Eigenvalues on Riemannian Manifolds - IMPA .Publica§µes Matemticas Eigenvalues on Riemannian

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  • Eigenvalues on Riemannian Manifolds

  • Publicaes Matemticas

    Eigenvalues on Riemannian Manifolds

    Changyu Xia UnB

    29o Colquio Brasileiro de Matemtica

  • Copyright 2013 by Changyu Xia

    Impresso no Brasil / Printed in Brazil

    Capa: Noni Geiger / Srgio R. Vaz

    29o Colquio Brasileiro de Matemtica

    Anlise em Fractais Milton Jara Asymptotic Models for Surface and Internal Waves - Jean-Claude Saut Bilhares: Aspectos Fsicos e Matemticos - Alberto Saa e Renato de S

    Teles Controle timo: Uma Introduo na Forma de Problemas e Solues -

    Alex L. de Castro Eigenvalues on Riemannian Manifolds - Changyu Xia

    Equaes Algbricas e a Teoria de Galois - Rodrigo Gondim, Maria Eulalia de Moraes Melo e Francesco Russo

    Ergodic Optimization, Zero Temperature Limits and the Max-Plus Algebra - Alexandre Baraviera, Renaud Leplaideur e Artur Lopes

    Expansive Measures - Carlos A. Morales e Vctor F. Sirvent Funes de Operador e o Estudo do Espectro - Augusto Armando de

    Castro Jnior Introduo Geometria Finsler - Umberto L. Hryniewicz e Pedro A. S.

    Salomo Introduo aos Mtodos de Crivos em Teoria dos Nmeros - Jlio

    Andrade Otimizao de Mdias sobre Grafos Orientados - Eduardo Garibaldi e

    Joo Tiago Assuno Gomes ISBN: 978-85-244-0354-5

    Distribuio: IMPA Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ E-mail: ddic@impa.br http://www.impa.br

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    Contents

    1 Eigenvalue problems on Riemannian manifolds 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Some estimates for the first eigenvalue of the Laplacian 5

    2 Isoperimetric inequalities for eigenvalues 142.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 142.2 The Faber-Krahn Inequality . . . . . . . . . . . . . . . 162.3 The Szego-Weinberger Inequality . . . . . . . . . . . . 172.4 The Ashbaugh-Benguria Theorem . . . . . . . . . . . 192.5 The Hersch Theorem . . . . . . . . . . . . . . . . . . . 23

    3 Universal Inequalities for Eigenvalues 293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Eigenvalues of the Clamped Plate Problem . . . . 343.3 Eigenvalues of the Polyharmonic Operator . . . 463.4 Eigenvalues of the Buckling Problem . . . . . . . . . . 57

    4 Polya Conjecture and Related Results 734.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 734.2 The Krogers Theorem . . . . . . . . . . . . . . . . . . 774.3 A generalized Polya conjecture by Cheng-Yang . . 814.4 Another generalized Polya conjecture . . . . . . . . . . 85

    5 The Steklov eigenvalue problems 945.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 945.2 Estimates for the Steklov eigenvalues . . . . . . . . . . 95

    3

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    4 CONTENTS

    Bibliography 103

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    Chapter 1

    Eigenvalue problems on

    Riemannian manifolds

    1.1 Introduction

    Let (M, g) be an n-dimensional Reimannian manifold with boundary(possibly empty). The most important operator on M is the Lapla-cian . In local coordinate system {xi}ni=1, the Laplacian is givenby

    =1G

    n

    i,j=1

    xi

    (Ggij

    xj

    )

    where (gij) is the inverse matrix (gij)1, gij = g(

    xi

    , xj ) are the

    coefficients of the Riemannian metric in the local coordinates, andG = det(gij). In local coordinates, the Riemannian measure dv on(M, g) is given by

    dv =Gdx1...dxn.

    Let C(M) and set

    ||||21 =

    M

    ||2 +

    M

    ||2.

    1

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    2 [CAP. 1: EIGENVALUE PROBLEMS ON RIEMANNIAN MANIFOLDS

    Here and in the future, the integrations on M are always taken withrespect to the Riemannian measure on M . Let us denote by H21 (M)

    ando

    H21 (M) the completion of C(M) and C0 (M) with respect to

    || ||. The theory of Sobolev spaces tells us that H21 (M) =o

    H21 (M)when M is complete. Our purpose is to study some eigenvalue prob-lems associated to the Laplacian operator on a compact manifold M .When M = , we consider the closed eigenvalue problem:

    u+ u = 0. (1.1)

    When M 6= , we are interested in the following eigenvalue prob-lems.

    The Dirichlet problem:{

    u = u in M,u|M = 0. (1.2)

    The Neumannn problem:{

    u = u in M,u

    M

    = 0,(1.3)

    where is the unit outward normal to M .

    The clamped plate problem:{

    2u = u in M,u|M = u

    M

    = 0,(1.4)

    The buckling problem:{

    2u = u in M,u|M = u

    M

    = 0,(1.5)

    The eigenvalue problem of poly-harmonic operator:{

    ()lu = u in M,u|M = u

    M

    = = l1ul1

    M

    = 0, l 2. (1.6)

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    [SEC. 1.1: INTRODUCTION 3

    The buckling problem of arbitrary order:{

    ()lu = u in M,u|M = u

    M

    = = l1ul1

    M

    = 0, l 2. (1.7)

    The Steklov problem of second order:{

    u = 0 in M,u = u on M.

    (1.8)

    The Steklov problem of fourth order:{

    2u = 0 in M,u = u u = 0 on M.

    (1.9)

    Let us denote by 1 the first non-zero eigenvalue of the above prob-lems. We can arrange the eigenvalues of these problems as follows:

    0 < 1 2 +.

    For many reasons in Mathematics and Physics, it is important toobtain nice estimates for the s. We will concentrate our attentionon this problem. Let us list some basic facts in this direction.

    Theorem 1.1 (Weyls asymptotic formula, [97]). In each of theeigenvalue problems (1.1), (1.2), (1.3), let N() be the number ofeigenvalues, counted with multiplicity, . Then

    N() n|M |n/2/(2)n (1.10)

    as , where n is the volume of the unit ball in Rn and |M | isthe volume of M . In particular,

    n/2k {(2)n/n}k/|M | (1.11)

    as +.

    There are similar asymptotic formulas for the other eigenvalueproblems above (Cf. [1], [79], [80]).

    Define a space H as follows:

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    4 [CAP. 1: EIGENVALUE PROBLEMS ON RIEMANNIAN MANIFOLDS

    For the closed eigenvalue problem (1.1),

    H =

    {f H21 (M)

    M

    f = 0

    }. (1.12)

    For the Dirichlet eigenvalue problem (1.2),

    H =o

    H21 (M). (1.13)

    For the Neumann eigenvalue problem (1.3),

    H =

    {f H21 (M)

    M

    f = 0

    }. (1.14)

    A fundamental fool in the theory of eigenvalues is the

    Mini-Max principle. We can find a countable orthonormal ba-sis {fi}, fi C(M) for the problems (1.1), (1.2) and (1.3) suchthat

    1 = inf{

    M|f |2

    Mf2

    | f H},

    i = inf{

    M|f |2

    Mf2

    | f H,

    Mffj = 0, j = 1, , i 1

    }.

    (1.15)

    In particular, we have the

    Poincare inequality:

    M

    |f |2 1

    M

    f2, f H. (1.16)

    For other eigenvalue problems above, similar mini-max principlesalso hold.

    Theorem 1.2 (The Co-Area formula, [17]). Let M be a compactRiemannian manifold with boundary, f H1(M). Then for anynon-negative function g on M ,

    M

    g =

    (

    {f=}

    g

    |f |

    )d (1.17)

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    [SEC. 1.2: SOME ESTIMATES FOR THE FIRST EIGENVALUE OF THE LAPLACIAN 5

    1.2 Some estimates for the first eigenvalue

    of the Laplacian

    In this section, we will prove some estimates for the first eigenvalueof the Laplacian.

    Theorem 1.3 ([73]). Let M be an n-dimensional complete Rie-mannian manifold with Ricci curvature RicM n 1. Then thefirst non-zero eigenvalue of the closed eigenvalue problem (1.1) of Msatisfies 1(M) n.

    The proof of Theorem 1.3 can be carried out by substituting afirst eigenfunction into the Bochner formula and integrating on Mthe resulted equality (see the proof of theorem 1.6 below).

    An important classical result about eigenvalue is the following

    Theorem 1.4 (Chengs Comparison Theorem, [18]). Let M bean n-dimensional complete Riemannian manifold with Ricci curva-ture satisfying RicM (n 1)c and let BR(p) be an open geodesicball of radius R around a point p in M , where R < /

    c, when

    c > 0. Then the first eigenvalue of the Dirichlet problem (1.2) ofBR(p) satisfies

    1(BR(p)) 1(BR(c)), (1.18)

    with equality holding if and only if BR(p) is isometric to BR(c), whereBR(c) is a geodesic ball of radius R in a complete simply connectedRiemannian manifold of constant curvature c and of dimension n.

    An immediate application of Chengs eigenvalue comparison the-orem is a rigidity theorem for compact manifolds of positive Riccicurvature.

    Theorem 1.5 (The Maximal Diameter Theorem, [18]). Let Mbe an n-dimensional complete Riemannian manifold with Ricci cur-vature RicM n 1. If the diameter of M satisfies d(M) , thenM is isometric to an n-dimensional unit sphere.

    Proof. Take two points p, q M so that d(p, q) ; thenB/2(p)B/2(q) = . Let f and g be the first eigenfunctions corre-

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    6 [CAP. 1: EIGENVALUE PROBLEMS ON RIEMANNIAN MANIFOLDS

    sponding to the first Dirichlet eigenvalues of B/2(p) and B/2(q), re-spectively. We extend f and g on the wholeM by setting f |M\B/2(p) =g|M\B/2(q) = 0 and take two non-zero constants a and