### Degree Theory in Analysis and Applications

Oxford University Press, 1995

**Back Cover**

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.

### Table of Contents

**Introduction****Degree theory for continuous functions**

1.1 Topological degree for C^{1} functions

1.2 Topological degree for continuous functions

1.3 Generalization of the degree

1.4 Exercises

**Degree theory in finite-dimensional spaces**

2.1 Dependence of the degree on φ and *p*

2.2 Dependence of the degree on the domain *D*

2.3 The multiplication theorem

2.4 An application of Hopf's theorem

2.5 Degree and winding number

2.6 Exercises

**Some applications of the degree theory to topology**

3.1 The Brouwer Fixed Point Theorem

3.2 Odd mappings

3.3 The Jordon Separation Theorem

3.4 Exercises

**Measure Theorem and Sobolov spaces**

4.1 Review of Measure theory

4.2 Hausdorff measures

4.3 Overview of Sobolov spaces

4.4 *p*-capacity

**Properties of the degree for Sobolev functions**

5.1 Results of weakly differential mappings

5.2 Weakly monotone functions

5.3 Change of variables via the multiplicity function

5.4 Change of variables via the degree

5.5 Change of variables for Sobolev functions

**Local invertibility of Sobolev functions and applications**

6.1 Local invertibility in *W*^{1,N}

6.2 Energy functionals involving variation of the domain

**Degree in infinite-dimensional spaces**

7.1 Introduction to the Leray-Schauder degree

7.2 Properties of the Leray-Schauder degree

7.3 Fixed point theorems

7.4 An application of the degree theory to ODEs

7.5 First application of the degree theory to ODEs

7.6 Second application of the degree theory to ODEs

7.7 Exercises

**References****Index**