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In the last 40 years semi-linear elliptic equations became a central subject of study in the theory of nonlinear partial differential equations. On the one hand, the interest in this area is of a theoretical nature, due to its deep relations to other branches of mathematics, especially linear and nonlinear harmonic analysis, dynamical systems, differential geometry and probability. On the other hand, this study is of interest because of its applications. Equations of this type come up in various areas such as problems of physics and astrophysics, curvature problems in Riemannian geometry, logistic problems related for instance to population models and, most importantly, the study of branching processes and superdiffusions in the theory of probability.
The aim of this book is to present a comprehensive study of boundary value problems for linear and semi-linear second order elliptic equations with measure data. We are particularly interested in semi-linear equations with absorption. The interactions between the diffusion operator and the absorption term give rise to a large class of nonlinear phenomena in the study of which singularities and boundary trace play a central role. This book is accessible to graduate students and researchers with a background in real analysis and partial differential equations.
Moshe Marcus, Technion, Haifa, Israel; Laurent Véron, Université François Rabelais, Tours, France.
1 Linear second order elliptic equations with measure data 5
1.1 Linear boundary value problems with L1 data. . . . . . . . . . . . . 5
1.2 Measure data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 M-boundary trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 The Herglotz – Doob theorem . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Sub-solutions, super-solutions and Kato’s inequality. . . . . . . . . . 26
1.6 Boundary Harnack principle. . . . . . . . . . . . . . . . . . . . . . . 36
1.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 Nonlinear second order elliptic equations with measure data 43
2.1 Semilinear problems with L1 data . . . . . . . . . . . . . . . . . . . . 43
2.2 Semilinear problems with bounded measure data . . . . . . . . . . . 47
2.3 Subcritical non-linearities . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.1 Weak Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.2 Continuity of G and P relative to Lp
w norm . . . . . . . . . . 59
2.3.3 Continuity of a superposition operator. . . . . . . . . . . . . 61
2.3.4 Weak continuity of Sg
. . . . . . . . . . . . . . . . . . . . . . . 65
2.3.5 Weak continuity of Sg
@
. . . . . . . . . . . . . . . . . . . . . . 69
2.4 The structure of Mg. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.5 Remarks on unbounded domains . . . . . . . . . . . . . . . . . . . . 80
2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3 The boundary trace and associated boundary value problems. 83
3.1 The boundary trace . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.1 Moderate solutions . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.2 Positive solutions . . . . . . . . . . . . . . . . . . . . . . . . . 883.1.3 Unbounded domains . . . . . . . . . . . . . . . . . . . . . . . 98
3.2 Maximal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.3 The boundary value problem with rough trace. . . . . . . . . . . . . 101
3.4 A problem with fading absorption. . . . . . . . . . . . . . . . . . . . 108
3.4.1 The similarity transformation and an extension of the Keller
– Osserman estimate. . . . . . . . . . . . . . . . . . . . . . . 109
3.4.2 Barriers and maximal solutions. . . . . . . . . . . . . . . . . . 111
3.4.3 The critical exponent. . . . . . . . . . . . . . . . . . . . . . . 116
3.4.4 The very singular solution. . . . . . . . . . . . . . . . . . . . 119
3.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4 Isolated singularities 133
4.1 Universal upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.1.1 The Keller-Osserman estimates . . . . . . . . . . . . . . . . . 133
4.1.2 Applications to model cases . . . . . . . . . . . . . . . . . . 138
4.2 Isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.2.1 Removable singularities . . . . . . . . . . . . . . . . . . . . . 140
4.2.2 Isolated positive singularities . . . . . . . . . . . . . . . . . . 142
4.2.3 Isolated signed singularities . . . . . . . . . . . . . . . . . . . 151
4.3 Boundary singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.3.1 Upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.3.2 The half space case . . . . . . . . . . . . . . . . . . . . . . . . 160
4.3.3 The case of a general domain . . . . . . . . . . . . . . . . . . 167
4.4 Boundary singularities with fading absorption . . . . . . . . . . . . . 176
4.4.1 Power-type degeneracy . . . . . . . . . . . . . . . . . . . . . . 176
4.4.2 A strongly fading absorption . . . . . . . . . . . . . . . . . . 180
4.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.5.1 General results of isotropy . . . . . . . . . . . . . . . . . . . . 187
4.5.2 Isolated singularities of super-solutions . . . . . . . . . . . . 188
4.6 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5 Classical theory of maximal and large solutions 195
5.1 Maximal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.1.1 Global conditions . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.1.2 Local conditions . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.2 Large solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.2.1 General nonlinearities . . . . . . . . . . . . . . . . . . . . . . 2015.2.2 The power and exponential cases . . . . . . . . . . . . . . . . 206
5.3 Uniqueness of large solutions . . . . . . . . . . . . . . . . . . . . . . 210
5.3.1 General uniqueness results . . . . . . . . . . . . . . . . . . . . 211
5.3.2 Applications to power and exponential types nonlinearities . 219
5.4 Equations with forcing term . . . . . . . . . . . . . . . . . . . . . . . 221
5.4.1 Maximal and minimal large solutions . . . . . . . . . . . . . . 222
5.4.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
5.5 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6 Further results on singularities and large solutions 233
6.1 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6.1.1 Internal singularities . . . . . . . . . . . . . . . . . . . . . . . 233
6.1.2 Boundary singularities . . . . . . . . . . . . . . . . . . . . . . 244
6.2 Symmetries of large solutions . . . . . . . . . . . . . . . . . . . . . . 259
6.3 Sharp blow-up rate of large solutions . . . . . . . . . . . . . . . . . . 268
6.3.1 Estimates in an annulus . . . . . . . . . . . . . . . . . . . . . 269
6.3.2 Curvature secondary effects . . . . . . . . . . . . . . . . . . . 275
6.4 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 279