A Lower Bound for the First Steklov Eigenvalue on a Domain

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1 Revista Colombiana de atemáticas Volumen 49(0151, páginas A Lowe Bound fo the Fist Steklov Eigenvalue on a Domain na cota infeio paa el pime valo popio de Steklov en un dominio Gonzalo Gacía, Ósca ontaño nivesidad del Valle, Cali, Colombia Abstact. In this pape we povide a lowe bound fo the fist eigenvalue of the Steklov poblem in a sta-shaped bounded domain in R n. This esult extends to highe dimensions a lowe estimate of Kuttle-Sigillito in a two dimensional sta-shaped bounded domain. Key wods and phases. Eigenvalue, Lowe bound, The Steklov poblem. 010 athematics Subject Classification. 35P15, 53C0, 53C4, 53C43. Resumen. En este tabajo poveemos una cota infeio paa el pime valo popio del poblema de Steklov en un dominio estellado acotado en R n. Este esultado extiende a dimensiones altas un estimativo infeio de Kuttle-Sigillito en un dominio estellado acotado dos dimensional. Palabas y fases clave. Valo popio, cota infeio, poblema de Steklov. 1. Intoduction Let Ω be a bounded domain of R n with smooth bounday Ω. The following poblem is called the Steklov poblem: ϕ = 0, in Ω ϕ η = ν, ϕ on Ω, (1 whee ν is a eal numbe. This poblem was studied by Steklov [7] fo bounded domains in the plane. The poblem has physical oigins; the function ϕ is a steady state tempeatue in Ω whee the flow ove the bounday, Ω, is popotional to the tempeatue. The set of eigenvalues fo the Steklov poblem 95

2 96 GONZALO GARCíA & ÓSCAR ONTAÑO is the same as the set of eigenvalues fo the Diichlet-Neumann function. This function associates to each function u defined on Ω, the nomal deivative of its hamonic extension û on Ω. The set of eigenvalues of the Steklov poblem consists of an inceasing sequence 0 = ν 0 < ν 1 < ν <, with ν k +. The fist non-zeo eigenvalue is known as the fist eigenvalue of the Steklov poblem; the vaiational chaacteization of this eigenvalue is { Ω ϕ dυ ν 1 = min ϕ Ω ϕ dσ : ϕ C ( Ω }, ϕ dσ = 0. ( Fo the unit ball B n R n, the eigenvalues of the Diichlet-Neumann function ae ν k = k, k = 0, 1,,..., and the eigenfunctions ae given by the space of hamonics homogeneous polynomials of degee k esticted to the bounday of the ball, the (n 1-dimensional unit sphee S n 1. The fist Steklov eigenvalue of the n-dimensional ball of adius > 0, B, is ν 1 (B = 1 and the coodinate functions {x 1,..., x n } ae the espective eigenfunctions. As in the Diichlet and the Neumann poblem, Steklov geometic estimates have been made in the Steklov poblem fo the fist eigenvalue. Fo bounded and simply connected domains in the plane xy, in 1954 Weinstock [8] poved that ν 1 π L, whee L epesents the peimete of the bounday cuve, with equality if and only if Ω is a disk. In 1970 fo convex domains in the plane, Payne [6] poved that ν 1 k o, whee k o is the minimum value of the cuvatue on the bounday of the domain. In 1997, Escoba [] genealized Payne s esult to -dimensional Riemannian manifolds with non-negative Gaussian cuvatue and with bounday such that the geodesic cuvatue k g is bounded below by a positive constant k o ; with these hypotheses Escoba showed that ν 1 k o. In highe dimensions, Escoba consideed compact manifolds with nonnegative Ricci cuvatue and again in the spiit of Payne s theoem poved the following esult: Theoem 1.1. If Ω is an n-dimensional compact Riemannian manifold (n 3 with nonnegative Ricci Cuvatue, with nonempty smooth bounday Ω and whose second fundamental fom π on Ω satisfies π ki fo some positive constant k, then ν 1 > k. Fo otationally invaiant metics with nonnegative Ricci cuvatue in the n-dimensional ball B, ontaño [5] poved that ν 1 h whee h is the mean cuvatue on B. Fo otationally invaiant metics with nonpositive Ricci cuvatue in the n-dimensional ball B, ontaño [4] poved that ν 1 h whee h is the mean cuvatue on B. In both cases equality holds if and only if (B, g is isometic to the Euclidean ball. When Ω R n is a sta-shaped domain with espect to a point P, which, without loss of geneality we can assume that it is the oigin, Bamble and Ω Volumen 49, Númeo 1, Año 015

3 A LOWER BOND FOR THE FIRST STEKLOV EIGENVALE ON A DOAIN 97 Payne [1] poved that ν 1 an 1 h m, n+1 whee a is the adius of a ball centeed at the oigin contained in Ω, is the maximum distance fom P to bode of Ω, and h m is the minimum of the function h : Ω R, defined by with η a oute unit nomal to Ω. h(x = x, η, With a diffeent idea fo the -dimensional case Kuttle and Sigillito [3] poved that { } ν min ( R R max R + (R, whee R(θ = x fo any x Ω of the fom x = x e iθ. In this pape, following the idea of Sigillito and Kuttle we povide a lowe bound fo the fist Steklov eigenvalue in a sta-shaped bounded domain Ω R n with espect to a point P. This impovement of the Kuttle and Sigillito s esult depends on two esults which elate geometic quantities on Ω and on S n 1.. Peliminaies Conside on the Euclidean space (R n,, pola coodinates (, ω, whee R + and ω S n 1. If u : S n 1 is a local chat of the unit sphee,, wites as, = d + g ij du i du j, whee g ij ae the components of the standad ound metic on S n 1 in the chat u. Let Ω be a sta-shaped bounded domain of R n with espect to a point P, which, without loss of geneality we assume that it is the oigin (see Figue 1. Let Ω be the bounday of Ω with oute unit nomal vecto given by η. By the chaacte of sta-shaped of Ω thee exists a function R : S n 1 R + such that Ω = { (, ω R n {0} : 0 < < R(ω } and Ω = { (, ω R n {0} : = R(ω }. In this wok we assume that the bounday of Ω is smooth. In the following poposition we get a esult that elates the function R and its gadient, R, on the bounday of Ω. Revista Colombiana de atemáticas

98 GONZALO GARCíA & ÓSCAR ONTAÑO Figue 1. Sta-shaped domain Ω. Poposition.1. Let G = (g ij and G = ( g ij be espectively the matices of the fist fundamental foms fo Ω and S n 1, in the given coodinate systems.

4 98 GONZALO GARCíA & ÓSCAR ONTAÑO Figue 1. Sta-shaped domain Ω. Poposition.1. Let G = (g ij and G = ( g ij be espectively the matices of the fist fundamental foms fo Ω and S n 1, in the given coodinate systems. Then R R = tan (θ, (3 whee θ is the angle between the oute unit nomal vecto η and the adial vecto field. Poof. The map Ψ : S n 1 R n, given by Ψ(ω = ( R(ω, ω ealizes the embedding of Ω. The induced metic on Ω theefoe wite, in the chat u as Ψ, = Ψ ( d + g ij du i du j = dr + R g ij du i du j = ( R i R j + R g ij dui du j = g ij du i du j. Note that, setting t = log R, we can wite g ij = e t( t i t j + g ij. (4 ( The invese metic g ij theefoe wites as g ij = e t g ij ti t j W, whee W = 1 + t + =, t j = g jk t k and is the connection on ( S n 1, g ij. Since Ω is the zeo set of the function F (, ω = R(ω, then the nomal vecto η is the nomalized gadient F, hence η = 1 W ( Rj R j. It theefoe follows that 1 Rj cos(θ = η, = ( W R j, = 1 W. Volumen 49, Númeo 1, Año 015

5 A LOWER BOND FOR THE FIRST STEKLOV EIGENVALE ON A DOAIN 99 Solving W cos θ = 1 in tems of t we deduce that as claimed. R R = t = 1 cos θ cos θ = tan θ, It is natual to ask fo the elationship between the elements of aea g = det(g and g = det( G. In this diection we have the following compaison theoem; although its poof is standad, we include it fo the convenience of the eade. Poposition.. R g = R g n 1 R + 1. (5 Poof. Let us take a local chat of the unit sphee u : S n 1 such that the matix G = ( g ij of the fist fundamental fom in the given chat is diagonal. Conside the (n 1 (n 1 matix B = (b ij, whee b ij = g ij and the vecto t = ( t 1, t,..., t n 1 t. By fomula (4 we have g ij = e t( AA t ij, whee A is the (n 1 n matix whose enties ae A = [B t]. The deteminant det g can be theefoe obtained easily via Binet theoem fo det ( AA t ( det(g = e (n 1t det G n 1 t i det G + g i=1 ii ( = e (n 1t det G n t i g ii i=1 ( = e (n 1t det G n t i t i i=1 = e (n 1t det G (1 + t Revista Colombiana de atemáticas

6 100 GONZALO GARCíA & ÓSCAR ONTAÑO eaching the geometic elationship g = R g n 1 R R + 1. (6 3. ain Result sing the esults of the pevious section, we establish the following theoem, whee we find a lowe bound fo the fist eigenvalue of the Steklov poblem in a sta-shaped bounded domain. Theoem 3.1. Let Ω be a bounded sta-shaped domain of R n with smooth bounday Ω and oute unit nomal η. If 0 θ α < π, whee cos(θ = η,, then the fist eigenvalue of Steklov fo Ω, ν 1 (Ω, satisfies the inequality n m + a a + 4a } ν 1 (Ω, (7 whee a = tan (α, m := min R and := max R. Poof. Without loss of geneality let us assume that Ω is a sta-shaped domain with espect to the oigin, ξ : S n 1 is a standad paametization of S n 1 R n and y : Ω is the associated paametization to the bounday of Ω. If we define R : R fo R(u = y(u, then y = Rξ. Consideing the spheical coodinates x = ξ(u, Ω = { (u, : u, 0 < < R(u } and dx = g d du. Fo ϕ : Ω R, we have the Rayleigh s quotient ϕ dx Ω R[ϕ] = = ϕ dσ x Ω R(u 0 { ( ϕ } + 1 ϕ g du d ϕ, g du whee g ij = R R u i u j + R g ij. aking the change of vaiables u = u and = ρr(u we obtain R[ϕ] = ( 1 ρ ϕ g du { ρ R ( ( ϕ ρ ϕ ρ ϕ, R + ρ R Fom the Cauchy-Schwaz inequality, + 1 ρ ϕ + ( } ϕ R ρ n 1 R n g du dρ. ρ Volumen 49, Númeo 1, Año 015

7 A LOWER BOND FOR THE FIRST STEKLOV EIGENVALE ON A DOAIN 101 ρ R ϕ ϕ, R (γ ϕ ρ ( ϕ + R ρ γ R, ρ fo any function γ. Fom hee, 1 0 aking 1 γ γ { ( R n R 1 1 γ γ R = β, it follows that ( ϕ ( } 1 ρ 1 γ ϕ ϕ. g du { ( 1 R n 1 β R ( ϕ R 1 β ρ 1+β ϕ } 0 ϕ. g du Fom the equality (3, taking a = tan (α, we aive to = = { (1 R n β a ( ϕ } 1 β ϕ ρ 1+β ϕ g du { (1 R n β a ( ϕ } 1 β ϕ ρ 1+β ϕ ( g g g du { (1 R n β a ( ϕ 1 β ρ ϕ ( R R + 1 { (1 R n β a ( ϕ 1 β ρ 1+β ϕ } R n 1 g du } ϕ 1+β ϕ ( R n 1 g. du Since m := min R and := max R, we have Revista Colombiana de atemáticas

8 10 GONZALO GARCíA & ÓSCAR ONTAÑO m n 1 0 Solving fo β the equation 1 β a = Consequently, { (1 β a ( ϕ 1 β ρ } ϕ 1+β ϕ g. du β 1+β we obtain 1 β a = β 1 + β = + a a + 4a > 0. m n + a a + 4a } 1 0 m n + a a + 4a } = B 1(0 { ( ϕ } 1 ϕ ρ ϕ dx S n 1 ϕ dσ x. ϕ g du If ϕ is feasible fo the ball B 1 (0 then m n + a a + 4a }. ϕ 1 dσ x S If we take ϕ = ϕ 1 n 1 dσ x, whee ϕ 1 is a fist eigenfunction fo the Steklov S n 1 poblem on Ω we get hence R[ϕ] = R[ϕ 1 ϕ 1 ] Ω ϕ 1 dx Ω (ϕ 1 ϕ 1 dσ x Ω ϕ1 dx Ω ( ϕ 1 + ϕ 1 dσ x ( n m { + a a + 4a } n m + a a + 4a } n m + a a + 4a }, B 1 ϕ dx S n 1 ϕ dσ x Volumen 49, Númeo 1, Año 015

9 A LOWER BOND FOR THE FIRST STEKLOV EIGENVALE ON A DOAIN 103 and theefoe Ω ϕ1 dx ϕ 1 dσ x Ω whee a = tan (α. ν 1 (Ω Ω Ω ϕ1 dx ( ϕ 1 + ϕ 1 dσ x n m + a a + 4a }, n m + a a + 4a }, Obseve fom hee that, in the ball of adius, a = 0, and in this case ou estimative is shap. Acknowledgements. Gonzalo Gacia and Osca ontaño wee suppoted by the nivesidad del Valle, Cali, Colombia. Osca ontaño thanks to the nivesidad del Valle fo its suppot duing his Ph.D. studies. Authos ae vey gateful to the eviewes fo thei suggestions and comments that eally impoved the pape. Refeences [1] J. H. Bamble and L. E. Payne, Bounds in the Neumann Poblem fo Second Ode nifomly Elliptic Opeatos, Pacific Jounal of athematics 1 (196, no. 3, [] J. F. Escoba, The Geomety of the fist Non-Zeo Stekloff Eigenvalue, Jounal of functional analysis 150 (1997, [3] J. R. Kuttle and V. G. Sigillito, Lowe Bounds fo Stekloff and Fee embane Eigenvalues, SIA Review 10 (1968, [4] O. A. ontaño, Cota supeio paa el pime valo popio del poblema de steklov, Revista Integación 31 (013, no. 1, (es. [5], The Stekloff Poblem fo Rotationally Invaiant etics on the Ball, Revista Colombiana de atemáticas 47 (013, no., [6] L. E. Payne, Some Isopeimetic Inequalities fo Hamonic Functions, SIA J. ath. Anal. 1 (1970, [7]. W. Stekloff, Su les poblemes fondamentaux de la physique mathematique, Ann. Sci. Ecole Nom 19 (190, (f. Revista Colombiana de atemáticas

10 104 GONZALO GARCíA & ÓSCAR ONTAÑO [8] R. Weinstock, Inequalities fo a Classical Eigenvalue Poblem, Rational ech. Anal 3 (1954, (Recibido en febeo de 014. Aceptado en diciembe de 014 Depatamento de atemáticas nivesidad del Valle Facultad de Ciencias Caea 100, calle 13 Cali, Colombia gonzalo.gacia@coeounivalle.edu.co osca.montano@coeounivalle.edu.co Volumen 49, Númeo 1, Año 015

THE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2

THE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2 THE LAPLACE EQUATION The Laplace (o potential) equation is the equation whee is the Laplace opeato = 2 x 2 u = 0. in R = 2 x 2 + 2 y 2 in R 2 = 2 x 2 + 2 y 2 + 2 z 2 in R 3 The solutions u of the Laplace