4.2. Integral of a uniformly continuous function 87
4.3. Case where E is not a Neumann space 92
4.4. Properties of the integral 93
4.5. Dependence of the integral on the domain of integration 96
4.6. Additivity with respect to the domain of integration 99
4.7. Continuity of the integral 101
4.8. Differentiating under the integral sign 103
Chapter 5. Properties of the Measure of an Open Set 105
5.1. Additivity of the measure 105
5.2. Negligible sets 107
5.3. Determinant of d vectors 112
5.4. Measure of a parallelepiped 115
Chapter 6. Additional Properties of the Integral 119
6.1. Contribution of a negligible set to the integral 119
6.2. Integration and differentiation in one dimension 120
6.3. Integration of a function of functions 123
6.4. Integrating a function of multiple variables 125
6.5. Integration between graphs 130
6.6. Integration by parts and weak vanishing condition for a function 133
6.7. Change of variables in an integral 135
6.8. Some particular changes of variables in an integral 142
Chapter 7. Weighting and Regularization of Functions 147
7.1. Weighting 147
7.2. Properties of weighting 150
7.3. Weighting of differentiable functions 153
7.4. Local regularization 157
7.5. Global regularization 162
7.6. Partition of unity 166
7.7. Separability of K∞(Ω) 170
Chapter 8. Line Integral of a Vector Field Along a Path 173
8.1. Paths 173
8.2. Line integral of a field along a path 176
8.3. Line integral along a concatenation of paths 181
8.4. Tubular flow and the concentration theorem 183
8.5. Invariance under homotopy of the line integral of a local gradient 186
Chapter 9. Primitives of Continuous Functions 191
9.1. Explicit primitive of a field with line integral zero 191
9.2. Primitive of a field orthogonal to the divergence-free test fields 194
9.3. Gluing of local primitives on a simply connected open set 195
9.4. Explicit primitive on a star-shaped set: Poincaré’s theorem 197
9.5. Explicit primitive under the weak Poincaré condition 199
9.6. Primitives on a simply connected open set 203
9.7. Comparison of the existence conditions for a primitive 205
9.8. Fields with local primitives but no global primitive 208
9.9. Uniqueness of primitives 210
9.10. Continuous primitive mapping 211
Chapter 10. Additional Results: Integration on a Sphere 213
10.1. Surface integration on a sphere 213
10.2. Properties of the integral on a sphere 215
10.3. Radial calculation of integrals 218
10.4. Surface integral as an integral of dimension d − 1 220
10.5. A Stokes formula 224
Appendix 227
Bibliography 239
Index 243
This book is the second of a set dedicated to the mathematical tools used in partial differential equations derived from physics. It presents the properties of continuous functions, which are useful for solving partial differential equations, and, more particularly, for constructing distributions valued in a Neumann space. The author examines partial derivatives, the construction of primitives, integration and the weighting of value functions in a Neumann space. Many of them are new generalizations of classical properties for values in a Banach space. Simple methods, semi-norms, sequential properties and others are discussed, making these tools accessible to the greatest number of students – doctoral students, postgraduate students – engineers and researchers, without restricting or generalizing the results.
Jacques Simon is Honorary Research Director at CNRS. His research focuses on Navier-Stokes equations, particularly in shape optimization and in the functional spaces they use.