Sharp bounds for the first non-zero Stekloff eigenvalues

By Qiaoling Wang, C. Xia. 2009

76 citations recorded by the source corpus.

Abstract Let ( M , 〈 , 〉 ) be an n ( ⩾ 2 ) -dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problems Δ u = 0 in M , ∂ u ∂ ν = p u on ∂ M ; Δ 2 u = 0 in M , u = Δ u − q ∂ u ∂ ν = 0 on ∂ M ; where Δ is the Laplacian operator on M and ν denotes the outward unit normal on ∂ M . The first non-zero eigenvalues of the above problems will be denoted by p 1 and q 1 , respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂ M are bounded below by a positive constant c , then p 1 ⩽ λ 1 ( λ 1 + λ 1 − ( n − 1 ) c 2 ) / { ( n − 1 ) c } with equality holding if and only if Ω is isometric to an n -dimensional Euclidean ball of radius 1 c , here λ 1 denotes the first non-zero eigenvalue of the Laplacian of ∂ M . We also show that if the mean curvature of ∂ M is bounded below by a positive constant c then q 1 ⩾ n c with equality holding if and only if M is isometric to an n -dimensional Euclidean ball of radius 1 c . Finally, we show that q 1 ⩽ A / V and that if the equality holds and if there is a point x 0 ∈ ∂ M such that the mean curvature of ∂ M at x 0 is no less than A / { n V } , then M is isometric to an n -dimensional Euclidean ball, being A and V the area of ∂ M and the volume of M , respectively.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/1653cdbd9d8277affeed6849b68ac5f320549fb7

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In Applied Mathematics

1. A. Girouard
2. Ahmad El Soufi
3. B. Colbois
4. Iosif Polterovich
5. G. Kokarev
6. P. Sicbaldi
7. Asma Hassannezhad

In Computational Theory and Mathematics

1. A. Girouard
2. B. Colbois
3. Ahmad El Soufi
4. Iosif Polterovich
5. Asma Hassannezhad
6. P. Sicbaldi

Citations in the corpus

• The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization (2011, unknown): “Recently, Wang-Xia [33] have extended Theorem 4.5 to compact manifolds with boundary.”
• Isoperimetric control of the Steklov spectrum (2011, background): “In comparison with [26], we make no assumption on the convexity of Ω.”