| Preface | 7 |
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| How to use this book in courses | 21 |
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| Acknowledgment | 25 |
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| Notation | 27 |
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| 1 Schwartz distributions | 39 |
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| 1.1 Introduction: Dirac’s delta function d(x) and its properties | 39 |
| 39 | 39 |
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| 1.2 Test space D (O) of Schwartz | 44 |
| 1.2.1 Support of a continuous function | 44 |
| 1.2.2 Space D (O) | 44 |
| 47 | 44 |
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| 1.2.3 Space Dm(O | 44 |
| 51 | 44 |
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| 1.2.4 Space DK (O) | 44 |
| 51 | 44 |
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| 1.2.5 Properties of D (O) | 44 |
| 52 | 44 |
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| 1.3 Space D'(O) of (Schwartz) distributions | 63 |
| 1.3.1 Algebraic dual space D*(O) | 63 |
| 1.3.2 Distributions and the space D'(O) of distributions on O | 64 |
| 1.3.3 Characterization, order and extension of a distribution | 65 |
| 1.3.4 Examples of distributions | 67 |
| 1.3.5 Distribution defined on test space D(O) of complex-valued functions | 78 |
| 1.4 Some more examples of interesting distributions | 79 |
| 1.5 Multiplication of distributions by C8-functions | 79 |
| 89 | 79 |
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| 1.6 Problem of division of distributions | 92 |
| 1.7 Even, odd and positive distributions | 95 |
| 1.8 Convergence of sequences of distributions in D'(O) | 97 |
| 1.9 Convergence of series of distributions in D'(O) | 97 |
| 105 | 97 |
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| 1.10 Images of distributions due to change of variables, homogeneous, invariant, spherically symmetric, constant distributions | 106 |
| 1.10.1 Periodic distributions | 113 |
| 1.11 Physical distributions versus mathematical distributions | 122 |
| 1.11.1 Physical interpretation of mathematical distributions | 122 |
| 1.11.2 Load intensity | 123 |
| 1.11.3 Electrical charge distribution | 126 |
| 1.11.4 Simple layer and double layer distributions | 128 |
| 1.11.5 Relation with probability distribution [7] | 132 |
| 2 Differentiation of distributions and application of distributional derivatives | 134 |
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| 2.1 Introduction: an integral definition of derivatives of C1-functions | 134 |
| 2.2 Derivatives of distributions | 138 |
| 2.2.1 Higher-order derivatives of distributions T | 139 |
| 2.3 Derivatives of functions in the sense of distribution | 140 |
| 2.4 Conditions under which the two notions of derivatives of functions coincide | 157 |
| 2.5 Derivative of product aT with T . D'(O) and a . C8(O) | 159 |
| 2.6 Problem of division of distribution revisited | 163 |
| 2.7 Primitives of a distribution and differential equations | 169 |
| 2.8 Properties of distributions whose distributional derivatives are known | 179 |
| 2.9 Continuity of differential operator .a : D'(O) . D'(O) | 180 |
| 2.10 Delta-convergent sequences of functions in D'(Rn) | 187 |
| 2.11 Term-by-term differentiation of series of distributions | 192 |
| 2.12 Convergence of sequences of Ck(O¯) (resp. Ck,.(O¯)) in D'(O) | 211 |
| 2.13 Convergence of sequences of Lp (O), 1 = p = 8, in D'(O) | 211 |
| 2.14 Transpose (or formal adjoint) of a linear partial differential operator | 213 |
| 2.15 Applications: Sobolev spaces Hm(O),Wm,p(O) | 215 |
| 2.15.1 Sobolev Spaces | 215 |
| 2.15.2 Space Hm(O) | 216 |
| 2.15.3 Examples of functions belonging to or not belonging to Hm(O) | 220 |
| 2.15.4 Separability of Hm(O) | 222 |
| 2.15.5 Generalized Poincaré inequality in Hm(O) | 224 |
| 2.15.6 Space H0m(O) | 225 |
| 2.15.7 Space H–m(O) | 229 |
| 2.15.8 Quotient space Hm(O)/M | 229 |
| 2.15.9 Quotient space Hm(O)/Pm-1 | 231 |
| 2.15.10 Other equivalent norms in Hm(O) | 232 |
| 2.15.11 Density results | 233 |
| 2.15.12 Algebraic inclusions (.) and imbedding (.) results | 233 |
| 2.15.13 Space Wm,p(O) with m . N, 1 = p | 233 |
| 234 | 233 |
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| 2.15.14 Space W0m,p(O), 1 = p | 233 |
| 238 | 233 |
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| 2.15.15 Space W-m,q (O) | 241 |
