Elliptic Partial Differential Equations: Existence and Regularity of Distributional Solutions

✍ Scribed by Lucio Boccardo; Gisella Croce

• Publisher: De Gruyter
• Year: 2013
• Tongue: English
• Leaves: 204
• Series: De Gruyter Studies in Mathematics; 55
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Notations
1 Introduction
Part I
2 Some fixed point theorems
2.1 Introduction
2.2 Banach–Caccioppoli theorem
2.3 Brouwer’s theorem
2.4 Schauder’s theorem
3 Preliminaries of real analysis
3.1 Introduction
3.2 Nemitski’s composition theorem
3.3 Marcinkiewicz spaces
3.4 Appendix
4 Linear and semilinear elliptic equations
4.1 Introduction
4.2 The Lax–Milgramand Stampacchia’s theorems
4.3 Linear equations
4.4 Some semilinear monotone equations
4.5 Sub and supersolutions method
4.6 Appendix
5 Nonlinear elliptic equations
5.1 Introduction
5.2 Surjectivity theorem
5.3 The Leray–Lions existence theorem
6 Summability of the solutions
6.1 Introduction
6.2 Preliminaries
6.3 Sources in Lebesgue spaces
6.4 Sources in Marcinkiewicz spaces
6.5 Sources in divergence form
7 H2 regularity for linear problems
7.1 Introduction
7.2 Preliminaries
7.3 H2(Ω) regularity of the solutions
8 Spectral analysis for linear operators
8.1 Introduction
8.2 Eigenvalues of linear elliptic operators
8.3 Applications to some semilinear equations
8.4 Appendix
9 Calculus of variations and Euler’s equation
9.1 Introduction
9.2 Direct methods in the calculus of variations
9.3 Euler equation
9.4 Summability of minimizers of integral functionals
9.5 The Ekeland variational principle
9.6 Appendix
Part II
10 Natural growth problems
10.1 Introduction
10.2 A problem with bounded solutions
10.3 A problem with unbounded solutions
11 Problems with low summable sources
11.1 Introduction
11.2 A priori estimates
11.3 Distributional solutions
11.4 The linear case: a different proof
11.5 Entropy solutions
11.6 A comparison between entropy solutions and distributional solutions
11.7 Measure sources
11.8 The regularizing effects of a lower order term
11.9 T-minima
11.10 Appendix
12 Uniqueness
12.1 Introduction
12.2 Monotone elliptic operators
12.3 A nonmonotone elliptic operator
12.4 A uniqueness result for measure sources
13 A problem with polynomial growth
13.1 Introduction
13.2 Existence results
14 A problem with degenerate coercivity
14.1 Introduction
14.2 The case 0 < θ < 1
14.3 The case θ > 1: existence and nonexistence
14.4 The regularizing effects of a lower order term
Bibliography
Index 191