TY  - JOUR
AU  - Xia, Chao
AU  - Xiong, Changwei
PY  - 2024
TI  - Escobar’s Conjecture on a Sharp Lower Bound for the First Nonzero Steklov Eigenvalue
JO  - Peking Mathematical Journal
SN  - 2096-6075
SP  - 759
EP  - 778
VL  - 7
IS  - 2
AB  - 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.
UR  - https://doi.org/10.1007/s42543-023-00068-2
DO  - 10.1007/s42543-023-00068-2