Escobar's Conjecture on a sharp lower bound for the first nonzero Steklov eigenvalue

Autor:	Xia, Chao, Xiong, Changwei
Rok vydání:	2019
Předmět:	Mathematics - Differential Geometry Mathematics - Analysis of PDEs Mathematics - Spectral Theory
Druh dokumentu:	Working Paper
Popis:	It was conjectured by Escobar [J. Funct. Anal. 165 (1999), 101--116] that for an $n$-dimensional ($n\geq 3$) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by $c>0$, the first nonzero Steklov eigenvalue is greater than or equal to $c$ with equality holding only on isometrically Euclidean balls with radius $1/c$. In this paper, we confirm this conjecture in the case of nonnegative sectional curvature. The proof is based on a combination of Qiu--Xia's weighted Reilly-type formula with a special choice of the weight function depending on the distance function to the boundary, as well as a generalized Pohozaev-type identity. Comment: Peking Math. J. (to appear)
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/1907.07340 Zobrazit plný text záznamu View this record from Arxiv