Upper bounds for Steklov eigenvalues on surfaces

We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Hersch-Payne-Schiffer, whose approach is used in the present paper.

[1]  Menahem Schiffer,et al.  Some inequalities for Stekloff eigenvalues , 1974 .

[2]  A. Calderón,et al.  On an inverse boundary value problem , 2006 .

[3]  Nicholas J. Korevaar.  Upper bounds for eigenvalues of conformal metrics , 1993 .

[4]  Shing-Tung Yau,et al.  Eigenvalues of the laplacian of compact Riemann surfaces and minimal submanifolds , 1980 .

[5]  L. Ahlfors.  Open Riemann surfaces and extremal problems on compact subregions , 1950 .

[6]  Matti Lassas,et al.  The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary , 2003 .

[7]  C. Bandle.  Isoperimetric inequalities and applications , 1980 .

[8]  Peter Buser,et al.  On the bipartition of graphs , 1984, Discret. Appl. Math..

[9]  A. E. Soufi,et al.  Le volume conforme et ses applications d'après Li et Yau , 1984 .

[10]  Ahmad El Soufi,et al.  Isoperimetric control of the Steklov spectrum , 2011, 1103.2863.

[11]  A. Gabard.  Sur la représentation conforme des surfaces de Riemann à bord et une caractérisation des courbes séparantes , 2006 .

[12]  A. Henrot,et al.  Some isoperimetric inequalities with application to the Stekloff problem , 2008, 0803.4242.

[13]  Isoperimetric Bounds on the Lowest Nonzero Stekloff Eigenvalue for Plane Strip Domains , 1976 .

[14]  José F. Escobar.  An Isoperimetric Inequality and the First Steklov Eigenvalue , 1999 .

[15]  Friedemann Brock,et al.  An Isoperimetric Inequality for Eigenvalues of the Stekloff Problem , 2001 .

[16]  Об оценках Херша - Пэйна - Шиффера для собственных значений задачи Стеклова@@@On the Hersch - Payne - Schiffer inequalities for Steklov eigenvalues , 2010 .

[17]  Richard Schoen,et al.  The first Steklov eigenvalue, conformal geometry, and minimal surfaces , 2009, 0912.5392.

[18]  S. Yau,et al.  Eigenvalues of elliptic operators and geometric applications , 2004 .

[19]  Eran Makover,et al.  Riemann surfaces with large first eigenvalue , 2001 .

[20]  Asma Hassannezhad.  Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem , 2011 .

[21]  G. Kokarev.  Variational aspects of Laplace eigenvalues on Riemannian surfaces , 2011, 1103.2448.

[22]  Robert Weinstock,et al.  Inequalities for a Classical Eigenvalue Problem , 1954 .