El problema de Steklov sobre el cono

ÓSCAR ANDRÉS MONTAÑO

Universidad del Valle, Departamento de Matemáticas, Cali, Colombia.

Dedicado a Rafael Isaacs, a quien recuerdo siempre con mucho cariño e
infinita gratitud por su valioso aporte en mi formación académica.

Resumen. Sea (Mⁿ, g) un cono de altura 0 ≤ x_n+1 ≤ 1 en ℝⁿ⁺¹, dotado con una métrica rotacionalmente invariante 2ds² + ƒ²(s)dw², donde dw² representa la métrica estándar sobre S^n-1, la esfera unitaria (n - 1)-dimensional. Supongamos que Ric(g) ≥ 0. En este artículo demostramos que si h > 0 es la curvatura media sobre ∂M y v₁ es el primer valor propio del problema de Steklov, entonces v₁ ≥ h.
Palabras Claves: Problema de Steklov, cono, curvatura media.
MSC2010: 35P15, 53C20, 53C42, 53C43

The Steklov problem on the cone

Abstract. Let (Mⁿ, g) be a cone of height 0 ≤ x_n+1 ≤ 1 en ℝⁿ⁺¹, endowed with a rotationally invariant metric 2ds² + ƒ²(s)dw², where dw² represents the standard metric on S^n-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 v₁ is the first eigenvalue of the Steklov problem, then v₁ ≥ h.
Keywords: Steklov problem, cone, mean curvature.

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Referencias

[1] Calderón A.P., "On an Inverse Boundary Value Problem", Comput. Appl. Math. 25 (2006), no. 2-3, 133–138.

[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.

^*E-mail: oscar.montano@correounivalle.edu.co
Recibido: 13 de junio de 2012, Aceptado: 10 de septiembre de 2012.