Preface ......................................................... v
1 Linear second order elliptic equations with measure data ..... 1
1.1 Linear boundary value problems with L1 data ............. 1
1.2 Measure data ............................................ 3
1.3 M-boundary trace ....................................... 13
1.4 The Herglotz-Doob theorem .............................. 18
1.5 Subsolutions, supersolutions and Kato's inequality ..... 20
1.6 Boundary Harnack principle ............................. 28
1.7 Notes .................................................. 32
2 Nonlinear second order elliptic equations with measure
data ........................................................ 33
2.1 Semilinear problems with L1 data ....................... 33
2.2 Semilinear problems with bounded measure data .......... 36
2.3 Subcritical nonlinearities ............................. 43
2.3.1 Weak Lp spaces .................................. 44
2.3.2 Continuity of  and  relative to Lωp norm ..... 47
2.3.3 Continuity of a superposition operator .......... 48
2.3.4 Weak continuity of SΩg .......................... 52
2.3.5 Weak continuity of S∂Ωg ......................... 56
2.4 The structure of  g .................................... 59
2.5 Remarks on unbounded domains ........................... 63
2.6 Notes .................................................. 64
3 The boundary trace and associated boundary value problems ... 66
3.1 The boundary trace ..................................... 66
3.1.1 Moderate solutions .............................. 66
3.1.2 Positive solutions .............................. 70
3.1.3 Unbounded domains ............................... 78
3.2 Maximal solutions ...................................... 78
3.3 The boundary value problem with rough trace ............ 81
3.4 A problem with fading absorption ....................... 87
3.4.1 The similarity transformation and an extension
of the Keller-Osserman estimate ................. 88
3.4.2 Barriers and maximal solutions .................. 89
3.4.3 The critical exponent ........................... 94
3.4.4 The very singular solution ...................... 96
3.5 Notes ................................................. 107
4 Isolated singularities ..................................... 108
4.1 Universal upper bounds ................................ 108
4.1.1 The Keller-Osserman estimates .................. 108
4.1.2 Applications to model cases .................... 113
4.2 Isolated singularities ................................ 114
4.2.1 Removable singularities ........................ 114
4.2.2 Isolated positive singularities ................ 116
4.2.3 Isolated signed singularities .................. 124
4.3 Boundary singularities ................................ 130
4.3.1 Upper bounds ................................... 130
4.3.2 The half-space case ............................ 131
4.3.3 The case of a general domain ................... 138
4.4 Boundary singularities with fading absorption ......... 147
4.4.1 Power-type degeneracy .......................... 147
4.4.2 A strongly fading absorption ................... 150
4.5 Miscellaneous ......................................... 156
4.5.1 General results of isotropy .................... 156
4.5.2 Isolated singularities of supersolutions ....... 157
4.6 Notes and comments .................................... 159
5 Classical theory of maximal and large solutions ............ 162
5.1 Maximal solutions ..................................... 162
5.1.1 Global conditions .............................. 162
5.1.2 Local conditions ............................... 166
5.2 Large solutions ....................................... 166
5.2.1 General nonlinearities ......................... 166
5.2.2 The power and exponential cases ................ 171
5.3 Uniqueness of large solutions ......................... 175
5.3.1 General uniqueness results ..................... 175
5.3.2 Applications to power and exponential types
of nonlinearities .............................. 182
5.4 Equations with a forcing term ......................... 184
5.4.1 Maximal and minimal large solutions ............ 184
5.4.2 Uniqueness ..................................... 188
5.5 Notes and comments .................................... 192
6 Further results on singularities and large solutions ...... 195
6.1 Singularities ......................................... 195
6.1.1 Internal singularities ......................... 195
6.1.2 Boundary singularities ......................... 205
6.2 Symmetries of large solutions ......................... 217
6.3 Sharp blow up rate of large solutions ................. 226
6.3.1 Estimates in an annulus ........................ 227
6.3.2 Curvature secondary effects .................... 231
6.4 Notes and comments .................................... 235
Bibliography .................................................. 239
Index ......................................................... 247
|
