论文标题
罗宾板的第二个特征值的尖锐等等不平等现象
A Sharp Isoperimetric Inequality for the Second Eigenvalue of the Robin Plate
论文作者
论文摘要
在所有$ c^{\ infty} $相等体积的有界域中，我们表明，只要罗宾参数位于特定的负值范围内，罗宾板的第二个特征值就被开放的球唯一最大化。我们的方法结合了Freitas和Laugesen引入的最新技术，研究了Robin膜问题的第二个特征值和Chasman使用的技术来研究自由板问题。特别是，我们选择球的本征函数作为通用域的瑞利商中的试验。这样的本征函数由超球面和修改的贝塞尔功能组成。我们的大部分工作取决于对这些特殊功能的微妙特性的理解，这可能引起独立的兴趣。
Among all $C^{\infty}$ bounded domains with equal volume, we show that the second eigenvalue of the Robin plate is uniquely maximized by an open ball, so long as the Robin parameter lies within a particular range of negative values. Our methodology combines recent techniques introduced by Freitas and Laugesen to study the second eigenvalue of the Robin membrane problem and techniques employed by Chasman to study the free plate problem. In particular, we choose eigenfunctions of the ball as trial functions in the Rayleigh quotient for a general domain; such eigenfunctions are comprised of ultraspherical Bessel and modified Bessel functions. Much of our work hinges on developing an understanding of delicate properties of these special functions, which may be of independent interest.
