Bounds for the Steklov eigenvalues

Some sharp bounds for Steklov eigenvalues

“…Here R m and R M are defined as above. In Theorem 2.2, we obtain a lower bound similar to [12], for all Steklov eigenvalues on a starshaped domain Ω in hypersurface of revolution centered at pole. In Theorem 3.1, we prove a result for a star-shaped domain in a paraboloid in R 3 analogous to the above.…”

“…Following the idea of Kuttler and Sigillito [10], Garcia and Montano [7] and the first author [12] obtained a similar bound for the first nonzero Steklov eigenvalue on a star-shaped domain in R n and S n , respectively. Let Ω be a star-shaped bounded domain with smooth boundary ∂Ω centered at a point p and ν be the outward unit normal to ∂Ω.…”

“…Let Ω ⊂ R n . Then with the above notations, the first nonzero eigenvalue of the Steklov problem µ 2 (Ω) satisfies 12]). Let Ω be a star-shaped bounded domain in S n such that Ω ⊂ S n \ {−p}.…”

“…Remark 2.3. In [7] and [12], authors obtained a lower bound for the first nonzero Steklov eigenvalue on a star-shaped bounded domain in R n and S n , respectively. Using the above idea, a similar bound can be obtained for all nonzero Steklov eigenvalues on a star-shaped bounded domain in R n and S n .…”

An isoperimetric inequality for harmonic mean of Steklov eigenvalues in Hyperbolic space

“…The interplay between the geometry of manifold and the Steklov eigenvalues has recently attracted substantial attention. See [3,4,5,7,9,10] and the references therein for recent development. The problem of finding a domain under some geometric constraints, which optimizes eigenvalues (or some combination of eigenvalues) is a classical question in spectral geometry.…”