### Abstract

Let (M^{n}, g) be a smooth compact Riemannian manifold with boundary ∂M ≠ 0. In this article we discuss the first positive eigenvalue of the Stekloff eigenvalue problem {(-Δ + q)u(x) = 0 in M ∂u/∂v = λu on ∂M, where q(x) is a C^{2} function defined on M, ∂v_{g} is the normal derivative with respect to the unit outward normal vector on the boundary ∂M. In particular, when the boundary ∂M satisfies the "interior rolling R-ball" condition, we obtain a positive lower bound for the first nonzero eigenvalue in terms of n, the diameter of M, R, the lower bound of the Ricci curvature, the lower bound of the second fundamental form elements, and the tangential derivatives of the second fundamental form elements.

Original language | English |
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Pages (from-to) | 277-296 |

Number of pages | 20 |

Journal | Pacific Journal of Mathematics |

Volume | 195 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2000 Oct |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*195*(2), 277-296. https://doi.org/10.2140/pjm.2000.195.277

}

*Pacific Journal of Mathematics*, vol. 195, no. 2, pp. 277-296. https://doi.org/10.2140/pjm.2000.195.277

**On Stekloff eigenvalue problem.** / Chen, Roger; Sung, Chiung Jue.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On Stekloff eigenvalue problem

AU - Chen, Roger

AU - Sung, Chiung Jue

PY - 2000/10

Y1 - 2000/10

N2 - Let (Mn, g) be a smooth compact Riemannian manifold with boundary ∂M ≠ 0. In this article we discuss the first positive eigenvalue of the Stekloff eigenvalue problem {(-Δ + q)u(x) = 0 in M ∂u/∂v = λu on ∂M, where q(x) is a C2 function defined on M, ∂vg is the normal derivative with respect to the unit outward normal vector on the boundary ∂M. In particular, when the boundary ∂M satisfies the "interior rolling R-ball" condition, we obtain a positive lower bound for the first nonzero eigenvalue in terms of n, the diameter of M, R, the lower bound of the Ricci curvature, the lower bound of the second fundamental form elements, and the tangential derivatives of the second fundamental form elements.

AB - Let (Mn, g) be a smooth compact Riemannian manifold with boundary ∂M ≠ 0. In this article we discuss the first positive eigenvalue of the Stekloff eigenvalue problem {(-Δ + q)u(x) = 0 in M ∂u/∂v = λu on ∂M, where q(x) is a C2 function defined on M, ∂vg is the normal derivative with respect to the unit outward normal vector on the boundary ∂M. In particular, when the boundary ∂M satisfies the "interior rolling R-ball" condition, we obtain a positive lower bound for the first nonzero eigenvalue in terms of n, the diameter of M, R, the lower bound of the Ricci curvature, the lower bound of the second fundamental form elements, and the tangential derivatives of the second fundamental form elements.

UR - http://www.scopus.com/inward/record.url?scp=0040778226&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040778226&partnerID=8YFLogxK

U2 - 10.2140/pjm.2000.195.277

DO - 10.2140/pjm.2000.195.277

M3 - Article

AN - SCOPUS:0040778226

VL - 195

SP - 277

EP - 296

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -