Olivier Druet

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

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 Greens Function on Compact Manifolds 201 Appendix B. Coercivity Is a Necessary Condition 209 Bibliography 213

Blow-up analysis in dimension 2 and a sharp form of trudinger-moser inequality

Abstract This paper deals with an improvement of the Trudinger–Moser inequality associated to the embedding of the standard Sobolev space into Orlicz spaces for Ω a smooth bounded domain in ℝ2. The inequality proved here gives in particular precise informations on a previous result obtained by Lions and can be very useful in the study of lack of compactness of the embedding of into {exp(4πu 2) ∈ L 1(Ω)}. We also provide a general asymptotic analysis for sequences of solutions to elliptic PDEs with critical Sobolev growth which blow up at some point. We obtain in particular a result which is well-known in higher dimensions: the concentration points are located at critical points of the regular part of the Green function of the linear operator involved in the equation.

The program in geometric analysis: sharp Sobolev inequalities and related problems

Euclidean background Statement of the

Sharp local isoperimetric inequalities involving the scalar curvature

AB

Bubbling phenomena for fourth-order four-dimensional pdes with exponential growth

program Some historical motivations The

Blow-up examples for second order elliptic PDEs of critical Sobolev growth

H^2_1

Generalized scalar curvature type equations on compact Riemannian manifolds

-inequality--Part I The

Isoperimetric Inequalities on Compact Manifolds

H^2_1

EXISTENCE AND A PRIORI BOUNDS FOR ELECTROSTATIC KLEIN–GORDON–MAXWELL SYSTEMS IN FULLY INHOMOGENEOUS SPACES

-inequality--Part II PDE methods The isoperimetric inequality The

Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature

H^p_1