标题： 加权特征值问题在退化环形区域中的四阶算子 显示英文标题

标题： Weighted Eigenvalue Problems for Fourth-Order Operators in Degenerating Annuli

摘要： 我们得到了在所有维度中退化环形区域（这是泡树分析中的核心对象）中与双调和算子相关的两个自然加权问题的第一特征值的近乎最优估计（以及一个四阶椭圆算子连续族在维度$2$中的第一特征值的估计）。该估计仅依赖于环形区域的共形类。我们还证明了在维度$2$和维度$4$中，如果环形区域的共形类足够大，那么第一个问题的第一个特征函数绝不会是径向的。另一个结果是针对这些四阶算子在环形区域上的加权Poincaré型不等式。给出了对莫尔斯理论的应用。 显示英文摘要

摘要： We obtain a nigh optimal estimate for the first eigenvalue of two natural weighted problems associated to the bilaplacian (and of a continuous family of fourth-order elliptic operators in dimension $2$) in degenerating annuli (that are central objects in bubble tree analysis) in all dimension. The estimate depends only on the conformal class of the annulus. We also show that in dimension $2$ and dimension $4$, the first eigenfunction (of the first problem) is never radial provided that the conformal class of the annulus is large enough. The other result is a weighted Poincar\'e-type inequality in annuli for those fourth-order operators. Applications to Morse theory are given.