Escobar’s Conjecture on a Sharp Lower Bound for the First Nonzero Steklov Eigenvalue  

作　　者：Chao Xia Changwei Xiong 

机构地区：[1]School of Mathematical Sciences,Xiamen University,Xiamen 361005,China [2]School of Mathematics,Sichuan University,Chengdu 610065,China

出　　处：《Peking Mathematical Journal》2024年第2期759-778,共20页北京数学杂志（英文）

基　　金：supported by NSFC(Grant nos.11871406,12271449);supported by Australian Laureate Fellowship FL150100126 of the Australian Research Council;National Key R and D Program of China 2021YFA1001800 and NSFC(Grant no.12171334).

摘　　要：It was conjectured by Escobar(J Funct Anal 165:101–116,1999)that for an ndimensional(n≥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.

关 键 词：Stekloveigenvalue Laplacian eigenvalue Sharp bound Nonnegative sectional curvature 

分 类 号：O17[理学—数学]