Details
Distributions
1. Aufl.
126,99 € |
|
Verlag: | Wiley |
Format: | EPUB |
Veröffentl.: | 19.08.2022 |
ISBN/EAN: | 9781394165353 |
Sprache: | englisch |
Anzahl Seiten: | 416 |
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Beschreibungen
This book presents a simple and original theory of distributions, both real and vector, adapted to the study of partial differential equations. It deals with value distributions in a Neumann space, that is, in which any Cauchy suite converges, which encompasses the Banach and Fréchet spaces and the same “weak” spaces. Alongside the usual operations – derivation, product, variable change, variable separation, restriction, extension and regularization – Distributions presents a new operation: weighting.<br /><br />This operation produces properties similar to those of convolution for distributions defined in any open space. Emphasis is placed on the extraction of convergent sub-sequences, the existence and study of primitives and the representation by gradient or by derivatives of continuous functions. Constructive methods are used to make these tools accessible to students and engineers.
<p>Introduction ix</p> <p>Notations xv</p> <p><b>Chapter 1 Semi-Normed Spaces and Function Spaces 1</b></p> <p>1.1. Semi-normed spaces 1</p> <p>1.2. Comparison of semi-normed spaces 4</p> <p>1.3. Continuous mappings 6</p> <p>1.4. Differentiable functions 8</p> <p>1.5. Spaces C<sup>m</sup> (Ω; E), C<sup>m</sup><sub>b</sub> (Ω; E) and C<sup>m</sup><sub>b</sub> (Ω; E) 11 </p> <p>1.6. Integral of a uniformly continuous function 14</p> <p><b>Chapter 2 Space of Test Functions 17</b></p> <p>2.1. Functions with compact support 17</p> <p>2.2. Compactness in their whole of support of functions 19</p> <p>2.3. The space D(Ω) 21</p> <p>2.4. Sequential completeness of D(Ω) 24</p> <p>2.5. Comparison of D(Ω) to various spaces 26</p> <p>2.6. Convergent sequences in D(Ω) 28</p> <p>2.7. Covering by crown-shaped sets and partitions of unity 33</p> <p>2.8. Control of the CK m (Ω)-norms by the semi-norms of D(Ω) 35</p> <p>2.9. Semi-norms that are continuous on all the CK ∞ (Ω) 38</p> <p><b>Chapter 3 Space of Distributions 41</b></p> <p>3.1. The space D ′ (Ω; E) 41</p> <p>3.2. Characterization of distributions 46</p> <p>3.3. Inclusion of C(Ω; E) into D ′ (Ω; E) 48</p> <p>3.4. The case where E is not a Neumann space 53</p> <p>3.5. Measures 57</p> <p>3.6. Continuous functions and measures 63</p> <p><b>Chapter 4 Extraction of Convergent Subsequences 65</b></p> <p>4.1. Bounded subsets of D ′ (Ω; E) 65</p> <p>4.2. Convergence in D ′ (Ω; E) 67</p> <p>4.3. Sequential completeness of D ′ (Ω; E) 69</p> <p>4.4. Sequential compactness in D ′ (Ω; E) 71</p> <p>4.5. Change of the space E of values 74</p> <p>4.6. The space E-weak 76</p> <p>4.7. The space D ′ (Ω; E-weak) and extractability 78</p> <p><b>Chapter 5 Operations on Distributions 81</b></p> <p>5.1. Distributions fields 81</p> <p>5.2. Derivatives of a distribution 84</p> <p>5.3. Image under a linear mapping 91</p> <p>5.4. Product with a regular function 94</p> <p>5.5. Change of variables 100</p> <p>5.6. Some particular changes of variables 107</p> <p>5.7. Positive distributions 109</p> <p>5.8. Distributions with values in a product space 113</p> <p><b>Chapter 6 Restriction, Gluing and Support 117</b></p> <p>6.1. Restriction 117</p> <p>6.2. Additivity with respect to the domain 121</p> <p>6.3. Local character 122</p> <p>6.4. Localization-extension 125</p> <p>6.5. Gluing 128</p> <p>6.6. Annihilation domain and support 130</p> <p>6.7. Properties of the annihilation domain and support 133</p> <p>6.8. The space DK ′ (Ω; E) 137</p> <p><b>Chapter 7 Weighting 141</b></p> <p>7.1. Weighting by a regular function 141</p> <p>7.2. Regularizing character of the weighting by a regular function 144</p> <p>7.3. Derivatives and support of distributions weighted by a regular weight 148</p> <p>7.4. Continuity of the weighting by a regular function 150</p> <p>7.5. Weighting by a distribution 153</p> <p>7.6. Comparison of the definitions of weighting 156</p> <p>7.7. Continuity of the weighting by a distribution 159</p> <p>7.8. Derivatives and support of a weighted distribution 161</p> <p>7.9. Miscellanous properties of weighting 165</p> <p><b>Chapter 8 Regularization and Applications 169</b></p> <p>8.1. Local regularization 169</p> <p>8.2. Properties of local approximations 174</p> <p>8.3. Global regularization 175</p> <p>8.4. Convergence of global approximations 178</p> <p>8.5. Properties of global approximations 180</p> <p>8.6. Commutativity and associativity of weighting 183</p> <p>8.7. Uniform convergence of sequences of distributions 188</p> <p><b>Chapter 9 Potentials and Singular Functions 191</b></p> <p>9.1. Surface integral over a sphere 191</p> <p>9.2. Distribution associated with a singular function 193</p> <p>9.3. Derivatives of a distribution associated with a singular function 196</p> <p>9.4. Elementary Newtonian potential 197</p> <p>9.5. Newtonian potential of order n 201</p> <p>9.6. Localized potential 208</p> <p>9.7. Dirac mass as derivatives of continuous functions 210</p> <p>9.8. Heaviside potential 214</p> <p>9.9. Weighting by a singular weight 217</p> <p><b>Chapter 10 Line Integral of a Continuous Field 221</b></p> <p>10.1. Line integral along a C<sup>1</sup> path 221</p> <p>10.2. Change of variable in a path 225</p> <p>10.3. Line integral along a piecewise C<sup>1</sup> path 228</p> <p>10.4. The homotopy invariance theorem 231</p> <p>10.5. Connectedness and simply connectedness 235</p> <p><b>Chapter 11 Primitives of Functions 237</b></p> <p>11.1. Primitive of a function field with a zero line integral 237</p> <p>11.2. Tubular flows and concentration theorem 239</p> <p>11.3. The orthogonality theorem for functions 243</p> <p>11.4. Poincaré’s theorem 244</p> <p><b>Chapter 12 Properties of Primitives of Distributions 247</b></p> <p>12.1. Representation by derivatives 247</p> <p>12.2. Distribution whose derivatives are zero or continuous 251</p> <p>12.3. Uniqueness of a primitive 253</p> <p>12.4. Locally explicit primitive 254</p> <p>12.5. Continuous primitive mapping 256</p> <p>12.6. Harmonic distributions, distributions with a continuous Laplacian 261</p> <p><b>Chapter 13 Existence of Primitives 265</b></p> <p>13.1. Peripheral gluing 266</p> <p>13.2. Reduction to the function case 268</p> <p>13.3. The orthogonality theorem 270</p> <p>13.4. Poincaré’s generalized theorem 274</p> <p>13.5. Current of an incompressible two dimensional field 277</p> <p>13.6. Global versus local primitives 279</p> <p>13.7. Comparison of the existence conditions of a primitive 282</p> <p>13.8. Limits of gradients 283</p> <p><b>Chapter 14 Distributions of Distributions 285</b></p> <p>14.1. Characterization 285</p> <p>14.2. Bounded sets 288</p> <p>14.3. Convergent sequences 289</p> <p>14.4. Extraction of convergent subsequences 293</p> <p>14.5. Change of the space of values 294</p> <p>14.6. Distributions of distributions with values in E-weak 295</p> <p><b>Chapter 15 Separation of Variables 297</b></p> <p>15.1. Tensor products of test functions 297</p> <p>15.2. Decomposition of test functions on a product of sets 301</p> <p>15.3. The tensorial control theorem 303</p> <p>15.4. Separation of variables 309</p> <p>15.5. The kernel theorem 311</p> <p>15.6. Regrouping of variables 317</p> <p>15.7. Permutation of variables 318</p> <p><b>Chapter 16 Banach Space Valued Distributions 323</b></p> <p>16.1. Finite order distributions 323</p> <p>16.2. Weighting of a finite order distribution 326</p> <p>16.3. Finite order distribution as derivatives of continuous functions 328</p> <p>16.4. Finite order distribution as derivative of a single function 333</p> <p>16.5. Distributions in a Banach space as derivatives of functions 335</p> <p>16.6. Non-representability of distributions with values in a Fréchet space 339</p> <p>16.7. Extendability of distributions with values in a Banach space 342</p> <p>16.8. Cancellation of distributions with values in a Banach space 347</p> <p>Appendix 349</p> <p>Bibliography 367</p> <p>Index 371</p>
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.