Elliptic Partial Differential Equations

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