加权拉普拉斯算子 Robin 特征值的 Faber - Krahn 不等式
Faber Krahn inequality of Robin eigenvalue of the Weighted Laplacian
AI总结：
研究\(\mathbb{R}^{n}\)和\(\mathbb{H}^{n}\)上带 Robin 边界条件的加权拉普拉斯算子第一特征值，通过证明两个不等式得出最优区域是以原点为中心的球这一结论。
AI中文摘要：
本文证明了在\(\mathbb{R}^{n}\)和\(\mathbb{H}^{n}\)上带 Robin 边界条件的加权拉普拉斯算子第一特征值的两个 Faber - Krahn 型不等式。两种情形下，最优区域均为以原点为中心的球。
英文摘要：
In this paper, we study the isoperimetric inequalities for the Robin eigenvalues of the weighted Laplacian with positive Robin parameter in the Euclidean space $\R^n$ and the hyperbolic space $\mathbb{H}^n$, respectively. More precisely, we prove that among all bounded Lipschitz domains with fixed weighted volume, the geodesic ball centered at the origin minimizes the first Robin eigenvalue of the weighted Laplacian, provided that the Robin parameter and the radial log-convex density satisfy suitable conditions. Furthermore, we show that the second Robin eigenvalue is bounded below by the first Robin eigenvalue of the geodesic ball centered at the origin with half the weighted volume. Our results extend classical Faber-Krahn inequalities to the setting of weighted spaces with log-convex densities. We also derive a lower bound for the second Robin eigenvalue in terms of the first eigenvalue of the centered ball with half the weighted volume.
