A Sharp Bound for the First Robin–Dirichlet Eigenvalue
研究了双连通区域上拉普拉斯算子的第一个特征值，当外边界为Robin条件、内边界为Dirichlet条件时，证明了在固定测度、外周长和内(n-1)次quermassintegral的条件下，球壳使该特征值达到最大。
Abstract In this paper, we study the first eigenvalue of the Laplacian on doubly connected domains when Robin and Dirichlet conditions are imposed on the outer and the inner part of the boundary, respectively. We provide that the spherical shell reaches the maximum of the first eigenvalue of this problem among a suitable class of domains when the measure, the outer perimeter and inner $$(n-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> th quermassintegral are fixed.