Title: Towards the optimality of the ball for the Rayleigh Conjecture concerning the clamped plate Show Chinese title

Title: 针对固定板的Rayleigh猜想中球体最优性的研究

Abstract: In 1995, Nadirashvili and subsequently Ashbaugh and Benguria proved the Rayleigh Conjecture concerning the first eigenvalue of the bilaplacian with clamped boundary conditions in dimension $2$ and $3$. Since then, the conjecture has remained open in dimension $d>3$. In this document, we contribute in answering the conjecture under a particular assumption regarding the critical values of the optimal eigenfunction. More precisely, we prove that if the optimal eigenfunction has no critical value except its minimum and maximum, then the conjecture holds. This is performed thanks to an improvement of Talenti's comparison principle, made possible after a fine study of the geometry of the eigenfunction's nodal domains. Show Chinese abstract

Abstract: 1995年，Nadirashvili随后Ashbaugh和Benguria证明了关于夹紧边界条件下双调和算子第一个特征值的Rayleigh猜想，在维度$2$和$3$中。 自那时起，该猜想在维度$d>3$中仍然未被解决。 在本文中，我们在关于最优特征函数临界值的一个特定假设下，对这一猜想做出了贡献。 更准确地说，我们证明了如果最优特征函数除了其最小值和最大值外没有其他临界值，那么该猜想成立。 这是通过改进Talenti的比较原理实现的，这在对特征函数的节点区域几何进行细致研究之后成为可能。