Research and Innovation Articles
Published 2012-11-28
Keywords
- Steklov problem,
- cone,
- mean curvature
How to Cite
Montaño Carreño, O. A. (2012). The Steklov problem on the cone. Revista Integración, Temas De matemáticas, 30(2), 121–128. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/2901
Abstract
Let (Mn, g) be a cone of height 0 ≤ xn+1 ≤ 1 in Rn+1, endowed with a rotationally invariant metric 2ds2 + f2(s)dw2, where dw2 represents the standard metric on Sn−1, the (n − 1)-dimensional unit sphere. Assume Ric(g) ≥ 0. In this paper we prove that if h > 0 is the mean curvature on ∂M and ν1 is the first eigenvalue of the Steklov problem, then ν1 ≥ h.
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References
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[2] Escobar J.F., “Conformal Deformation of a Riemannian Metric to a Scalar Flat Metric with Constant Mean Curvature on the Boundary”, Ann. of Math. 136 (1992), no. 2, 1–50.
[3] Escobar J.F., “The Yamabe problem on manifolds with Boundary”, J. Differential Geom. 35 (1992) no. 1, 21–84.
[4] Escobar J.F., “The Geometry of the first Non-Zero Stekloff Eigenvalue”, J. Funct. Anal. 150 (1997), no. 2, 544–556.
[5] Escobar J.F., “An isoperimetric inequality and the first Steklov Eigenvalue”, J. Funct. Anal. 165 (1999), no. 1, 101–116.
[6] Escobar J.F., “A comparison theorem for the first non-zero Steklov Eigenvalue”, J. Funct. Anal. 178 (2000), no. 1, 143–155.
[7] Escobar J.F., “Topics in PDE’s and Differential Geometry”, XII Escola de Geometria Diferencial. [XII School of Differential Geometry] Universidade Federal de Goiás, Goiânia, 2002. viii+88 pp.
[8] Montaño O.A., “The first non-zero Stekloff eigenvalue for conformal metrics on the ball”, preprint.
[9] Reilly R.C., “Aplications of the Hessian operator in a Riemannian manifold”, Indiana Univ. Math. J. 26 (1977), no. 3, 459–472.
[10] Payne L.E., “Some isoperimetric inequalities for harmonic functions”, SIAM J. Math. Anal. 1 (1970), 354–359.
[11] Steklov W., “Sur les problemes fondamentaux de la physique mathématique”, Ann. Sci. École Norm. Sup. 19 (1902), no.3, 455–490.
[12] Weinstock R., “Inequalities for a classical eigenvalue problem”, J. Rational Mech. Anal. 3 (1954), 745–753.
[13] Wang Q., Xia C., “Sharp bounds for the first non-zero Stekloff eigenvalues”, J. Funct. Anal. 257 (2009), no. 8, 2635–2644.
[14] Xia C., “Rigidity of compact manifolds with boundary and nonnegative Ricci curvature”, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1801–1806.