p-厄米特算子的法伯-克拉恩不等式

中文摘要 AI 辅助

我们证明了，在具有正罗宾参数的罗宾边界条件下，\n$\mathbb{R}^n$中利普希茨域上的p-厄米特算子（带高斯权重的加权p-拉普拉斯算子）的第一特征值满足法伯-克拉恩不等式。主要结果表明，在所有具有给定高斯测度的域中，半空间使第一特征值最小，且仅对半空间取等。这将p-拉普拉斯算子的经典法伯-克拉恩不等式（见文献[BucurCV]）推广到p-厄米特算子，并将线性情形（见文献[ChiacchioMathann]）推广到全非线性区域$p>1$。

英文摘要

We prove a Faber-Krahn inequality for the first eigenvalue of the $p$-Hermite operator (the weighted $p$-Laplacian with Gaussian weight) on Lipschitz domains in $\R^n$ under Robin boundary conditions with positive Robin parameter. The main result states that, among all domains of given Gaussian measure, the first eigenvalue is minimized by a half-space, and equality holds only for half-spaces. This extends the classical Faber-Krahn inequalities for the $p$-Laplacian \cite{BucurCV} to the $p$-Hermite operator and generalizes the linear case \cite{ChiacchioMathann} to the full nonlinear regime $p>1$.

发表机构

• College of informatics, Huazhong Agricultural University（华中农业大学信息学院）
• School of Mathematical Sciences, Soochow University（苏州大学数学科学学院）
• School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）

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