| Preface |
|
vii | |
| How to use this book in courses |
|
xxi | |
| Acknowledgment |
|
xxv | |
| Notation |
|
xxvii | |
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1 | (95) |
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1.1 Introduction: Dirac's delta function δ(x) and its properties |
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1 | (5) |
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1.2 Test space D (Ω) of Schwartz |
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6 | (19) |
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1.2.1 Support of a continuous function |
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6 | (3) |
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9 | (4) |
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13 | (1) |
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13 | (1) |
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1.2.5 Properties of D (Ω) |
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14 | (11) |
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1.3 Space D' (Ω) of (Schwartz) distributions |
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25 | (16) |
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1.3.1 Algebraic dual space D* (Ω) |
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25 | (1) |
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1.3.2 Distributions and the space D' (Ω) of distributions on Ω |
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26 | (1) |
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1.3.3 Characterization, order and extension of a distribution |
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27 | (2) |
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1.3.4 Examples of distributions |
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29 | (11) |
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1.3.5 Distribution defined on test space D (Ω) of complex-valued functions |
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40 | (1) |
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1.4 Some more examples of interesting distributions |
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41 | (10) |
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1.5 Multiplication of distributions by C∞-functions |
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51 | (3) |
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1.6 Problem of division of distributions |
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54 | (3) |
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1.7 Even, odd and positive distributions |
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57 | (2) |
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1.8 Convergence of sequences of distributions in D' (Ω) |
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59 | (8) |
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1.9 Convergence of series of distributions in D' (Ω) |
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67 | (1) |
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1.10 Images of distributions due to change of variables, homogeneous, invariant, spherically symmetric, constant distributions |
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68 | (16) |
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1.10.1 Periodic distributions |
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75 | (9) |
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1.11 Physical distributions versus mathematical distributions |
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84 | (12) |
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1.11.1 Physical interpretation of mathematical distributions |
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84 | (1) |
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85 | (3) |
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1.11.3 Electrical charge distribution |
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88 | (2) |
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1.11.4 Simple layer and double layer distributions |
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90 | (4) |
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1.11.5 Relation with probability distribution [ 7] |
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94 | (2) |
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2 Differentiation of distributions and application of distributional derivatives |
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96 | (115) |
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2.1 Introduction: an integral definition of derivatives of C1-functions |
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96 | (4) |
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2.2 Derivatives of distributions |
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100 | (2) |
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2.2.1 Higher-order derivatives of distributions T |
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101 | (1) |
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2.3 Derivatives of functions in the sense of distribution |
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102 | (17) |
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2.4 Conditions under which the two notions of derivatives of functions coincide |
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119 | (2) |
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2.5 Derivative of product αT with T D' (Ω) and α C∞ (Ω) |
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121 | (4) |
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2.6 Problem of division of distribution revisited |
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125 | (6) |
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2.7 Primitives of a distribution and differential equations |
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131 | (10) |
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2.8 Properties of distributions whose distributional derivatives are known |
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141 | (1) |
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2.9 Continuity of differential operator α D (Ω) → D' (Ω) |
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142 | (7) |
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2.10 Delta-convergent sequences of functions in D' (Rn) |
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149 | (5) |
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2.11 Term-by-term differentiation of series of distributions |
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154 | (19) |
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2.12 Convergence of sequences of Ck (Ω) (resp. Ck,λ (Ω) in D' (Ω) |
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173 | (1) |
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2.13 Convergence of sequences of Lp (Ω), 1 ≤ p ≤ ∞, in D' (Ω) |
|
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173 | (2) |
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2.14 Transpose (or formal adjoint) of a linear partial differential operator |
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175 | (2) |
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2.15 Applications: Sobolev spaces Hm(Ω), Wm,p(Ω) |
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177 | (34) |
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177 | (1) |
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178 | (4) |
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2.15.3 Examples of functions belonging to or not belonging to Hm(Ω) |
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182 | (2) |
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2.15.4 Separability of Hm(Ω) |
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184 | (2) |
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2.15.5 Generalized Poincare inequality in Hm(Ω) |
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186 | (1) |
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187 | (4) |
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191 | (1) |
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2.15.8 Quotient space Hm(Ω)/M |
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191 | (2) |
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2.15.9 Quotient space Hm(Ω)/Pm-1 |
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193 | (1) |
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2.15.10 Other equivalent norms in Hm (Ω) |
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194 | (1) |
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195 | (1) |
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2.15.12 Algebraic inclusions (⊂) and imbedding (→) results |
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195 | (1) |
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2.15.13 Space Wm,p(Ω) with m N, 1 ≤ p ≤ ∞ |
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196 | (4) |
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2.15.14 Space Wm'p(Ω), 1 ≤ p < ∞ |
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200 | (3) |
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203 | (1) |
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2.15.16 Quotient space Wm,p(Ω)/M for m N, 1 ≤ p ≤ ∞ |
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203 | (4) |
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207 | (1) |
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2.15.18 A non-density result |
|
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208 | (1) |
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2.15.19 Algebraic inclusion ⊂ and imbedding (→) results |
|
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209 | (1) |
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2.15.20 Space Ws,p(Ω) for arbitrary s R |
|
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209 | (2) |
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3 Derivatives of piecewise smooth functions, Green's formula, elementary solutions, applications to Sobolev spaces |
|
|
211 | (52) |
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3.1 Distributional derivatives of piecewise smooth functions |
|
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211 | (24) |
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3.1.1 Case of single variable (n = 1) |
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211 | (4) |
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3.1.2 Case of two variables (n = 2) |
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215 | (15) |
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3.1.3 Case of three variables (n = 3) |
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230 | (5) |
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3.2 Unbounded domain Ω ⊂ Rn, Green's formula |
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235 | (3) |
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238 | (19) |
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257 | (6) |
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4 Additional properties of D' (Ω) |
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263 | (17) |
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4.1 Reflexivity of D (Ω) and density of D (Ω) in D' (Ω) |
|
|
263 | (2) |
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4.2 Continuous imbedding of dual spaces of Banach spaces in D' (Ω) |
|
|
265 | (4) |
|
4.3 Applications: Sobolev spaces H-m(Ω), W-m,q (Ω) |
|
|
269 | (11) |
|
4.3.1 Space W-m,q (Ω), 1 < q ≤ ∞, m N |
|
|
273 | (7) |
|
5 Local properties, restrictions, unification principle, space ε'(Rn) of distributions with compact support |
|
|
280 | (18) |
|
5.1 Null distribution in an open set |
|
|
280 | (1) |
|
5.2 Equality of distributions in an open set |
|
|
280 | (1) |
|
5.3 Restriction of a distribution to an open set |
|
|
280 | (3) |
|
5.4 Unification principle |
|
|
283 | (2) |
|
5.5 Support of a distribution |
|
|
285 | (1) |
|
5.6 Distributions with compact support |
|
|
286 | (1) |
|
5.7 Space ε'(Rn) of distributions with compact support |
|
|
287 | (9) |
|
|
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287 | (1) |
|
|
|
288 | (8) |
|
5.8 Definition of (T, ø) for ø C ∞ (Rn) and T D' (Rn) with non-compact support |
|
|
296 | (2) |
|
6 Convolution of distributions |
|
|
298 | (85) |
|
|
|
298 | (5) |
|
6.2 Convolution of functions |
|
|
303 | (12) |
|
6.3 Convolution of two distributions |
|
|
315 | (12) |
|
6.4 Regularization of distributions by convolution |
|
|
327 | (2) |
|
6.5 Approximation of distributions by C∞-functions |
|
|
329 | (2) |
|
6.6 Convolution of several distributions |
|
|
331 | (2) |
|
6.7 Derivatives of convolutions, convolution of distributions on a circle Γ and their Fourier series representations on Γ |
|
|
333 | (16) |
|
|
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349 | (15) |
|
6.9 Convolution equations (see also Section 8.7,
Chapter 8) |
|
|
364 | (11) |
|
6.10 Application of convolutions in electrical circuit analysis and heat flow problems |
|
|
375 | (8) |
|
6.10.1 Electric circuit analysis problem [ 7] |
|
|
375 | (5) |
|
6.10.2 Excitations and responses defined by several functions or distributions [ 7] |
|
|
380 | (3) |
|
7 Fourier transforms of functions of L1(Rn) and S(Rn) |
|
|
383 | (40) |
|
7.1 Fourier transforms of integrable functions in L1 (Rn) |
|
|
383 | (22) |
|
7.2 Space S(Rn) of infinitely differentiable functions with rapid decay at infinity |
|
|
405 | (7) |
|
|
|
407 | (5) |
|
7.3 Continuity of linear mapping from S(Rn) into S(Rn) |
|
|
412 | (1) |
|
|
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413 | (2) |
|
|
|
415 | (2) |
|
7.6 Fourier transform of functions of S(Rn) |
|
|
417 | (1) |
|
7.7 Fourier inversion theorem in S(Rn) |
|
|
418 | (5) |
|
8 Fourier transforms of distributions and Sobolev spaces of arbitrary order Hs(Rn) |
|
|
423 | (289) |
|
8.1 Motivation for a possible definition of the Fourier transform of a distribution |
|
|
423 | (1) |
|
8.2 Space S'(Rn) of tempered distributions |
|
|
424 | (11) |
|
8.2.1 Tempered distributions |
|
|
424 | (2) |
|
|
|
426 | (1) |
|
8.2.3 Examples of tempered distributions of S'(Rn) |
|
|
426 | (3) |
|
8.2.4 Convergence of sequences in S'(Rn) |
|
|
429 | (3) |
|
8.2.5 Derivatives of tempered distributions |
|
|
432 | (3) |
|
8.3 Fourier transform of tempered distributions |
|
|
435 | (10) |
|
8.3.1 Fourier transforms of Dirac distributions and their derivatives |
|
|
438 | (2) |
|
8.3.2 Inversion theorem for Fourier transforms on S'(Rn) |
|
|
440 | (1) |
|
8.3.3 Fourier transform of even and odd tempered distributions |
|
|
441 | (4) |
|
8.4 Fourier transform of distributions with compact support |
|
|
445 | (5) |
|
8.5 Fourier transform of convolution of distributions |
|
|
450 | (8) |
|
8.5.1 Fourier transforms of convolutions |
|
|
451 | (7) |
|
8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives of tempered distributions |
|
|
458 | (18) |
|
8.7 Fourier transform methods for differential equations and elementary solutions in S'(Rn) |
|
|
476 | (16) |
|
8.8 Laplace transform of distributions on R |
|
|
492 | (10) |
|
|
|
492 | (4) |
|
8.8.2 Distribution T-1 D'+ (see also convolution algebra A = D'+ (6.9.15b)) |
|
|
496 | (1) |
|
8.8.3 Inverse L-1 of Laplace transform L |
|
|
497 | (5) |
|
|
|
502 | (44) |
|
8.9.1 Sobolev spaces Hs(Rn) |
|
|
502 | (1) |
|
|
|
503 | (4) |
|
8.9.3 Sobolev spaces Hm(Rn) of integral order m on Rn |
|
|
507 | (5) |
|
8.9.4 Sobolev's Imbedding Theorem (see also imbedding results in Section 8.12) |
|
|
512 | (9) |
|
8.9.5 Imbedding result: S(Rn) Hs(Rn) |
|
|
521 | (1) |
|
8.9.6 Density results Hs(Rn) |
|
|
522 | (1) |
|
8.9.7 Dual space (Hs(Rn))' |
|
|
523 | (3) |
|
8.9.8 Trace properties of elements of Hs(Rn) |
|
|
526 | (20) |
|
8.10 Sobolev spaces on Ω ≠ Rn revisited |
|
|
546 | (59) |
|
8.10.1 Space Hs(Ω) with s R, Ω Rn |
|
|
546 | (4) |
|
8.10.2 m-extension property of Ω |
|
|
550 | (8) |
|
8.10.3 m-extension property of Rn+ |
|
|
558 | (11) |
|
8.10.4 m-extension property of Cm-regular domains Ω |
|
|
569 | (4) |
|
8.10.5 Space Hs(Ω) with s R+, Ω Rn |
|
|
573 | (5) |
|
8.10.6 Density results in Hs(Ω) |
|
|
578 | (1) |
|
|
|
579 | (1) |
|
8.10.8 Space Hs0(Ω) with s > 0 |
|
|
579 | (1) |
|
8.10.9 Space H-s(Ω) with s > 0 |
|
|
580 | (1) |
|
8.10.10 Space Ws,p(Ω) for real s > 0 and 1 ≤ p < ∞ |
|
|
580 | (5) |
|
8.10.11 Space Hs00(Ω) with s > 0 |
|
|
585 | (6) |
|
8.10.12 Dual space (Hs00 (Ω))' for s > 0 |
|
|
591 | (1) |
|
8.10.13 Space Ws,p00 (Ω) for s > 0, 1 < p < ∞ |
|
|
591 | (2) |
|
8.10.14 Restrictions of distributions in Sobolev spaces |
|
|
593 | (5) |
|
8.10.15 Differentiation of distributions in Hs(Ω) with s R |
|
|
598 | (3) |
|
8.10.16 Differentiation of distributions u Hs(Ω) with s > 0 |
|
|
601 | (4) |
|
8.11 Compactness results in Sobolev spaces |
|
|
605 | (12) |
|
8.11.1 Compact imbedding results in Hs (Ω), Hs0(Ω) and Hs00(Ω) |
|
|
616 | (1) |
|
8.12 Sobolev's imbedding results |
|
|
617 | (17) |
|
8.12.1 Compact imbedding results |
|
|
632 | (2) |
|
8.13 Sobolev spaces Hs (Γ), Ws,p(Γ) on a manifold boundary Γ |
|
|
634 | (17) |
|
8.13.1 Surface integrals on boundary Γ of bounded Ω ⊂ Rn |
|
|
634 | (3) |
|
8.13.2 Alternative definition of Hs (Γ) with Γ Cm-class (resp. C∞-class) |
|
|
637 | (1) |
|
8.13.3 Space Hs(Γ) (s > 0) with Γ in Cm-class (resp. C∞-class) |
|
|
638 | (3) |
|
8.13.4 Sobolev spaces on boundary curves Γ in R2 |
|
|
641 | (10) |
|
8.13.5 Spaces Hs0(Γi), Hs00(Γi) for polygonal sides Γi C∞-class, 1 ≤ i ≤ N |
|
|
651 | (1) |
|
8.14 Trace results in Sobolev spaces on Ω Rn |
|
|
651 | (61) |
|
8.14.1 Trace results in Hm(Rn+) |
|
|
652 | (2) |
|
8.14.2 Trace results in Hm∞ with bounded domain Ω Rn |
|
|
654 | (16) |
|
8.14.3 Trace results in Ws,p-spaces |
|
|
670 | (2) |
|
8.14.4 Trace results for polygonal domains Ω ⊂ R2 |
|
|
672 | (13) |
|
8.14.5 Trace results for bounded domains with curvilinear polygonal boundary Γ in R2 |
|
|
685 | (1) |
|
8.14.6 Traces of normal components in Lp (div; Ω) |
|
|
686 | (5) |
|
8.14.7 Trace theorems based on Green's formula |
|
|
691 | (19) |
|
|
|
710 | (2) |
|
9 Vector-valued distributions |
|
|
712 | (19) |
|
|
|
712 | (1) |
|
9.2 Vector-valued functions |
|
|
712 | (3) |
|
9.3 Spaces of vector-valued functions |
|
|
715 | (3) |
|
9.4 Vector-valued distributions |
|
|
718 | (5) |
|
9.5 Derivatives of vector-valued distributions |
|
|
723 | (1) |
|
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|
724 | (7) |
|
|
|
725 | (1) |
|
9.6.2 Hilbert space W1(0,T;V) |
|
|
725 | (3) |
|
9.6.3 Hilbert space W2(0,T;V) |
|
|
728 | (1) |
|
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|
729 | (2) |
|
A Functional analysis (basic results) |
|
|
731 | (40) |
|
|
|
731 | (10) |
|
A.0.1 An important result on logical implication (⇒) and non-implication () |
|
|
731 | (1) |
|
A.0.2 Supremum (l.u.b.) and infimum (g.l.b.) |
|
|
732 | (1) |
|
A.0.3 Metric spaces and important results therein |
|
|
732 | (3) |
|
A.0.4 Important subsets of a metric space X ≡ (X, d) |
|
|
735 | (2) |
|
A.0.5 Compact sets in Rn with the usual metric d2 |
|
|
737 | (1) |
|
A.0.6 Elementary properties of functions of real variables |
|
|
738 | (1) |
|
A.0.7 Limit of a function at a cluster point x0 Rn |
|
|
738 | (1) |
|
A.0.8 Limit superior and limit inferior of a sequence in R |
|
|
739 | (1) |
|
A.0.9 Pointwise and uniform convergence of sequences of functions |
|
|
740 | (1) |
|
A.0.10 Continuity and uniform continuity of f F(Ω) |
|
|
740 | (1) |
|
A.1 Important properties of continuous functions |
|
|
741 | (2) |
|
A.1.1 Some remarkable properties on compact sets in Rn |
|
|
741 | (1) |
|
A.1.2 C∞0 (Ω)-partition of unity on compact set K ⊂⊂ Ω ⊂ Rn |
|
|
741 | (1) |
|
A.1.3 Continuous extension theorems |
|
|
741 | (2) |
|
A.2 Finite and infinite dimensional linear spaces |
|
|
743 | (5) |
|
|
|
743 | (3) |
|
|
|
746 | (1) |
|
|
|
747 | (1) |
|
|
|
748 | (2) |
|
|
|
748 | (2) |
|
A.3.2 Closed subspace, dense subspace, Banach space and its separability |
|
|
750 | (1) |
|
A.4 Banach spaces of continuous functions |
|
|
750 | (3) |
|
A.4.1 Banach spaces C0(Ω), Ck(Ω) |
|
|
750 | (3) |
|
A.5 Banach spaces C0'λ(Ω), 0 < λ < 1, of Holder continuous functions |
|
|
753 | (3) |
|
A.5.1 Holder continuity and Lipschitz continuity |
|
|
753 | (1) |
|
A.5.2 Holder space C0,λ(Ω) |
|
|
754 | (1) |
|
A.5.3 Space Ck,λ (Ω), 0 < λ ≤ 1 |
|
|
754 | (2) |
|
|
|
756 | (1) |
|
A.7 Continuous linear functionals on normed linear spaces |
|
|
756 | (4) |
|
|
|
756 | (1) |
|
A.7.2 Hahn-Banach extension of linear functionals in analytic form |
|
|
757 | (1) |
|
A.7.3 Consequences of the Hahn-Banach theorem in normed linear spaces |
|
|
758 | (2) |
|
A.8 Continuous linear operators on normed linear spaces |
|
|
760 | (3) |
|
|
|
760 | (1) |
|
A.8.2 Continuous extension of continuous linear operators by density |
|
|
761 | (1) |
|
A.8.3 Isomorphisms and isometric isomorphisms |
|
|
762 | (1) |
|
A.8.4 Graph of an operator A L(V; W) and graph norm |
|
|
762 | (1) |
|
A.9 Reflexivity of Banach spaces |
|
|
763 | (1) |
|
A.10 Strong, weak and weak-* convergence in Banach space V |
|
|
763 | (1) |
|
A.10.1 Strong convergence ↑ |
|
|
763 | (1) |
|
A.10.2 Weak convergence ↑ |
|
|
764 | (1) |
|
A.10.3 Weak-* convergence ↑ * in Banach space V' |
|
|
764 | (1) |
|
A.11 Compact linear operators in Banach spaces |
|
|
764 | (1) |
|
|
|
765 | (3) |
|
A.13 Dual space V' of a Hilbert space V, reflexivity of V |
|
|
768 | (1) |
|
A.14 Strong, weak and weak-* convergences in a Hilbert space |
|
|
769 | (1) |
|
A.15 Self-adjoint and unitary operators in Hilbert space V |
|
|
769 | (1) |
|
A.16 Compact linear operators in Hilbert spaces |
|
|
769 | (2) |
|
|
|
771 | (32) |
|
B.1 Lebesgue measure μ on Rn |
|
|
771 | (5) |
|
B.1.1 Lebesgue-measurable sets in Rn |
|
|
771 | (1) |
|
B.1.2 Sets with zero (Lebesgue) measure in Rn |
|
|
772 | (3) |
|
B.1.3 Property P holds almost everywhere (a.e.) on Ω |
|
|
775 | (1) |
|
B.2 Space M(Ω) of Lebesgue-measurable functions on Ω |
|
|
776 | (2) |
|
B.2.1 Measurable functions and space M(Ω) |
|
|
776 | (2) |
|
B.2.2 Pointwise convergence a.e. on Ω |
|
|
778 | (1) |
|
B.3 Lebesgue integrals and their important properties |
|
|
778 | (10) |
|
B.3.1 Lebesgue integral of a bounded function on bounded domain Ω |
|
|
778 | (2) |
|
B.3.2 Important properties of Lebesgue integrals (Kolmogorov and Fomin [ 20]) |
|
|
780 | (4) |
|
B.3.3 Some important approximation and density results in L1 (Ω) |
|
|
784 | (4) |
|
B.4 Spaces Lp(Ω), 1 ≤ p ≤ ∞ |
|
|
788 | (15) |
|
|
|
788 | (6) |
|
B.4.2 Dual space (Lp(Ω))' of Lp(Ω) for 1 ≤ p ≤ ∞ |
|
|
794 | (3) |
|
|
|
797 | (1) |
|
B.4.4 Some negative properties of L∞(Ω) |
|
|
798 | (1) |
|
B.4.5 Some nice properties of L∞(Ω) |
|
|
799 | (1) |
|
B.4.6 Space Lploc(Ω) inclusion results |
|
|
799 | (4) |
|
C Open cover and partition of unity |
|
|
803 | (5) |
|
C.1 C∞0(Ω)-partition of unity theorem for compact sets |
|
|
803 | (5) |
|
|
|
808 | (11) |
|
|
|
808 | (4) |
|
D.1.1 Locally one-sided and two-sided bounded domains (Ω) |
|
|
808 | (1) |
|
D.1.2 Star-shaped domain (Ω) |
|
|
808 | (1) |
|
D.1.3 Cone property and uniform cone property |
|
|
809 | (2) |
|
|
|
811 | (1) |
|
D.2 Continuity and differential properties of a boundary |
|
|
812 | (4) |
|
D.2.1 Continuity and differential properties |
|
|
812 | (1) |
|
D.2.2 Open cover {Γr}Nr=1 of Γ, local coordinate systems {ri}n i =1 and mappings {ør}N r=1 |
|
|
813 | (1) |
|
D.2.3 Properties of the mappings ør: Rn-1 → R, 1 ≤ r ≤ N |
|
|
814 | (2) |
|
D.3 Alternative definition of locally one-sided domain |
|
|
816 | (1) |
|
D.4 Alternative definition of continuity and differential properties of (Ω) as a manifold in Rn |
|
|
817 | (1) |
|
D.5 Atlas/local charts of Γ |
|
|
818 | (1) |
| Bibliography |
|
819 | (4) |
| Index |
|
823 | |
Lisainfo e-raamatute kohta