Weighted Eigenvalue Problems for Fourth-Order Operators in Degenerating Annuli

2023/06/07 by Alexis Michelat, Michelat, Alexis, Tristan Rivière +1 · 1 citation
Computer Science · Mathematics · #15A18 #15A42 #34L15 #35A15 #35J20 #35P15 #58E05 #58J05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2306.04609

openalex publication_date 2023/06/07 · openalex created_date 2023/06/09 · openalex updated_date 2026/07/28

Abstract

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 Poincaré-type inequality in annuli for those fourth-order operators. Applications to Morse theory are given.

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