Preface ....................................................... vii
1. Introduction ................................................ 1
1.1. Singularity locus as parameter ........................ 1
1.2. The main steps of reduction ........................... 2
1.3. A few definitions ..................................... 4
1.4. An algorithm in eight steps ........................... 4
1.5. Simple examples of reduced Fuchsian equations ......... 5
1.6. Reduction and applications ........................... 13
PART I. FUCHSIAN REDUCTION
2. Formal Series .............................................. 23
2.1. The Operator D and its first properties .............. 24
2.2. The space Al and its variants ........................ 27
2.3. Formal series with variable exponents ................ 35
2.4. Relation of Ag to the invariant theory of
binary forms ......................................... 39
Problems ................................................... 42
3. General Reduction Methods .................................. 45
3.1. Reduction of a single equation ....................... 45
3.2. Introduction of several time variables and second
reduction ............................................ 51
3.3. Semilinear systems ................................... 52
3.4. Structure of the formal series with several
time variables ....................................... 54
3.5. Resonances, instability, and group invariance ........ 58
3.6. Stability and parameter dependence ................... 64
Problems ................................................... 65
PART II. THEORY OF FUCHSIAN PARTIAL DIFFERENTIAL EQUATIONS
4. Convergent Series Solutions of Fuchsian Initial-Value
Problems ................................................... 69
4.1. Theory of linear Fuchsian ODEs ....................... 69
4.2. Initial-value problem for Fuchsian PDEs with
analytic data ........................................ 71
4.3. Generalized Fuchsian systems ......................... 75
4.4. Notes ................................................ 82
Problems ................................................... 83
5. Fuchsian Initial-Value Problems in Sobolev Spaces .......... 85
5.1. Singular systems of ODEs in weighted spaces .......... 86
5.2. A generalized Fuchsian ODE ........................... 89
5.3. Fuchsian PDEs: abstract results ...................... 90
5.4. Optimal regularity for Fuchsian PDEs ................. 97
5.5. Reduction to a symmetric system ..................... 101
Problems .................................................. 104
6. Solution of Fuchsian Elliptic Boundary-Value Problems ..... 105
6.1. Basic LP results for equations with degenerate
characteristic form ................................. 106
6.2. Schauder regularity for Fuchsian problems ........... 109
6.3. Solution of a model Fuchsian operator ............... 113
Problems .................................................. 118
PART III. APPLICATIONS
7. Applications in Astronomy ................................. 121
7.1. Notions on stellar modeling ......................... 121
7.2. Polytropic model .................................... 123
7.3. Point-source model .................................. 124
Problems .................................................. 126
8. Applications in General Relativity ........................ 129
8.1. The big-bang singularity and AVD behavior ........... 129
8.2. Gowdy space-times ................................... 131
8.3. Space-times with twist .............................. 137
Problems .................................................. 142
9. Applications in Differential Geometry ..................... 143
9.1. Fefferman-Graham metrics ............................ 143
9.2. First Fuchsian reduction and construction
of formal solutions ................................. 147
9.3. Second Fuchsian reduction and convergence
of formal solutions ................................. 149
9.4. Propagation of constraint equations ................. 151
9.5. Special cases ....................................... 153
9.6. Conformal changes of metric ......................... 154
9.7. Loewner-Nirenberg metrics ........................... 156
Problems .................................................. 161
10. Applications to Nonlinear Waves ........................... 163
10.1. From blowup time to blowup pattern .................. 163
10.2. Semilinear wave equations ........................... 167
10.3. Nonlinear optics and lasers ......................... 181
10.4. Weak detonations .................................... 198
10.5. Soliton theory ...................................... 202
10.6. The Liouville equation .............................. 209
10.7. Nirenberg's example ................................. 213
Problems .................................................. 214
11. Boundary Blowup for Nonlinear Elliptic Equations .......... 217
11.1. A renormalized energy for boundary blowup ........... 218
11.2. Hardy-Trudinger inequalities ........................ 219
11.3. Variational characterization of solutions
with boundary blowup ................................ 223
11.4. Construction of the partition of unity .............. 225
Problems .................................................. 227
PART IV. BACKGROUND RESULTS
12. Distance Function and Holder Spaces ....................... 231
12.1. The distance function ............................... 231
12.2. Holder spaces on C2+a domains ....................... 233
12.3. Interior estimates for the Laplacian ................ 239
12.4. Perturbation of coefficients ........................ 243
13. Nash—Moser Inverse Function Theorem ....................... 247
13.1. Nash-Moser theorem without smoothing ................ 247
13.2. Nash-Moser theorem with smoothing ................... 249
Solutions ..................................................... 253
References .................................................... 277
Index ......................................................... 287
|
