A Lower Bound for the First Steklov Eigenvalue on a Domain

In this paper we provide a lower bound for the first eigenvalue of the Steklov problem in a star-shaped bounded domain in R. This result extends to higher dimensions a lower estimate of Kuttler-Sigillito in a two dimensional star-shaped bounded domain.


Introduction
Let Ω be a bounded domain of R n with smooth boundary ∂Ω.The following problem is called the Steklov problem: where ν is a real number.This problem was studied by Steklov [7] for bounded domains in the plane.The problem has physical origins; the function ϕ is a steady state temperature in Ω where the flow over the boundary, ∂Ω, is proportional to the temperature.The set of eigenvalues for the Steklov problem is the same as the set of eigenvalues for the Dirichlet-Neumann function.This function associates to each function u defined on ∂Ω, the normal derivative of its harmonic extension u on Ω.The set of eigenvalues of the Steklov problem consists of an increasing sequence 0 = ν 0 < ν 1 < ν 2 < • • • , with ν k → +∞.
The first non-zero eigenvalue is known as the first eigenvalue of the Steklov problem; the variational characterization of this eigenvalue is For the unit ball B n ⊂ R n , the eigenvalues of the Dirichlet-Neumann function are ν k = k, k = 0, 1, 2, . .., and the eigenfunctions are given by the space of harmonics homogeneous polynomials of degree k restricted to the boundary of the ball, the (n − 1)-dimensional unit sphere S n−1 .The first Steklov eigenvalue of the n-dimensional ball of radius r > 0, B r , is ν 1 (B r ) = 1 r and the coordinate functions {x 1 , . . ., x n } are the respective eigenfunctions.L , where L represents the perimeter of the boundary curve, with equality if and only if Ω is a disk.In 1970 for convex domains in the plane, Payne [6] proved that ν 1 ≥ k o , where k o is the minimum value of the curvature on the boundary of the domain.In 1997, Escobar [2] generalized Payne's result to 2-dimensional Riemannian manifolds with non-negative Gaussian curvature and with boundary such that the geodesic curvature k g is bounded below by a positive constant k o ; with these hypotheses Escobar showed that ν 1 ≥ k o .In higher dimensions, Escobar considered compact manifolds with nonnegative Ricci curvature and again in the spirit of Payne's theorem proved the following result: Theorem 1.1.If Ω is an n-dimensional compact Riemannian manifold (n ≥ 3) with nonnegative Ricci Curvature, with nonempty smooth boundary ∂Ω and whose second fundamental form π on ∂Ω satisfies π ≥ kI for some positive constant k, then

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For rotationally invariant metrics with nonnegative Ricci curvature in the n-dimensional ball B r , Montaño [5] proved that ν 1 ≥ h where h is the mean curvature on ∂B r .For rotationally invariant metrics with nonpositive Ricci curvature in the n-dimensional ball B r , Montaño [4] proved that ν 1 ≤ h where h is the mean curvature on ∂B r .In both cases equality holds if and only if (B r , g) is isometric to the Euclidean ball.When Ω ⊂ R n is a star-shaped domain with respect to a point P , which, without loss of generality we can assume that it is the origin, Bramble and Payne [1] proved that where a is the radius of a ball centered at the origin contained in Ω, r M is the maximum distance from P to border of Ω, and h m is the minimum of the function h : ∂Ω → R, defined by with η a outer unit normal to ∂Ω.
With a different idea for the 2-dimensional case Kuttler and Sigillito [3] proved that where R(θ) = |x| for any x ∈ ∂Ω of the form x = |x|e iθ .
In this paper, following the idea of Sigillito and Kruttler we provide a lower bound for the first Steklov eigenvalue in a star-shaped bounded domain Ω ⊂ R n with respect to a point P.This improvement of the Kuttler and Sigillito's result depends on two results which relate geometric quantities on ∂Ω and on S n−1 .

Preliminaries
Consider on the Euclidean space (R n , , ) polar coordinates (r, ω), where r ∈ R + and ω ∈ S n−1 .If u : U → S n−1 is a local chart of the unit sphere, , writes as where g ij are the components of the standard round metric on S n−1 in the chart u.Let Ω be a star-shaped bounded domain of R n with respect to a point P , which, without loss of generality we assume that it is the origin (see Figure 1Star-shaped domain Ω.figure.caption.1).Let ∂Ω be the boundary of Ω with outer unit normal vector given by η.By the character of star-shaped of Ω there exists a function R : In this work we assume that the boundary of Ω is smooth.In the following proposition we get a result that relates the function R and its gradient, ∇R, on the boundary of Ω.Then

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where θ is the angle between the outer unit normal vector η and the radial vector field ∂ r .
Proof.The map Ψ : S n−1 → R n , given by Ψ(ω) = R(ω), ω realizes the embedding of ∂Ω.The induced metric on ∂Ω therefore write, in the chart u as Note that, setting t = log R, we can write The inverse metric g ij therefore writes as , t j = g jk t k and ∇ is the connection on S n−1 , g ij .
Since ∂Ω is the zero set of the function F (r, ω) = r − R(ω), then the normal vector η is the normalized gradient ∇F , hence It therefore follows that Volumen 49, Número 1, Año 2015 Solving W 2 cos 2 θ = 1 in terms of ∇t 2 we deduce that It is natural to ask for the relationship between the elements of area √ g = det(G) and g = det( G).In this direction we have the following comparison theorem; although its proof is standard, we include it for the convenience of the reader.
Proof.Let us take a local chart of the unit sphere u : U → S n−1 such that the matrix G = g ij of the first fundamental form in the given chart is diagonal.
By formula (4Preliminariesequation.2.4) we have where A is the (n − 1) × n matrix whose entries are The determinant det g can be therefore obtained easily via Binet theorem for det

Main Result
Using the results of the previous section, we establish the following theorem, where we find a lower bound for the first eigenvalue of the Steklov problem in a star-shaped bounded domain.
Theorem 3.1.Let Ω be a bounded star-shaped domain of R n with smooth boundary ∂Ω and outer unit normal η.If 0 ≤ θ ≤ α < π 2 , where cos(θ) = η, ∂r , then the first eigenvalue of Steklov for Ω, ν 1 (Ω), satisfies the inequality where a = tan 2 (α), r m := min where Making the change of variables u = u and r = ρR(u) we obtain Volumen 49, Número From the Cauchy-Schwarz inequality, for any function γ 2 .From here, From the equality (3equation.2.3), taking a = tan 2 (α), we arrive to Cali, Colombia e-mail: gonzalo.garcia@correounivalle.edu.coe-mail: oscar.montano@correounivalle.edu.co Dirichlet and the Neumann problem, Steklov geometric estimates have been made in the Steklov problem for the first eigenvalue.For bounded and simply connected domains in the plane xy, in 1954 Weinstock [8] proved that ν 1 ≤ 2π FOR THE FIRST STEKLOV EIGENVALUE ON A DOMAIN 97
FOR THE FIRST STEKLOV EIGENVALUE ON A DOMAIN 99

UR
and r M := max U R. Proof.Without loss of generality let us assume that Ω is a star-shaped domain with respect to the origin, ξ : U → S n−1 is a standard parametrization of S n−1 ⊆ R n and y : U → ∂Ω is the associated parametrization to the boundary of Ω.If we define R : U → R for R(u) = |y(u)|, then y = Rξ.Considering the spherical coordinates x = rξ(u), Ω = (u, r) : u ∈ U, 0 < r < R(u) and dx = r n−1 g dr du.For ϕ : Ω → R, we have the Rayleigh's quotient R. Weinstock, Inequalities for a Classical Eigenvalue Problem, Rational Mech.Anal 3 (1954), 745-753.(Recibido en febrero de 2014.Aceptado en diciembre de 2014) Departamento de Matemáticas Universidad del Valle Facultad de Ciencias Carrera 100, calle 13