们的证明方法依赖于适合的P函数和关联的Pohozaev类型身份的最大原理。