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 bound...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.展开更多
基金supported by NSFC(Grant nos.11871406,12271449)supported by Australian Laureate Fellowship FL150100126 of the Australian Research CouncilNational Key R and D Program of China 2021YFA1001800 and NSFC(Grant no.12171334).
文摘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.
