I Hardy-Sobolev inequalities with non smooth boundary 49

Contents

7 ^{Chapter 1} Introduction

27 ^{Chapter 2} Introduction (English version)

I Hardy-Sobolev inequalities with non smooth boundary 49

51

Chapter 3

Hardy-Sobolev inequalities with non smooth boundary, I

3.1 Introduction 52

3.2 The best Hardy constant and Hardy Sobolev Inequality 58

3.3 Regularity and approximate solutions 65

3.4 Symmetry of the extremals for µ

( R

) 67 3.5 Existence of extremals: the case of small val-

ues of γ 72

3.6 Proof of Theorem 3.1.3 96

99

Chapter 4

Hardy-Sobolev inequalities with non smooth boundary, II

4.1 Introduction 99

4.2 Definition of the generalized curvature and the mass 103

4.3 Some background results 104

5

6 Contents

4.4 Test-functions estimates for the mass: proof of Theorem 4.1.2 106

4.5 Examples of mass 116

4.6 Proof of Theorem 4.1.3: functional background for the perturbed equation 117

4.7 Proof of Theorem 4.1.3: Test-Functions esti- mates 123

II The second best constant for the Hardy-Sobolev inequal-

ity 131

133

Chapter 5

The second best constant for the Hardy-Sobolev inequality

5.1 Introduction 133

5.2 Preliminary blow-up analysis 137

5.3 Refined blowup analysis: proof of Theorem 5.1.3 144 5.4 Direct consequences of Theorem 5.1.3 153

5.5 Pohozaev identity and proof of Theorem 5.1.4 161 5.6 Proof of Theorem 5.1.2 177

5.7 Appendix 179

III Paneitz-Branson type equation 181

183 ^{Chapter 6} Paneitz-Branson type equation

6.1 Introduction 183 6.2 Preliminaries 189

6.3 A relation between Σ

( R

) and S 191 6.4 Asymptotic estimates 196

6.5 A Sobolev inequality of second order 205 6.6 Minimizing solutions for small α 209

213 ^{Chapter 7} Bibliography