Degree Theory in Analysis and Applications (Oxford Lecture Series in Mathematics and Its Applications, 2)

In recent years the need to extend the notion of degree to nonsmooth functions has been triggered by developments in nonlinear analysis and some of its applications. This new study relates several approaches to degree theory for continuous functions and incorporates newly obtained results for Sobolev functions. These results are put to use in the study of variational principles in nonlinear elasticity. Several applications of the degree are illustrated in the theories of ordinary and partial differential equations. Other topics include multiplication theorem, Hopf's theorem, Brower's fixed point theorem, odd mappings, and Jordan's separation theorem, all suitable for graduate courses in degree theory and application. Cover ......Page 1 Back ......Page 2 Title ......Page 5 Preface ......Page 7 Contents ......Page 9 Introduction ......Page 11 1.1 Topological degree for C1 functions ......Page 15 1.2 Topological degree for continuous functions ......Page 26 1.3 Generalization of the degree ......Page 30 1.4 Exercises ......Page 35 2.1 Dependence of the degree on ? and ? ......Page 40 2.2 Dependence of the degree on the domain D ......Page 42 2.3 The multiplication theorem ......Page 45 2.4 An application of Hopf's theorem ......Page 49 2.5 Degree and winding number ......Page 51 2.6 Exercises ......Page 56 3.1 The Brouwer Fixed Point Theorem ......Page 58 3.2 Odd mappings ......Page 64 3.3 The Jordan Separation Theorem ......Page 74 3.4 Exercises ......Page 81 4.1 Review of measure theory ......Page 84 4.2 Hausdorff measures ......Page 88 4.3 Overview of Sobolev spaces ......Page 97 4.4 p-capacity ......Page 102 5 Properties of the degree for Sobolev functions ......Page 116 5.1 Results of weakly differential mappings ......Page 117 5.2 Weakly monotone functions ......Page 129 5.3 Change of variables via the multiplicity function ......Page 141 5.4 Change of variables via the degree ......Page 145 5.5 Change of variables for Sobolev functions ......Page 150 6.1 Local invertibility in W^l,N ......Page 159 6.2 Energy functional involving variation of the domain ......Page 170 7.1 Introduction to the Leray-Schauder degree ......Page 182 7.2 Properties of the Leray-Schauder degree ......Page 187 7.3 Fixed point theorems ......Page 195 7.4 An application of the degree theory to ODEs ......Page 200 7.5 First application of the degree theory to PDEs ......Page 202 7.6 Second application of the degree theory to PDEs ......Page 209 7.7 Exercises ......Page 213 References ......Page 215 Index ......Page 219 This text examines degree theory and some of its applications in analysis. Topics described include: degree theory for continuous functions; the multiplication theorem; Hopf's theorem; Brower's fixed point theorem; odd mappings; and Jordan's separation theorem.