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, 201 Seiten

Reihe: ISSN

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.