Local And Global Aspects Of Quasilinear Degenerate Elliptic Equations: Quasilinear Elliptic Singular Problems

by Laurent Veron

Overview

This book is devoted to the study of elliptic second-order degenerate quasilinear equations, the model of which is the p-Laplacian, with or without dominant lower order reaction term. Emphasis is put on three aspects:

Product Details

ISBN-13:	9789814730327
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/14/2017
Pages:	476
Product dimensions:	6.00(w) x 9.10(h) x 1.20(d)

Table of Contents

Preface vii

1 Regularity 1

1.1 Notations 1

1.1.1 Domains and flow coordinates 1

1.1.2 Riemannian manifolds 2

1.2 Distributions, Sobolev and Besov spaces 6

1.2.1 L^p and Lorentz spaces 6

1.2.2 Hausdorff measure 9

1.2.3 Weak derivatives and distributions 10

1.2.3.1 Distributions 10

1.2.3.2 Weak derivatives 13

1.2.3.3 Sobolev spaces 13

1.2.3.4 Fractional Sobolev spaces 16

1.2.4 Bessel potentials 17

1.2.4.1 Bessel potential spaces 18

1.2.4.2 Bessel-Lorentz potential spaces 18

1.2.4.3 Bessel-Lorentz capacities 19

1.2.4.4 Riesz and Sobolev capacities 21

1.3 General quasilinear elliptic equations 22

1.3.1 The Serrin's results 23

1.3.2 Regularity 29

1.3.2.1 Local regularity 29

1.3.2.2 Boundary regularity 31

1.3.3 Maximum principle 31

1.4 Existence result 33

1.4.1 Method of monotonicity 33

1.4.2 Method of super and sub solutions 37

1.5 The p-Laplace operator 52

1.5.1 Comparison principles 53

1.5.2 Isotropic singularities of p-harmonic functions 56

1.5.3 Rigidity theorems for p-harmonic functions 60

1.5.4 Isolated singularities of p-superharmonic functions 63

1.6 Notes and open problems 64

2 Separable solutions 65

2.1 Eigenvalue and eigenfunctions 65

2.1.1 Dirichlet problem 65

2.1.2 The case p → ∞ 70

2.1.3 Higher order eigenvalues 74

2.1.4 Eigenvalues on a compact manifold 77

2.2 Spherical p-harmonic functions 87

2.2.1 The spherical p-harmonic eigenvalue problem 87

2.2.2 The 2-dimensional case 99

2.2.3 The case p → ∞ 106

2.2.3.1 The spherical infinite harmonic eigenvalue problem 106

2.2.3.2 The 1-dim equation 111

2.3 Boundary singularities of p-harmonic functions 113

2.3.1 Boundary singular solutions 113

2.3.2 Rigidity results for singular p-harmonic functions in the half-space 121

2.4 Notes and open problems 124

3 Quasilinear equations with absorption 129

3.1 Singular solutions with power absorption in R^N 129

3.1.1 Removable singularities in R^N 130

3.1.2 Classification of isolated singularities 134

3.1.3 Global solutions in R^N 151

3.1.4 Large solutions 154

3.2 Singular solutions in R^N₊ 167

3.2.1 Separable solutions in R^N₊ 167

3.3 Classification of boundary singularities 173

3.3.1 The general gradient estimate near a singularity 174

3.3.2 Removable singularities 178

3.3.3 Classification of boundary singularities 184

3.4 Singularities of quasilinear Hamilton-Jacobi type equations 196

3.4.1 Comparison principles 196

3.4.2 Radial solutions 201

3.4.3 Gradient estimates and applications 203

3.4.4 Isolated singularities 207

3.4.4.1 Removable isolated singularities 207

3.4.4.2 Isolated singularities of positive solutions 211

3.4.4.3 Global singular solutions 216

3.4.4.4 Isolated singularities of negative solutions 217

3.4.5 Geometric estimates 222

3.4.5.1 Gradient estimates on a Riemannian manifold 222

3.4.5.2 Growth of solutions and Liouville type results 226

3.5 Boundary singularities of quasilinear Hamilton-Jacobi typo equations 231

3.5.1 Separable solutions 231

3.5.2 Boundary isolated singularities 233

3.5.2.1 Removable singularities 233

3.5.2.2 Construction of singular solutions 235

3.5.2.3 Global solutions 241

3.5.2.4 Entire singular solutions 245

3.6 Notes and open problems 247

4 Quasilinear equations with measure data 249

4.1 Equations with measure data: the framework 249

4.2 p-superharmonic functions 250

4.2.1 The notion of p-superharmonicity 250

4.2.2 The p-harmonic measure 254

4.2.3 The Peron's method 256

4.3 Renormalized solutions 260

4.3.1 Locally renormalized solutions 260

4.3.2 The stability theorem 267

4.3.3 Further stability results 286

4.4 Notes and open problems 288

5 Quasilinear equations with absorption and measure data 291

5.1 The subcritical case 291

5.1.1 General nonlinearity with data in W^-1,p' (Ω) 291

5.1.2 Subcritical nonlinearity with measure data 297

5.1.3 Applications 304

5.2 Removable singularities 305

5.2.1 Local estimates for locally renormalized solutions 305

5.2.2 The power case 311

5.2.3 The Hamilton-Jacobi equation 317

5.3 Supercritical equations 320

5.3.1 Estimates on potentials 320

5.3.2 Approximation of measures 334

5.4 Solvability with measure data revisited 335

5.4.1 The general case 335

5.4.2 Power growth nonlinearities 340

5.4.3 Exponential type nonlinearities 342

5.5 Large solutions revisited 343

5.5.1 The maximal solution 344

5.5.2 Potential estimates 346

5.5.3 Applications to large solutions 354

5.5.4 The case p = 2 356

5.6 Notes and open problems 365

6 Quasilinear equations with source 367

6.1 Singularities of quasilinear Lane-Emden equations 367

6.1.1 Separable solutions in R^N 367

6.1.2 Isolated singularities 369

6.1.3 Rigidity theorems 374

6.1.4 Separable solutions in R^N₊ 388

6.1.5 Boundary singularities 398

6.2 Equations with measure data 403

6.2.1 The case p = 2 403

6.2.2 The subcritical case 404

6.2.2.1 Sufficient condition 405

6.2.2.2 The case 1 < p < N 409

6.2.2.3 The case p = N 415

6.2.3 The supercritical case 425

6.2.34 Necessary and sufficient conditions 425

6.2.3.2 Removable singularities 428

6.3 Quasilinear Hamilton-Jacobi type equations 428

6.3.1 Separable solutions of quasilinear Hamilton-Jacobi equations 428

6.3.2 Quasilinear Hamilton-Jacobi equations with measure data 432

6.4 Notes and open problems 440

Bibliography 443

List of symbols 453

Glossary 455