Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45): [NOOK Book]

Overview

Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the well-known Yamabe type. They involve Schrödinger operators on the left hand side and a critical nonlinearity on the right hand side.

A significant development in the study of such equations occurred in the 1980s. It was discovered that the sequence splits into a...

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Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45):

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Overview

Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the well-known Yamabe type. They involve Schrödinger operators on the left hand side and a critical nonlinearity on the right hand side.

A significant development in the study of such equations occurred in the 1980s. It was discovered that the sequence splits into a solution of the limit equation--a finite sum of bubbles--and a rest that converges strongly to zero in the Sobolev space consisting of square integrable functions whose gradient is also square integrable. This splitting is known as the integral theory for blow-up. In this book, the authors develop the pointwise theory for blow-up. They introduce new ideas and methods that lead to sharp pointwise estimates. These estimates have important applications when dealing with sharp constant problems (a case where the energy is minimal) and compactness results (a case where the energy is arbitrarily large). The authors carefully and thoroughly describe pointwise behavior when the energy is arbitrary.

Intended to be as self-contained as possible, this accessible book will interest graduate students and researchers in a range of mathematical fields.

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What People Are Saying

William Beckner
This is an important and original work. It develops critical new ideas and methods for the analysis of elliptic PDEs on compact manifolds, especially in the framework of the Yamabe equation, critical Sobolev embedding, and blow-up techniques. This volume will have an important influence on current research.
William Beckner, University of Texas at Austin
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Product Details

  • ISBN-13: 9781400826162
  • Publisher: Princeton University Press
  • Publication date: 1/10/2009
  • Series: Mathematical Notes
  • Sold by: Barnes & Noble
  • Format: eBook
  • Edition description: Course Book
  • Pages: 224
  • File size: 16 MB
  • Note: This product may take a few minutes to download.

Meet the Author

Olivier Druet is Researcher at CNRS, Ecole Normale Superieure de Lyon. Emmanuel Hebey is Professor at Universite de Cergy-Pontoise. Frederic Robert is Associate Professor at Universite de Nice Sophia-Antipolis.
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Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45)


By Olivier Druet Emmanuel Hebey Frédéric Robert

Princeton University Press

Copyright © 2004 Princeton University Press
All right reserved.

ISBN: 978-0-691-11953-3


Chapter One

Background Material

We recall in this chapter basic facts concerning Riemannian geometry and nonlinear analysis on manifolds. For reasons of length, we are obliged to be succinct and partial. Possible references are Chavel, do Carmo, Gallot-Hulin-Lafontaine, Hebey, Jost, Kobayashi-Nomizu, Sakai, and Spivak. As a general remark, we mention that Einstein's summation convention is adopted: an index occurring twice in a product is to be summed. This also holds for the rest of this book.

1.1 RIEMANNIAN GEOMETRY

We start with a few notions in differential geometry. Let M be a Hausdorff topological space. We say that M is a topological manifold of dimension n if each point of M possesses an open neighborhood that is homeomorphic to some open subset of the Euclidean space [[??].sup.n]. A chart of M is then a couple ([OMEGA], [phi]) where [OMEGA] is an open subset of M, and [phi] is a homeomorphism of [OMEGA] onto some open subset of [[??].sup.n]. For y [member of] [OMEGA], the coordinates of [phi](y) in [[??].sup.n] are said to be the coordinates of y in ([OMEGA], [phi]). Anatlas of M is a collection of charts [([[OMEGA].sub.i], [[phi].sub.i]), i [member of] I, such that M = [U.sub.i]member of]I] [[OMEGA].sub.i]. Given an atlas [([[OMEGA].sub.i], [[phi].sub.i]).sub.i]member of]I, the transition functions are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with the obvious convention that we consider [[phi].sub.j] [omicron] [[phi].sup.-1.sub.i] if and only if [[OMEGA].sub.i] [intersection] [[OMEGA].sub.j] [not equal to] 0. The atlas is then said to be of class [C.sup.k] if the transition functions are of class [C.sup.k], and it is said to be [C.sup.]k-complete if it is not contained in a (strictly) larger atlas of class [C.sup.k]. As one can easily check, every atlas of class [C.sup.k] is contained in a unique [C.sup.k]-complete atlas. For our purpose, we will always assume in what follows that k = + [infinity] and that M is connected. One then gets the following definition of a smooth manifold: A smooth manifold M of dimension n is a connected topological manifold M of dimension n together with a [ITLITL.sup.[infinity]]-complete atlas. Classical examples of smooth manifolds are the Euclidean space [[??].sup.n] itself, the torus [T.sup.n], the unit sphere [S.sup.n] of [[??].sup.n+1], and the real projective space [[??].sup.n]([??]).

Given two smooth manifolds, M and N, and a smooth map f : M [right arrow] N from M to N, we say that f is differentiable (or of class [C.sup.k]) if for any charts ([OMEGA], [phi]) and ([??], [??]) of M and N such that f([OMEGA]) [subset] [??], the map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is differentiable (or of class [C.sup.k]). In particular, this allows us to define the notion of diffeomorphism and the notion of diffeomorphic manifolds.

We refer to the above definition of a manifold as the abstract definition of a smooth manifold. As a surface gives the idea of a two-dimensional manifold, a more concrete approach would have been to define manifolds as submanifolds of Euclidean space. According to a well-known result of Whitney, any paracompact (abstract) manifold of dimension n can be seen as a submanifold of some Euclidean space.

Let us now say some words about the tangent space of a manifold. Given M a smooth manifold and x [member of] M, let [F.sub.x] be the vector space of functions f : M [right arrow] [??] which are differentiable at x. For f [member of] [F.sub.x], we say that f is flat at x if for some chart ([OMEGA], [phi]) of M at x, D[(f [omicron] [[phi].sup.-1]).sub.[??]](x)] = 0. Let [N.sub.x] be the vector space of such functions. A linear form X on [F.sub.x] is then said to be a tangent vector of M at x if [N.sub.x] [subset] KerX. We let [T.sub.x](M) be the vector space of such tangent vectors. Given ([OMEGA], [phi]) some chart at x, of associated coordinates [x.sup.i], we define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by, for any f [member of] F.sub.x,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a simple remark, one gets that the [([partial derivative]/[partial derivative][x.sub.i]).sub.x]'s form a basis of [T.sub.x](M). Now, one defines the tangent bundle of M as the disjoint union of the [T.sub.x](M)'s, x [member of] M. If M is n-dimensional, one can show that T(M) possesses a natural structure of a 2n-dimensional smooth manifold. Given a chart ([OMEGA], [??]) of M,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a chart of T(M), where for X [member of] [T.sub.x](M), x [member of] [OMEGA],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[the coordinates of x in ([OMEGA], [phi]) and the components of X in ([OMEGA], [phi]), that is, the coordinates of X in the basis of [T.sub.x](M) associated with ([OMEGA], [phi]) by the process described above]. By definition, a vector field on M is a map X : M [right arrow] T(M) such that for any x [member of] M, X(x) [member of] [T.sub.x](M). Since M and T(M) are smooth manifolds, the notion of a vector field of class [C.sup.k] makes sense. A manifold M of dimension n is said to be parallelizable if there exist n smooth vector fields [X.sub.i], i = 1,..., n, such that for any x [member of] M, the [X.sub.i](x)'s, i = 1,..., n, define a basis of [T.sub.x](M).

Given two smooth manifolds, M and N, a point x in M, and a differentiable map f : M [right arrow] N at x, the tangent linear map of f at x (or the differential map of f at x), denoted by [f.sub.*] (x), is the linear map from [T.sub.x](M) to [T.sub.f(x)](N) defined, for X [member of] [T.sub.x](M) and g : N [right arrow] [??] differentiable at f(x), by

([f.sub.*](x) · (X) · (g) = X(g [omicron] f).

By extension, if f is differentiable on M, one gets the tangent linear map of f, denoted by [f.sub.*]. That is the map [f.sub.*] : T(M) [right arrow] T(N) defined, for X [member of] [T.sub.x](M), by [f.sub.*](X) = [f.sub.*](x).(X). As one can easily check, [f.sub.*] is [C.sup.k-1] if f is [C.sup.k]. Similar to the construction of the tangent bundle, one can define the cotangent bundle of a smooth manifold M as the disjoint union of the [T.sub.x][(M).sup.*]'s, x [member of] M. In a more general way, one can define [T.sup.q.sub.p](M) as the disjoint union of the [T.sup.q.sub.p]([T.sub.x](M))'s, where [T.sup.q.sub.p]([T.sub.x](M)) is the space of (p, q)-tensors on [T.sub.x](M). Then [T.sup.q.sub.p](M) possesses a natural structure of a smooth manifold of dimension n(1 + [n.sup.p+q-1]). A map T : M [right arrow] T.sup.q.sub.p](M) is then said to be a (p, q)-tensor field on M if for any x [member of] M, T(x) [member of] [T.sup.q.sub.p]([T.sub.x](M)). It is said to be of class [C.sup.k] if it is of class [C.sup.k] from the manifold M to the manifold [T.sup.q.sub.p](M). Given two manifolds M and N, a map f : M [right arrow] N of class [C.sup.k+1], and a (p, 0)-tensor field T of class [C.sup.k] on N, one can define the pullback [f.sup.*]T of T by f, that is, the (p, 0)-tensor field of class [C.sup.k] on M defined for x [member of] M and [X.sub.1],..., [X.sub.p] [member of] [T.sub.x](M), by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now define the notion of a linear connection. Denote by [GAMMA](M) the space of differentiable vector fields on M. A linear connection D on M is a map D : T(M) × [GAMMA](M) [right arrow] T(M) which satisfies a certain number of propositions. In local coordinates, given a chart ([OMEGA], [??]), this is equivalent to having [n.sup.3] smooth functions [[GAMMA].sup.k.sub.ij] : [OMEGA] [right arrow] [??], that we refer to as the Christoffel symbols of the connection in ([OMEGA], [phi]). They characterize the connection in the sense that for X [member of] [T.sub.x](M), x [member of] [OMEGA], and Y [member of] [GAMMA](M),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the [X.sup.i]'s and [Y.sup.i]'s denote the components of X and Y in the chart ([OMEGA], [phi]), and for f : M [right arrow] [??] differentiable at x,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As one can easily check, the [[GAMMA].sup.k.sub.ij]'s are not the components of a (2, 1)-tensor field. An important remark is that linear connections have natural extensions to differentiable tensor fields. Given a differentiable (p, q)-tensor field, T, a point x in M, X [member of] [T.sub.x](M), and a chart ([OMEGA], [phi]) of M at x, [D.sub.X](T) is the (p, q)-tensor on [T.sub.x](M) defined by [D.sub.X](T) = [X.sup.i]([[nabla].sub.I]T)(x), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The covariant derivative commutes with the contraction in the sense that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sup.[k.sub.2].sub.[k.sub.1]] T stands for the contraction of T of order ([k.sub.1], [k.sub.2]). Given a (p, q)-tensor field of class [C.sup.k+1], T, we let [nabla]T be the (p + 1, q)-tensor field of class [C.sup.k] whose components in a chart are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By extension, one can then define [[nabla].sup.2]T, [[nabla].sup.3]T, and so on. For f : M [right arrow] [??] a smooth function, one has that [nabla] f = df and, in any chart ([OMEGA], [phi]) of M,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the Riemannian context, [[nabla].sup.2] f is called the Hessian of f and is sometimes denoted by Hess(f).

The torsion T of a linear connection D can be seen as the smooth (2, 1)-tensor field on M whose components in any chart are given by the relation [T.sup.k.sub.ij] = [[GAMMA].sup.k.sub.ij]-[[GAMMA].sup.k.wub.ji]. One says that the connection is torsion-free if T [equivalent to] 0. The curvature R of D can be seen as the smooth (3, 1)-tensor field on M whose components in any chart are given by the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As one can easily check, R.sup.l.sub.ijk] = -[R.sup.l.sub.ikj]. Moreover, when the connection is torsion-free, one has that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Such relations are referred to as the first Bianchi and second Bianchi identities.

We now discuss Riemannian geometry. Let M be a smooth manifold. A Riemannian metric g on M is a smooth (2, 0)-tensor field on M such that for any x [member of] M, g(x) is a scalar product on [T.sub.x](M). A smooth Riemannian manifold is a pair (M, g) where M is a smooth manifold and g a Riemannian metric on M. According to Whitney, for any paracompact smooth n-manifold there exists a smooth embedding f : M [right arrow] [[??].sup.2n+1]. One then gets that any smooth paracompact manifold possesses a Riemannian metric. Just think of g = [f.sup.*][xi], where [xi] is the Euclidean metric. Two Riemannian manifolds ([M.sub.1], [g.sub.1]) and ([M.sub.2], [g.sub.2]) are said to be isometric if there exists a diffeomorphism f : [M.sub.1] [right arrow] [M.sub.2] such that [f.sup.*][g.sub.2] = [g.sub.1].

Given a smooth Riemannian manifold (M, g), and [gamma] : [a, b] [right arrow] M a curve of class [ITLITL.sup.1], the length of [gamma] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [(d]gamma]/dt).sub.t] [member of] [T.sub.[gamma](t)](M) is such that [(d]gamma]/dt).sub.t] · f = (f [omicron] [gamma])'(t) for any f : M [right arrow] [??] differentiable at [gamma](t). If [gamma] is piecewise [ITLITL.sup.1], the length of [gamma] may be defined as the sum of the lengths of its [ITLITL.sup.1] pieces. For x and y in M, let [C.sub.xy] be the space of piecewise [ITLITL.sub.1] curves [gamma] : [a, b] [right arrow] M such that [gamma](a) = x and [gamma](b) = y. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

defines a distance on M whose topology coincides with the original one of M. In particular, by Stone's theorem, a smooth Riemannian manifold is paracompact. By definition, [d.sub.g] is the distance associated with g.

Let (M, g) be a smooth Riemannian manifold. There exists a unique torsion-free connection on M having the property that [nabla]g = 0. Such a connection is the Levi-Civita connection of g. In any chart ([OMEGA], [??]) of M, of associated coordinates [x.sup.i], and for any x [member of] [OMEGA], its Christoffel symbols are given by the relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the [g.sup.ij]'s are such that [g.sub.im][g.sup.mj] = [[delta].sup.j.sub.i]]. Let R be the curvature of the Levi-Civita connection as introduced above. One defines

1. the Riemann curvature [R.sub.mg] of g as the smooth (4, 0)-tensor field on M whose components in a chart are [R.sub.ijkl] = [g.sub.i[alpha]][R.sup.[alpha].sub.jkl,

2. the Ricci curvature R[c.sub.g] of g as the smooth (2, 0)-tensor field on M whose components in a chart are [R.sub.ij] = [R.sub.[alpha]I[beta]j][g.sup.[alpha][beta]], and

3. the scalar curvature [S.sub.g] of g as the smooth real-valued function on M whose expression in a chart is [S.sub.g] = [R.sub.ij][g.sup.ij].

As one can check, in any chart,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and the two Bianchi identities are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In particular, the Ricci curvature is symmetric, so that in any chart [R.sub.ij] = [R.sub.ji].

Given a smooth Riemannian manifold (M, g), and its Levi-Civita connection D, a smooth curve [gamma] : [a, b] [right arrow] M is said to be a geodesic, if for all t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This means again that in any chart, and for all k,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Continues...)



Excerpted from Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) by Olivier Druet Emmanuel Hebey Frédéric Robert Copyright © 2004 by Princeton University Press. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Preface vii
Chapter 1. Background Material 1
1.1 Riemannian Geometry 1
1.2 Basics in Nonlinear Analysis 7
Chapter 2. The Model Equations 13
2.1 Palais-Smale Sequences 14
2.2 Strong Solutions of Minimal Energy 17
2.3 Strong Solutions of High Energies 19
2.4 The Case of the Sphere 23
Chapter 3. Blow-up Theory in Sobolev Spaces 25
3.1 The H 2/1-Decomposition for Palais-Smale Sequences 26
3.2 Subtracting a Bubble and Nonnegative Solutions 32
3.3 The De Giorgi-Nash-Moser Iterative Scheme for Strong Solutions 45
Chapter 4. Exhaustion and Weak Pointwise Estimates 51
4.1 Weak Pointwise Estimates 52
4.2 Exhaustion of Blow-up Points 54
Chapter 5. Asymptotics When the Energy Is of Minimal Type 67
5.1 Strong Convergence and Blow-up 68
5.2 Sharp Pointwise Estimates 72
Chapter 6. Asymptotics When the Energy Is Arbitrary 83
6.1 A Fundamental Estimate: 1 88
6.2 A Fundamental Estimate: 2 143
6.3 Asymptotic Behavior 182
Appendix A. The Green's Function on Compact Manifolds 201
Appendix B. Coercivity Is a Necessary Condition 209
Bibliography 213
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