Foreword ........................................................ v
1 Introduction ................................................. 1
1.1 The Dirichlet problem ................................... 1
1.2 Periodic problems ....................................... 6
1.3 Problems in unbounded domains ........................... 9
1.4 Rescaling .............................................. 11
1.5 Eigenvalue problems .................................... 12
1.6 Calculus of variations problems ........................ 15
1.7 Parabolic problems ..................................... 19
1.8 Strongly nonlinear problems ............................ 24
1.9 Higher order operators ................................. 29
1.10 Higher order estimates ................................ 33
1.11 Global convergence .................................... 35
1.12 Correctors for the Dirichlet problem .................. 37
2 The Dirichlet problem in some unbounded domains ............. 47
2.1 The linear case ........................................ 47
2.2 A quasilinear case ..................................... 55
2.3 Pointwise convergence .................................. 64
3 The pure Neumann problem .................................... 71
3.1 Introduction ........................................... 71
3.2 Construction of a solution to the pure Neumann
problem ................................................ 76
3.3 The case of data independent of x1 ..................... 87
3.4 The case with a lower order term ....................... 92
4 Periodic problems .......................................... 101
4.1 A general theory ...................................... 101
4.2 Some degenerate case .................................. 110
4.3 Application to the periodic obstacle problem .......... 113
5 Anisotropic singular perturbation problems ................. 119
5.1 Introduction .......................................... 119
5.2 Anisotropic singular perturbation problems ............ 122
5.3 Estimates for the rate of convergence ................. 130
6 Eigenvalue problems ........................................ 133
6.1 Introduction .......................................... 133
6.2 Convergence of the eigenvalues ........................ 134
6.3 Convergence of the eigenfunctions ..................... 144
6.4 An application ........................................ 150
6.5 The case of Neumann boundary conditions ............... 155
7 Elliptic systems ........................................... 165
7.1 Abstract formulation .................................. 165
7.2 Some applications ..................................... 169
8 The Stokes problem ......................................... 177
8.1 Introduction and notation ............................. 177
8.2 Auxiliary lemmas ...................................... 179
8.3 Main results .......................................... 181
9 Variational inequalities ................................... 191
9.1 A simple result ....................................... 191
9.2 The obstacle problem in unbounded domains ............. 194
9.3 A general framework ................................... 205
9.4 Some applications ..................................... 211
10 Calculus of variations ..................................... 213
10.1 Monotonicity properties .............................. 213
10.2 A convergence result ................................. 217
10.3 The case of the g-Laplace operator ................... 230
Some concluding remarks ....................................... 237
Appendix A .................................................... 239
A.l Dictionary of the main notation ....................... 239
A.2 Some existence results ................................ 240
A.3 Poincare's Inequality ................................. 243
Bibliography .................................................. 245
Index ......................................................... 251
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