Nonlinear Second Order Elliptic Equations Involving Measures

by Moshe Marcus, Laurent Véron

Overview

In the last 40 years semi-linear elliptic equations became a central subject of study in the theory of nonlinear partial differential equations. On the one hand, the interest in this area is of a theoretical nature, due to its deep relations to other branches of mathematics, especially linear and nonlinear harmonic analysis, dynamical systems, differential geometry and probability. On the other hand, this study is of interest because of its applications. Equations of this type come up in various areas such as problems of physics and astrophysics, curvature problems in Riemannian geometry, logistic problems related for instance to population models and, most importantly, the study of branching processes and superdiffusions in the theory of probability.

The aim of this book is to present a comprehensive study of boundary value problems for linear and semi-linear second order elliptic equations with measure data. We are particularly interested in semi-linear equations with absorption. The interactions between the diffusion operator and the absorption term give rise to a large class of nonlinear phenomena in the study of which singularities and boundary trace play a central role. This book is accessible to graduate students and researchers with a background in real analysis and partial differential equations.

Product Details

ISBN-13:	9783110305159
Publisher:	De Gruyter
Publication date:	11/15/2013
Series:	De Gruyter Series in Nonlinear Analysis and Applications , #21
Pages:	261
Product dimensions:	6.69(w) x 9.45(h) x 0.03(d)
Age Range:	18 Years

About the Author

Moshe Marcus, Technion, Haifa, Israel; Laurent Véron, Universityé François Rabelais, Tours, France.

Table of Contents

Preface v

1 Linear second order elliptic equations with measure data 1

1.1 Linear boundary value problems with L¹ data 1

1.2 Measure data 3

1.3 M-boundary trace 13

1.4 The Herglotz-Doob theorem 18

1.5 Subsolutions, supersolutions and Kato's inequality 20

1.6 Boundary Harnack principle 28

1.7 Notes 32

2 Nonlinear second order elliptic equations with measure data 33

2.1 Semilinear problems with L¹ data 33

2.2 Semilinear problems with bounded measure data 36

2.3 Subcritical nonlinearities 43

2.3.1 Weak L^p spaces 44

2.3.2 Continuity of G and P relative to L^p_w norm 47

2.3.3 Continuity of a superposition operator 48

2.3.4 Weak continuity of S^g_ω 52

2.3.5 Weak continuity of S^g_ω 56

2.4 The structure of ^g 59

2.5 Remarks on unbounded domains 63

2.6 Notes 64

3 The boundary trace and associated boundary value problems 66

3.1 The boundary trace 66

3.1.1 Moderate solutions 66

3.1.2 Positive solutions 70

3.1.3 Unbounded domains 78

3.2 Maximal solutions 78

3.3 The boundary value problem with rough trace 81

3.4 A problem with fading absorption 87

3.4.1 The similarity transformation and an extension of the Keller-Osserman estimate 88

3.4.2 Barriers and maximal solutions 89

3.4.3 The critical exponent 94

3.4.4 The very singular solution 96

3.5 Notes 107

4 Isolated singularities 108

4.1 Universal upper bounds 108

4.1.1 The Keller-Osserman estimates 108

4.1.2 Applications to model cases 113

4.2 Isolated singularities 114

4.2.1 Removable singularities 114

4.2.2 Isolated positive singularities 116

4.2.3 Isolated signed singularities 124

4.3 Boundary singularities 130

4.3.1 Upper bounds 130

4.3.2 The half-space case 131

4.3.3 The case of a general domain 138

4.4 Boundary singularities with fading absorption 147

4.4.1 Power-type degeneracy 147

4.4.2 A strongly fading absorption 150

4.5 Miscellaneous 156

4.5.1 General results of isotropy 156

4.5.2 Isolated singularities of supersolutions 157

4.6 Notes and comments 159

5 Classical theory' of maximal and large solutions 162

5.1 Maximal solutions 162

5.1.1 Global conditions 162

5.1.2 Local conditions 166

5.2 Large solutions 166

5.2.1 General nonlinearities 166

5.2.2 The power and exponential cases 171

5.3 Uniqueness of large solutions 175

5.3.1 General uniqueness results 175

5.3.2 Applications to power and exponential types of nonlinearities 182

5.4 Equations with a forcing term 184

5.4.1 Maximal and minimal large solutions 184

5.4.2 Uniqueness 188

5.5 Notes and comments 192

6 Further results on singularities and large solutions 195

6.1 Singularities 195

6.1.1 Internal singularities 195

6.1.2 Boundary singularities 205

6.2 Symmetries of large solutions 217

6.3 Sharp blow up rate of large solutions 226

6.3.1 Estimates in an annulus 227

6.3.2 Curvature secondary effects 231

6.4 Notes and comments 235

Bibliography 239

Index 247

Editorial Reviews

"The book is self-contained and it is highly recommended to researchers and graduate students with a background in Real Analysis and Partial Differential Equations." Zentralblatt für Mathematik

From the Publisher