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Nonlinear Second Order Elliptic Equations Involving Measures

Moshe Marcus, Laurent Véron (Autoren)

Buch | Hardcover

XIII, 248 Seiten

2013
De Gruyter (Verlag)
978-3-11-030515-9 (ISBN)

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The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Please submit book proposals to Jürgen Appell.

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

"The book is self-contained and it is highly recommended to researchers and graduate students with a background in Real Analysis and Partial Differential Equations." Zentralblatt für Mathematik

Erscheint lt. Verlag	15.11.2013
Reihe/Serie	De Gruyter Series in Nonlinear Analysis and Applications ; 21
Verlagsort	Berlin/Boston
Sprache	englisch
Maße	170 x 240 mm
Gewicht	568 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Schlagworte	Boundary trace • Elliptic equations • elliptic equations, boundary trace, singularities, large solutions, subcritical nonlinearity. • Elliptic equations; Boundary trace; Singularities; Large solutions; Subcritical nonlinearity • Elliptische Gleichungen • Large solutions • Maßtheorie • Nichtlineare Analysis • Singularities • Subcritical nonlinearity
ISBN-10	3-11-030515-1 / 3110305151
ISBN-13	978-3-11-030515-9 / 9783110305159
Zustand	Neuware