On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. The aim of this paper is to show that this is false by analyzing the perturbation problem from the Neumann case. First, we classify all convex polyhedral domains on which the first variation of the ground state with respect to the Robin parameter at zero is not a concave function. Then, we conclude from this that the Robin ground state is not $\operatorname{log}$-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on convex domains with smooth boundary which approximate these in Hausdorff distance.

中文翻译：

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罗宾基态的非凹性

在凸有界欧几里德域上，具有Neumann边界条件的拉普拉斯算子的基态是一个常数，而Dirichlet基态是对数凹的。Robin特征值问题可以看作是Dirichlet和Neumann情况之间的插值，因此，Robin基态应该具有相似的凹度特性似乎很自然。本文的目的是通过分析诺伊曼案例的摄动问题来证明这是错误的。首先，我们对所有凸多面域进行分类，在这些凸多面域上，基态相对于Robin参数为零的第一个变化不是凹函数。然后，