Distributions

Jacques Simon is Emeritus Research Director at CNRS, France. His research focuses on the Navier–Stokes equations, particularly in shape optimization and in the functional spaces they use.

• Introduction ixNotations xvChapter 1 Semi-Normed Spaces and Function Spaces 11.1. Semi-normed spaces 11.2. Comparison of semi-normed spaces 41.3. Continuous mappings 61.4. Differentiable functions 81.5. Spaces Cm (Ω; E), Cmb (Ω; E) and Cmb (Ω; E) 11              1.6. Integral of a uniformly continuous function 14Chapter 2 Space of Test Functions 172.1. Functions with compact support 172.2. Compactness in their whole of support of functions 192.3. The space D(Ω) 212.4. Sequential completeness of D(Ω) 242.5. Comparison of D(Ω) to various spaces 262.6. Convergent sequences in D(Ω) 282.7. Covering by crown-shaped sets and partitions of unity 332.8. Control of the CK m (Ω)-norms by the semi-norms of D(Ω) 352.9. Semi-norms that are continuous on all the CK ∞ (Ω) 38Chapter 3 Space of Distributions 413.1. The space D ′ (Ω; E) 413.2. Characterization of distributions 463.3. Inclusion of C(Ω; E) into D ′ (Ω; E) 483.4. The case where E is not a Neumann space 533.5. Measures 573.6. Continuous functions and measures 63Chapter 4 Extraction of Convergent Subsequences 654.1. Bounded subsets of D ′ (Ω; E) 654.2. Convergence in D ′ (Ω; E) 674.3. Sequential completeness of D ′ (Ω; E) 694.4. Sequential compactness in D ′ (Ω; E) 714.5. Change of the space E of values 744.6. The space E-weak 764.7. The space D ′ (Ω; E-weak) and extractability 78Chapter 5 Operations on Distributions 815.1. Distributions fields 815.2. Derivatives of a distribution 845.3. Image under a linear mapping 915.4. Product with a regular function 945.5. Change of variables 1005.6. Some particular changes of variables 1075.7. Positive distributions 1095.8. Distributions with values in a product space 113Chapter 6 Restriction, Gluing and Support 1176.1. Restriction 1176.2. Additivity with respect to the domain 1216.3. Local character 1226.4. Localization-extension 1256.5. Gluing 1286.6. Annihilation domain and support 1306.7. Properties of the annihilation domain and support 1336.8. The space DK ′ (Ω; E) 137Chapter 7 Weighting 1417.1. Weighting by a regular function 1417.2. Regularizing character of the weighting by a regular function 1447.3. Derivatives and support of distributions weighted by a regular weight 1487.4. Continuity of the weighting by a regular function 1507.5. Weighting by a distribution 1537.6. Comparison of the definitions of weighting 1567.7. Continuity of the weighting by a distribution 1597.8. Derivatives and support of a weighted distribution 1617.9. Miscellanous properties of weighting 165Chapter 8 Regularization and Applications 1698.1. Local regularization 1698.2. Properties of local approximations 1748.3. Global regularization 1758.4. Convergence of global approximations 1788.5. Properties of global approximations 1808.6. Commutativity and associativity of weighting 1838.7. Uniform convergence of sequences of distributions 188Chapter 9 Potentials and Singular Functions 1919.1. Surface integral over a sphere 1919.2. Distribution associated with a singular function 1939.3. Derivatives of a distribution associated with a singular function 1969.4. Elementary Newtonian potential 1979.5. Newtonian potential of order n 2019.6. Localized potential 2089.7. Dirac mass as derivatives of continuous functions 2109.8. Heaviside potential 2149.9. Weighting by a singular weight 217Chapter 10 Line Integral of a Continuous Field 22110.1. Line integral along a C1 path 22110.2. Change of variable in a path 22510.3. Line integral along a piecewise C1 path 22810.4. The homotopy invariance theorem 23110.5. Connectedness and simply connectedness 235Chapter 11 Primitives of Functions 23711.1. Primitive of a function field with a zero line integral 23711.2. Tubular flows and concentration theorem 23911.3. The orthogonality theorem for functions 24311.4. Poincaré’s theorem 244Chapter 12 Properties of Primitives of Distributions 24712.1. Representation by derivatives 24712.2. Distribution whose derivatives are zero or continuous 25112.3. Uniqueness of a primitive 25312.4. Locally explicit primitive 25412.5. Continuous primitive mapping 25612.6. Harmonic distributions, distributions with a continuous Laplacian 261Chapter 13 Existence of Primitives 26513.1. Peripheral gluing 26613.2. Reduction to the function case 26813.3. The orthogonality theorem 27013.4. Poincaré’s generalized theorem 27413.5. Current of an incompressible two dimensional field 27713.6. Global versus local primitives 27913.7. Comparison of the existence conditions of a primitive 28213.8. Limits of gradients 283Chapter 14 Distributions of Distributions 28514.1. Characterization 28514.2. Bounded sets 28814.3. Convergent sequences 28914.4. Extraction of convergent subsequences 29314.5. Change of the space of values 29414.6. Distributions of distributions with values in E-weak 295Chapter 15 Separation of Variables 29715.1. Tensor products of test functions 29715.2. Decomposition of test functions on a product of sets 30115.3. The tensorial control theorem 30315.4. Separation of variables 30915.5. The kernel theorem 31115.6. Regrouping of variables 31715.7. Permutation of variables 318Chapter 16 Banach Space Valued Distributions 32316.1. Finite order distributions 32316.2. Weighting of a finite order distribution 32616.3. Finite order distribution as derivatives of continuous functions 32816.4. Finite order distribution as derivative of a single function 33316.5. Distributions in a Banach space as derivatives of functions 33516.6. Non-representability of distributions with values in a Fréchet space 33916.7. Extendability of distributions with values in a Banach space 34216.8. Cancellation of distributions with values in a Banach space 347Appendix 349Bibliography 367Index 371