El problema de Steklov sobre el cono

• Montaño Carreño, Oscar Andres ^[1]
  1. [1] Universidad del Valle (Colombia)

     Universidad del Valle (Colombia)

     Colombia

     
• Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 30, Nº. 2, 2012 (Ejemplar dedicado a: Revista Integración), págs. 121-128
• Idioma: español
• Títulos paralelos:
  • The Steklov problem on the cone
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    Sea (Mn, g) un cono de altura 0 ≤ xn+1 ≤ 1 en Rn+1, dotado con una métrica rotacionalmente invariante 2ds2 + f2(s)dw2, donde dw2 representa la métrica estándar sobre Sn−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 ν1 es el primer valor propio del problema de Steklov, entonces ν1 ≥ h.

  • English

    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.

• Referencias bibliográficas
  • Citas [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.
  • (1992) no. 1, 21–84.
  • [4] Escobar J.F., “The Geometry of the first Non-Zero Stekloff Eigenvalue”, J. Funct. Anal.
  • (1997), no. 2, 544–556.
  • [5] Escobar J.F., “An isoperimetric inequality and the first Steklov Eigenvalue”, J. Funct. Anal.
  • (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.
  • (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.
  • (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.