Overview

Full Product Details

Author:   Moshe Marcus ,  Laurent Véron ,  M Marcus
Publisher:   De Gruyter
Imprint:   De Gruyter
Volume:   21
Dimensions:   Width: 17.00cm , Height: 2.00cm , Length: 24.00cm
Weight:   0.580kg
ISBN:  

9783110305159

ISBN 10:   3110305151
Pages:   261
Publication Date:   15 November 2013
Recommended Age:   College Graduate Student
Audience:   Professional and scholarly ,  Professional & Vocational ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Available To Order  
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Table of Contents

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

Reviews

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 f r Mathematik

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 fur Mathematik

Author Information

Moshe Marcus, Technion, Haifa, Israel; Laurent Véron, Université François Rabelais, Tours, France.

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