On Stekloff eigenvalue problem

Roger Chen, Chiung Jue Sung

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)277-296
Number of pages20
JournalPacific Journal of Mathematics
Volume195
Issue number2
DOIs
Publication statusPublished - 2000 Oct

Fingerprint

Eigenvalue Problem
Second Fundamental Form
Lower bound
Eigenvalue
Unit normal vector
Derivative
Manifolds with Boundary
Ricci Curvature
Compact Manifold
Riemannian Manifold
Interior
Ball

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Chen, Roger ; Sung, Chiung Jue. / On Stekloff eigenvalue problem. In: Pacific Journal of Mathematics. 2000 ; Vol. 195, No. 2. pp. 277-296.
@article{2b377e461ebe4229a0b59fdfcb8a6e14,
title = "On Stekloff eigenvalue problem",
abstract = "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.",
author = "Roger Chen and Sung, {Chiung Jue}",
year = "2000",
month = "10",
doi = "10.2140/pjm.2000.195.277",
language = "English",
volume = "195",
pages = "277--296",
journal = "Pacific Journal of Mathematics",
issn = "0030-8730",
publisher = "University of California, Berkeley",
number = "2",

}

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

In: Pacific Journal of Mathematics, Vol. 195, No. 2, 10.2000, p. 277-296.

Research output: Contribution to journalArticle

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 -