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

Chao Xia; Changwei Xiong

doi:10.1007/s42543-023-00068-2

Peking Mathematical Journal ›› 2023, Vol. 7 ›› Issue (2) : 759 -778. DOI: 10.1007/s42543-023-00068-2

Original Article

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

Chao Xia ¹^,^a
, Changwei Xiong ²

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Abstract

It was conjectured by Escobar (J Funct Anal 165:101–116, 1999) that for an n-dimensional ($n\ge 3$) 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.

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Chao Xia, Changwei Xiong. Escobar’s Conjecture on a Sharp Lower Bound for the First Nonzero Steklov Eigenvalue. Peking Mathematical Journal, 2023, 7(2): 759-778 DOI:10.1007/s42543-023-00068-2