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Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Issue 2 (September 2020)   > Article

Sharp bounds for Steklov eigenvalues on star-shaped domains

Sur les valeurs propres de Sketlov pour certains *-forme domaines


Sheela Verma
Centre For Applicable Mathematics
India

G. Santhanam
Indian Institute of Technology Kanpur
India



Published on 3 September 2020   DOI : 10.21494/ISTE.OP.2020.0544

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In this article, we consider Steklov eigenvalue problem on star-shaped bounded domain Ω in hypersurface of revolution and paraboloid, P = {(x, y, z) ∈ ℝ3 : z = x2 + y2}. A sharp lower bound is derived for all Steklov eigenvalues of Ω in terms of the Steklov eigenvalues of the largest geodesic ball contained in Ω with the same center as Ω. This work is a generalization of a result given by Kuttler and Sigillito (SIAM Rev 10:368 − 370, 1968) on a star-shaped bounded domain in ℝ2.

In this article, we consider Steklov eigenvalue problem on star-shaped bounded domain Ω in hypersurface of revolution and paraboloid, P = {(x, y, z) ∈ ℝ3 : z = x2 + y2}. A sharp lower bound is derived for all Steklov eigenvalues of Ω in terms of the Steklov eigenvalues of the largest geodesic ball contained in Ω with the same center as Ω. This work is a generalization of a result given by Kuttler and Sigillito (SIAM Rev 10:368 − 370, 1968) on a star-shaped bounded domain in ℝ2.

Laplacian Steklov eigenvalue problem Star-shaped domain Rayleigh quotient

Laplacian Steklov eigenvalue problem Star-shaped domain Rayleigh quotient