Elliptic Partial Differential Equations (eBook)

Existence and Regularity of Distributional Solutions

Lucio Boccardo, Gisella Croce (Autoren)

eBook Download: PDF

2013
201 Seiten
De Gruyter (Verlag)
978-3-11-031542-4 (ISBN)

Infos

Elliptic partial differential equations is one of the main and most active areas in mathematics. In our book we study linear and nonlinear elliptic problems in divergence form, with the aim of providing classical results, as well as more recent developments about distributional solutions. For this reason the book is addressed to master's students, PhD students and anyone who wants to begin research in this mathematical field.

Lucio Boccardo, University of Rome, Italy; Gisella Croce, University of Le Havre, France.

Autor

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en">

Lucio Boccardo, University of Rome, Italy; Gisella Croce, University of Le Havre, France.

Inhalt

Notations 10
1 Introduction 11
Part I 13
2 Some fixed point theorems 15
2.1 Introduction 15
2.2 Banach–Caccioppoli theorem 15
2.3 Brouwer’s theorem 16
2.4 Schauder’s theorem 20
3 Preliminaries of real analysis 23
3.1 Introduction 23
3.2 Nemitski’s composition theorem 23
3.3 Marcinkiewicz spaces 28
3.4 Appendix 32
4 Linear and semilinear elliptic equations 34
4.1 Introduction 34
4.2 The Lax–Milgramand Stampacchia’s theorems 34
4.3 Linear equations 37
4.4 Some semilinear monotone equations 38
4.5 Sub and supersolutions method 41
4.6 Appendix 44
5 Nonlinear elliptic equations 47
5.1 Introduction 47
5.2 Surjectivity theorem 47
5.3 The Leray–Lions existence theorem 50
6 Summability of the solutions 57
6.1 Introduction 57
6.2 Preliminaries 58
6.3 Sources in Lebesgue spaces 61
6.4 Sources in Marcinkiewicz spaces 66
6.5 Sources in divergence form 68
7 H2 regularity for linear problems 71
7.1 Introduction 71
7.2 Preliminaries 71
7.3 H2(O) regularity of the solutions 73
8 Spectral analysis for linear operators 77
8.1 Introduction 77
8.2 Eigenvalues of linear elliptic operators 77
8.3 Applications to some semilinear equations 84
8.4 Appendix 92
9 Calculus of variations and Euler’s equation 94
9.1 Introduction 94
9.2 Direct methods in the calculus of variations 94
9.3 Euler equation 96
9.4 Summability of minimizers of integral functionals 100
9.5 The Ekeland variational principle 105
9.6 Appendix 112
Part II 113
10 Natural growth problems 115
10.1 Introduction 115
10.2 A problem with bounded solutions 116
10.3 A problem with unbounded solutions 122
11 Problems with low summable sources 131
11.1 Introduction 131
11.2 A priori estimates 133
11.3 Distributional solutions 140
11.4 The linear case: a different proof 140
11.5 Entropy solutions 142
11.6 A comparison between entropy solutions and distributional solutions 147
11.7 Measure sources 148
11.8 The regularizing effects of a lower order term 150
11.9 T-minima 153
11.10 Appendix 158
12 Uniqueness 160
12.1 Introduction 160
12.2 Monotone elliptic operators 160
12.3 A nonmonotone elliptic operator 162
12.4 A uniqueness result for measure sources 164
13 A problem with polynomial growth 167
13.1 Introduction 167
13.2 Existence results 168
14 A problem with degenerate coercivity 176
14.1 Introduction 176
14.2 The case 0 < . <
14.3 The case . > 1: existence and nonexistence
14.4 The regularizing effects of a lower order term 185
Bibliography 197
Index 191 201

Rezensionen

"[...] This self-contained book is very well written and organized. I highly recommend it to graduate students who study elliptic partial differential equations and beginning researchers in this area. It can also serve as a reference book for a broader community of mathematicians, physicists, and engineers." Mathematical Reviews

Produkt-Details

• Erscheint lt. Verlag: 29.10.2013
• Reihe/Serie: De Gruyter Studies in Mathematics
• Reihe/Serie: De Gruyter Studies in Mathematics
• Reihe/Serie: ISSN
• Reihe/Serie: ISSN
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Dirichlet Problems for Elliptic Equations • Distributional Solution • Elliptische Probleme • Entropy Solutions • Existenz • Leray-Lions Problem • Nonlinear Elliptic Partial Differential Equations in Divergence Form • Ordnung • Partielle Differentialgleichungen • Sobolev Space
• ISBN-10: 3-11-031542-4 / 3110315424
• ISBN-13: 978-3-11-031542-4 / 9783110315424