Research and Innovation Articles
Upper bound for the first eigenvalue of the Steklov problem
Published 2013-07-29
Keywords
- Sectional curvature,
- mean curvature,
- Steklov eigenvalue
How to Cite
Montaño Carreño, Óscar A. (2013). Upper bound for the first eigenvalue of the Steklov problem. Revista Integración, Temas De matemáticas, 31(1), 53–58. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/3383
Abstract
Let Br be an n-dimensional ball endowed with a rotationally invariant metric and with non-positive radial sectional curvatures. If is thefirst Steklov eigenvalue and h is the mean curvature on the boundary of the ball, we prove that h. Equality holds only when Br is the ball endowedwith the standard metric of Rn.
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References
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[3] Montaño O.A., “The First Non-zero Stekloff Eigenvalue for conformal metrics on the ball”, Preprint.
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[2] Escobar J.F., “A comparison theorem for the first non-zero Steklov Eigenvalue”, J. Funct. Anal. 178 (2000), no. 1, 143–155.
[3] Montaño O.A., “The First Non-zero Stekloff Eigenvalue for conformal metrics on the ball”, Preprint.
[4] Payne L.E., “Some isoperimetric inequalities for harmonic functions”, SIAM J. Math. Anal. 1 (1970), 354–359.
[5] Stekloff M.W., “Sur les problèmes fondamentaux de la physique mathématique”, Ann. Sci. École Norm. Sup. 19 (1902), 445–490.
[6] Schoen R. and Yau S.T., Lectures on Differential Geometry, International Press, 1994.
[7] Weinstock R., “Inequalities for a classical eigenvalue problem”, J. Rational Mech. Anal. 3 (1954), 745–753.