Boccardo / Croce Elliptic Partial Differential Equations

Existence and Regularity of Distributional Solutions
1. Auflage 2013
ISBN: 978-3-11-031542-4
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 55, 201 Seiten

Reihe: De Gruyter Studies in MathematicsISSN

Elliptic partial differential equations is one of the main and most active areas in mathematics. This book is devoted to the study of 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 this monograph is addressed to master's students, PhD students and anyone who wants to begin research in this mathematical field.

Zielgruppe

PhD students, young researchers in Mathematics, academic libraries

Autoren/Hrsg.

Boccardo, Lucio

Croce, Gisella

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

1;Notations;10
2;1 Introduction;11
3;Part I;13
3.1;2 Some fixed point theorems;15
3.1.1;2.1 Introduction;15
3.1.2;2.2 Banach–Caccioppoli theorem;15
3.1.3;2.3 Brouwer’s theorem;16
3.1.4;2.4 Schauder’s theorem;20
3.2;3 Preliminaries of real analysis;23
3.2.1;3.1 Introduction;23
3.2.2;3.2 Nemitski’s composition theorem;23
3.2.3;3.3 Marcinkiewicz spaces;28
3.2.4;3.4 Appendix;32
3.3;4 Linear and semilinear elliptic equations;34
3.3.1;4.1 Introduction;34
3.3.2;4.2 The Lax–Milgramand Stampacchia’s theorems;34
3.3.3;4.3 Linear equations;37
3.3.4;4.4 Some semilinear monotone equations;38
3.3.5;4.5 Sub and supersolutions method;41
3.3.6;4.6 Appendix;44
3.4;5 Nonlinear elliptic equations;47
3.4.1;5.1 Introduction;47
3.4.2;5.2 Surjectivity theorem;47
3.4.3;5.3 The Leray–Lions existence theorem;50
3.5;6 Summability of the solutions;57
3.5.1;6.1 Introduction;57
3.5.2;6.2 Preliminaries;58
3.5.3;6.3 Sources in Lebesgue spaces;61
3.5.4;6.4 Sources in Marcinkiewicz spaces;66
3.5.5;6.5 Sources in divergence form;68
3.6;7 H2 regularity for linear problems;71
3.6.1;7.1 Introduction;71
3.6.2;7.2 Preliminaries;71
3.6.3;7.3 H2(O) regularity of the solutions;73
3.7;8 Spectral analysis for linear operators;77
3.7.1;8.1 Introduction;77
3.7.2;8.2 Eigenvalues of linear elliptic operators;77
3.7.3;8.3 Applications to some semilinear equations;84
3.7.4;8.4 Appendix;92
3.8;9 Calculus of variations and Euler’s equation;94
3.8.1;9.1 Introduction;94
3.8.2;9.2 Direct methods in the calculus of variations;94
3.8.3;9.3 Euler equation;96
3.8.4;9.4 Summability of minimizers of integral functionals;100
3.8.5;9.5 The Ekeland variational principle;105
3.8.6;9.6 Appendix;112
4;Part II;113
4.1;10 Natural growth problems;115
4.1.1;10.1 Introduction;115
4.1.2;10.2 A problem with bounded solutions;116
4.1.3;10.3 A problem with unbounded solutions;122
4.2;11 Problems with low summable sources;131
4.2.1;11.1 Introduction;131
4.2.2;11.2 A priori estimates;133
4.2.3;11.3 Distributional solutions;140
4.2.4;11.4 The linear case: a different proof;140
4.2.5;11.5 Entropy solutions;142
4.2.6;11.6 A comparison between entropy solutions and distributional solutions;147
4.2.7;11.7 Measure sources;148
4.2.8;11.8 The regularizing effects of a lower order term;150
4.2.9;11.9 T-minima;153
4.2.10;11.10 Appendix;158
4.3;12 Uniqueness;160
4.3.1;12.1 Introduction;160
4.3.2;12.2 Monotone elliptic operators;160
4.3.3;12.3 A nonmonotone elliptic operator;162
4.3.4;12.4 A uniqueness result for measure sources;164
4.4;13 A problem with polynomial growth;167
4.4.1;13.1 Introduction;167
4.4.2;13.2 Existence results;168
4.5;14 A problem with degenerate coercivity;176
4.5.1;14.1 Introduction;176
4.5.2;14.2 The case 0 < . < 1;177
4.5.3;14.3 The case . > 1: existence and nonexistence;182
4.5.4;14.4 The regularizing effects of a lower order term;185
5;Bibliography;197
6;Index 191;201

Über Autor(innen)

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