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

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

Olivier Druet
Emmanuel Hebey
Frédéric Robert
Copyright Date: 2004
Pages: 224
https://www.jstor.org/stable/j.ctt7s38w
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  • Book Info
    Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45):
    Book Description:

    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.

    eISBN: 978-1-4008-2616-2
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-viii)
    The Authors
  4. Chapter One Background Material
    (pp. 1-12)

    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 [20], do Carmo [22], Gallot-Hulin-Lafontaine [36], Hebey [43], Jost [50], Kobayashi-Nomizu [53], Sakai [65], and Spivak [72]. 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.

    We start with a few notions in differential geometry. LetMbe a Hausdorff topological space. We say thatM...

  5. Chapter Two The Model Equations
    (pp. 13-24)

    Let$\left( {M,g} \right)$be a smooth compact Riemannian manifold of dimensionn ≥ 3. We let$\left( {{h_\alpha }} \right)$be a sequence of smooth functions onM, and consider equations like.

    ${\Delta _g}u + {h_\alpha }u = {u^{2* - 1}}$$\left( {E{}_\alpha } \right)$

    where${\Delta _g} = - di{v_g}\nabla $is the Laplace-Beltrami operator,$2{\kern 1pt} * = 2n/(n\; - \;2)$is the critical Sobolev exponent for the embedding of the Sobolev space$H_1^2\left( M \right)$into Lebesgue’s spaces${L^p}(M)$, anduis required to be positive. By standard regularity theory, as developed by Gilbarg-Trudinger [37], and thanks to the maximum principle, if$u \in H_1^2\left( M \right)$is a nonnegative solution of$\left( {{E_\alpha }} \right)$, thenuis smooth and either$u > 0$everywhere or$u \equiv 0$. As already mentioned, thanks to the...

  6. Chapter Three Blow-up Theory in Sobolev Spaces
    (pp. 25-50)

    An important result of the 1980s describes the asymptotic behavior of Palais-Smale sequences associated with equations like

    $\Delta u = {u^{2*\, - \,1}}$, (3.0.1)

    where$\Delta $is the Euclidean Laplacian, anduis required to vanish on the boundary of a smooth bounded open subset$\Omega $of the Euclidean space${\mathbb{R}^n}$. This result was proved by Struwe [73]. Related references are Brézis [11], Brézis-Coron [12, 13], Lions [58], Sacks-Uhlenbeck [64], Schoen [67], and Wente [76]. Let$D_1^2\left( {{\mathbb{R}^n}} \right)$be the homogeneous Sobolev space de ned as the completion of$C_0^\infty \left( {{\mathbb{R}^n}} \right)$, the space of smooth functions with compact support in${\mathbb{R}^n}$, with respect to the norm a...

  7. Chapter Four Exhaustion and Weak Pointwise Estimates
    (pp. 51-66)

    Let$\left({M,g}\right)$be a smooth compact Riemannian manifold of dimension$n\ge3$. We let$({h_\alpha})$be a sequence of smooth functions onM, and consider equations like

    ${\Delta_g}u\;+\;{h_\alpha}u={u^{2*\,-\;1}}\quad\quad\quad\quad\quad({E_\alpha})$

    whereuis required to be positive. We let$({u_\alpha})$be a sequence of solutions to$({E_\alpha})$, so that${u_\alpha}>0$and

    ${\Delta_g}{u_\alpha}+\;{h_\alpha}{u_\alpha}=u_\alpha^{2*\,-\;1}$

    for all$\alpha$. We describe in this chapter an exhaustion method for the blow-up behavior of the${u_\alpha}$’s. This method provides a constructive approach to Theorem 3.1, and weak (with respect to the material in Chapter 5 and 6) pointwise estimates on the${u_\alpha}$’s.

    We assume in what follows that there...

  8. Chapter Five Asymptotics When the Energy Is of Minimal Type
    (pp. 67-82)

    Let$\left({M,g}\right)$be a smooth compact Riemannian manifold of dimension$n\ge\;3$. We let$({h_\alpha})$be a sequence of smooth functions onM, and consider equations like

    ${\Delta_g}u\;+\;{h_\alpha}u\;=\;{u^{2*\,-\,1}}\quad\quad\quad\quad\quad\quad({E_\alpha})$

    whereuis required to be positive. We let$({u_\alpha})$be a sequence of solutions to$({E_\alpha})$, so that${u_\alpha}>\;0$and

    ${\Delta_g}{u_\alpha}+\;{h_\alpha}{u_\alpha}=u_\alpha^{2*\,-1}$

    for all$\alpha$. For the reader’s convenience, we describe in this chapter the${C^0}$-theory for the blow-up behavior of the${u_\alpha}$’s when the energy of the${u_\alpha}$’s is of minimal type. The general situation of arbitrary energies is treated in the next chapter.

    As in Chapter 4, we assume in what follows...

  9. Chapter Six Asymptotics When the Energy Is Arbitrary
    (pp. 83-200)

    Let$\left({M,g}\right)$be a smooth compact Riemannian manifold of dimension$n\ge\;3$. We let$({h_\alpha})$be a sequence of smooth functions onMand consider equations like

    ${\Delta_g}u\;+\;{h_\alpha}u\;=\;{u^{2*\,-\,1}}\quad\quad\quad\quad\quad\quad({E_\alpha})$

    whereuis required to be positive. We let$({u_\alpha})$be a sequence of solutions to$({E_\alpha})$, so that${u_\alpha}>0$and

    ${\Delta_g}{u_\alpha}+\;{h_\alpha}{u_\alpha}=u_\alpha^{2*\,-1}$

    for all$\alpha$. We describe in this chapter the${C^0}$-theory for the blow-up behavior of the${u_\alpha}$’s when the energy of the${u_\alpha}$’s is arbitrary.

    As in Chapters 4 and 5, we assume in what follows that there exist$0<\theta<1$and a smooth (or only${C^{0,\theta}}$) function${h_\infty}$onMsuch that

    the...

  10. Appendix A. The Green’s Function on Compact Manifolds
    (pp. 201-208)
  11. Appendix B. Coercivity Is a Necessary Condition
    (pp. 209-212)
  12. Bibliography
    (pp. 213-218)