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