Weighted Eigenvalue Problems for Fourth-Order Operators in Degenerating Annuli

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.