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Revista Integración

Print version ISSN 0120-419X

Integración - UIS vol.31 no.1 Bucaramanga Jan./June 2013

 

Cota superior para el primer valor propio del
problema de Steklov

ÓSCAR ANDRÉS MONTAÑO CARREÑO *

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


Resumen. Sea Br una bola n-dimensional dotada con una métrica rotacionalmente invariante y con curvaturas seccionales radiales no positivas. Si v es el primer valor propio de Steklov y h es la curvatura media sobre el borde de la bola, nosotros demostramos que vh con igualdad si y solo si Br es la bola con la métrica usual de ℝn.
Palabras claves: Curvatura seccional, curvatura media, valor propio de Steklov.
MSC2010: 35P15, 53C20, 53C42, 53C43


Upper bound for the first eigenvalue of the Steklov
problem

Abstract. Let Br be an n-dimensional ball endowed with a rotationally in- variant metric and with non-positive radial sectional curvatures. If v is the first Steklov eigenvalue and h is the mean curvature on the boundary of the ball, we prove that vh. Equality holds only when Br is the ball endowed with the standard metric of ℝn.
Keywords: Sectional curvature, mean curvature, Steklov eigenvalue.


Texto Completo disponible en PDF


Referencias

[1] Escobar J.F., "The Geometry of the first Non-Zero Stekloff Eigenvalue", J. Funct. Anal. 150 (1997), no. 2, 544-556.         [ Links ]

[2] Escobar J.F., "A comparison theorem for the first non-zero Steklov Eigenvalue", J. Funct. Anal. 178 (2000), no. 1, 143-155.         [ Links ]

[3] Montaño O.A., "The First Non-zero Stekloff Eigenvalue for conformal metrics on the ball", Preprint.         [ Links ]

[4] Payne L.E., "Some isoperimetric inequalities for harmonic functions", SIAM J. Math. Anal. 1 (1970), 354-359.         [ Links ]

[5] Stekloff M.W., "Sur les problèmes fondamentaux de la physique mathématique", Ann. Sci. École Norm. Sup. 19 (1902),         [ Links ] 445-490.

[6] Schoen R. and Yau S.T., Lectures on Differential Geometry, International Press, 1994.         [ Links ]

[7] Weinstock R., "Inequalities for a classical eigenvalue problem", J. Rational Mech. Anal. 3 (1954), 745-753.         [ Links ]


*E-mail: oscar.montano@correounivalle.edu.co.
Recibido: 16 de febrero de 2013, Aceptado: 21 de junio de 2013.