Distributions

1 - Preface [Seite 7]
2 - How to use this book in courses [Seite 21]
3 - Acknowledgment [Seite 25]
4 - Notation [Seite 27]
5 - 1 Schwartz distributions [Seite 39]
5.1 - 1.1 Introduction: Dirac's delta function d(x) and its properties [Seite 1.1 Introduction: Dirac's delta function d(x) and its properties]
- 39 [Seite 39]
5.2 - 1.2 Test space D (O) of Schwartz [Seite 44]
5.2.1 - 1.2.1 Support of a continuous function [Seite 44]
5.2.2 - 1.2.2 Space D (O) [Seite 1.2.2 Space D (O)]
- 47 [Seite 47]
5.2.3 - 1.2.3 Space Dm(O [Seite 1.2.3 Space Dm(O]
) - 51 [Seite 51]
5.2.4 - 1.2.4 Space DK (O) [Seite 1.2.4 Space DK (O)]
- 51 [Seite 51]
5.2.5 - 1.2.5 Properties of D (O) [Seite 1.2.5 Properties of D (O)]
- 52 [Seite 52]
5.3 - 1.3 Space D'(O) of (Schwartz) distributions [Seite 63]
5.3.1 - 1.3.1 Algebraic dual space D*(O) [Seite 63]
5.3.2 - 1.3.2 Distributions and the space D'(O) of distributions on O [Seite 64]
5.3.3 - 1.3.3 Characterization, order and extension of a distribution [Seite 65]
5.3.4 - 1.3.4 Examples of distributions [Seite 67]
5.3.5 - 1.3.5 Distribution defined on test space D(O) of complex-valued functions [Seite 78]
5.4 - 1.4 Some more examples of interesting distributions [Seite 79]
5.5 - 1.5 Multiplication of distributions by C8-functions [Seite 1.5 Multiplication of distributions by C8-functions]
- 89 [Seite 89]
5.6 - 1.6 Problem of division of distributions [Seite 92]
5.7 - 1.7 Even, odd and positive distributions [Seite 95]
5.8 - 1.8 Convergence of sequences of distributions in D'(O) [Seite 97]
5.9 - 1.9 Convergence of series of distributions in D'(O) [Seite 1.9 Convergence of series of distributions in D'(O)]
- 105 [Seite 105]
5.10 - 1.10 Images of distributions due to change of variables, homogeneous, invariant, spherically symmetric, constant distributions [Seite 106]
5.10.1 - 1.10.1 Periodic distributions [Seite 113]
5.11 - 1.11 Physical distributions versus mathematical distributions [Seite 122]
5.11.1 - 1.11.1 Physical interpretation of mathematical distributions [Seite 122]
5.11.2 - 1.11.2 Load intensity [Seite 123]
5.11.3 - 1.11.3 Electrical charge distribution [Seite 126]
5.11.4 - 1.11.4 Simple layer and double layer distributions [Seite 128]
5.11.5 - 1.11.5 Relation with probability distribution [7] [Seite 132]
6 - 2 Differentiation of distributions and application of distributional derivatives [Seite 134]
6.1 - 2.1 Introduction: an integral definition of derivatives of C1-functions [Seite 134]
6.2 - 2.2 Derivatives of distributions [Seite 138]
6.2.1 - 2.2.1 Higher-order derivatives of distributions T [Seite 139]
6.3 - 2.3 Derivatives of functions in the sense of distribution [Seite 140]
6.4 - 2.4 Conditions under which the two notions of derivatives of functions coincide [Seite 157]
6.5 - 2.5 Derivative of product aT with T . D'(O) and a . C8(O) [Seite 159]
6.6 - 2.6 Problem of division of distribution revisited [Seite 163]
6.7 - 2.7 Primitives of a distribution and differential equations [Seite 169]
6.8 - 2.8 Properties of distributions whose distributional derivatives are known [Seite 179]
6.9 - 2.9 Continuity of differential operator .a : D'(O) . D'(O) [Seite 180]
6.10 - 2.10 Delta-convergent sequences of functions in D'(Rn) [Seite 187]
6.11 - 2.11 Term-by-term differentiation of series of distributions [Seite 192]
6.12 - 2.12 Convergence of sequences of Ck(O¯) (resp. Ck,.(O¯)) in D'(O) [Seite 211]
6.13 - 2.13 Convergence of sequences of Lp (O), 1 = p = 8, in D'(O) [Seite 211]
6.14 - 2.14 Transpose (or formal adjoint) of a linear partial differential operator [Seite 213]
6.15 - 2.15 Applications: Sobolev spaces Hm(O),Wm,p(O) [Seite 215]
6.15.1 - 2.15.1 Sobolev Spaces [Seite 215]
6.15.2 - 2.15.2 Space Hm(O) [Seite 216]
6.15.3 - 2.15.3 Examples of functions belonging to or not belonging to Hm(O) [Seite 220]
6.15.4 - 2.15.4 Separability of Hm(O) [Seite 222]
6.15.5 - 2.15.5 Generalized Poincaré inequality in Hm(O) [Seite 224]
6.15.6 - 2.15.6 Space H0m(O) [Seite 225]
6.15.7 - 2.15.7 Space H-m(O) [Seite 229]
6.15.8 - 2.15.8 Quotient space Hm(O)/M [Seite 229]
6.15.9 - 2.15.9 Quotient space Hm(O)/Pm-1 [Seite 231]
6.15.10 - 2.15.10 Other equivalent norms in Hm(O) [Seite 232]
6.15.11 - 2.15.11 Density results [Seite 233]
6.15.12 - 2.15.12 Algebraic inclusions (.) and imbedding (.) results [Seite 233]
6.15.13 - 2.15.13 Space Wm,p(O) with m . N, 1 = p < 8 [Seite 234]
6.15.14 - 2.15.14 Space W0m,p(O), 1 = p < 8 [Seite 238]
6.15.15 - 2.15.15 Space W-m,q (O) [Seite 241]
6.15.16 - 2.15.16 Quotient space Wm,p (O)/M for m . N, 1 = p < 8 [Seite 241]
6.15.17 - 2.15.17 Density results [Seite 245]
6.15.18 - 2.15.18 A non-density result [Seite 246]
6.15.19 - 2.15.19 Algebraic inclusion . and imbedding (.) results [Seite 247]
6.15.20 - 2.15.20 Space Ws,p (O) for arbitrary s . R [Seite 247]
7 - 3 Derivatives of piecewise smooth functions, Green's formula, elementary solutions, applications to Sobolev spaces [Seite 249]
7.1 - 3.1 Distributional derivatives of piecewise smooth functions [Seite 249]
7.1.1 - 3.1.1 Case of single variable (n = 1) [Seite 249]
7.1.2 - 3.1.2 Case of two variables (n = 2) [Seite 253]
7.1.3 - 3.1.3 Case of three variables (n = 3) [Seite 268]
7.2 - 3.2 Unbounded domain O . Rn, Green's formula [Seite 273]
7.3 - 3.3 Elementary solutions [Seite 276]
7.4 - 3.4 Applications [Seite 295]
8 - 4 Additional properties of D'(O) [Seite 301]
8.1 - 4.1 Reflexivity of D(O) and density of D(O) in D'(O) [Seite 301]
8.2 - 4.2 Continuous imbedding of dual spaces of Banach spaces in D'(O) [Seite 303]
8.3 - 4.3 Applications: Sobolev spaces H-m(O), W-m,q (O) [Seite 307]
8.3.1 - 4.3.1 Space W-m,q (O), 1 < q = 8, m . N [Seite 311]
9 - 5 Local properties, restrictions, unification principle, space E'(Rn) of distributions with compact support [Seite 318]
9.1 - 5.1 Null distribution in an open set [Seite 318]
9.2 - 5.2 Equality of distributions in an open set [Seite 318]
9.3 - 5.3 Restriction of a distribution to an open set [Seite 318]
9.4 - 5.4 Unification principle [Seite 321]
9.5 - 5.5 Support of a distribution [Seite 323]
9.6 - 5.6 Distributions with compact support [Seite 324]
9.7 - 5.7 Space E'(Rn) of distributions with compact support [Seite 325]
9.7.1 - 5.7.1 Space E'(Rn) [Seite 325]
9.7.2 - 5.7.2 Space E'(Rn) [Seite 326]
9.8 - 5.8 Definition of for f . C8 (Rn) and T . D'(Rn) with non-compact support [Seite 334]
10 - 6 Convolution of distributions [Seite 336]
10.1 - 6.1 Tensor product [Seite 336]
10.2 - 6.2 Convolution of functions [Seite 341]
10.3 - 6.3 Convolution of two distributions [Seite 353]
10.4 - 6.4 Regularization of distributions by convolution [Seite 365]
10.5 - 6.5 Approximation of distributions by C8-functions [Seite 367]
10.6 - 6.6 Convolution of several distributions [Seite 369]
10.7 - 6.7 Derivatives of convolutions, convolution of distributions on a circle G and their Fourier series representations on G [Seite 371]
10.8 - 6.8 Applications [Seite 387]
10.9 - 6.9 Convolution equations (see also Section 8.7, Chapter 8) [Seite 402]
10.10 - 6.10 Application of convolutions in electrical circuit analysis and heat flow problems [Seite 413]
10.10.1 - 6.10.1 Electric circuit analysis problem [7] [Seite 413]
10.10.2 - 6.10.2 Excitations and responses defined by several functions or distributions [7] [Seite 418]
11 - 7 Fourier transforms of functions of L1 (Rn) and S(Rn) [Seite 421]
11.1 - 7.1 Fourier transforms of integrable functions in L1 (Rn) [Seite 421]
11.2 - 7.2 Space S(Rn) of infinitely differentiable functions with rapid decay at infinity [Seite 443]
11.2.1 - 7.2.1 Space S(Rn) [Seite 445]
11.3 - 7.3 Continuity of linear mapping from S(Rn) into S(Rn) [Seite 450]
11.4 - 7.4 Imbedding results [Seite 451]
11.5 - 7.5 Density results [Seite 453]
11.6 - 7.6 Fourier transform of functions of S(Rn) [Seite 455]
11.7 - 7.7 Fourier inversion theorem in S(Rn) [Seite 456]
12 - 8 Fourier transforms of distributions and Sobolev spaces of arbitrary order HS (Rn) [Seite 461]
12.1 - 8.1 Motivation for a possible definition of the Fourier transform of a distribution [Seite 461]
12.2 - 8.2 Space S'(Rn) of tempered distributions [Seite 462]
12.2.1 - 8.2.1 Tempered distributions [Seite 462]
12.2.2 - 8.2.2 Space S'(Rn) [Seite 464]
12.2.3 - 8.2.3 Examples of tempered distributions of S'(Rn) [Seite 464]
12.2.4 - 8.2.4 Convergence of sequences in S'(Rn) [Seite 467]
12.2.5 - 8.2.5 Derivatives of tempered distributions [Seite 470]
12.3 - 8.3 Fourier transform of tempered distributions [Seite 473]
12.3.1 - 8.3.1 Fourier transforms of Dirac distributions and their derivatives [Seite 476]
12.3.2 - 8.3.2 Inversion theorem for Fourier transforms on S'(Rn) [Seite 478]
12.3.3 - 8.3.3 Fourier transform of even and odd tempered distributions [Seite 479]
12.4 - 8.4 Fourier transform of distributions with compact support [Seite 483]
12.5 - 8.5 Fourier transform of convolution of distributions [Seite 488]
12.5.1 - 8.5.1 Fourier transforms of convolutions [Seite 489]
12.6 - 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives of tempered distributions [Seite 496]
12.7 - 8.7 Fourier transform methods for differential equations and elementary solutions in S'(Rn) [Seite 514]
12.8 - 8.8 Laplace transform of distributions on R [Seite 530]
12.8.1 - 8.8.1 Space D'+ [Seite 530]
12.8.2 - 8.8.2 Distribution T-1 . D'+ (see also convolution algebra A = D'+ (6.9.15b)) [Seite 534]
12.8.3 - 8.8.3 Inverse L-1 of Laplace transform L [Seite 535]
12.9 - 8.9 Applications [Seite 540]
12.9.1 - 8.9.1 Sobolev spaces Hs (Rn) [Seite 540]
12.9.2 - 8.9.2 Imbedding result [Seite 541]
12.9.3 - 8.9.3 Sobolev spaces Hm(Rn) of integral order m on Rn [Seite 545]
12.9.4 - 8.9.4 Sobolev's Imbedding Theorem (see also imbedding results in Section 8.12) [Seite 550]
12.9.5 - 8.9.5 Imbedding result: S(Rn) . HS (Rn) [Seite 559]
12.9.6 - 8.9.6 Density results HS (Rn) [Seite 560]
12.9.7 - 8.9.7 Dual space (Hs (Rn))' [Seite 561]
12.9.8 - 8.9.8 Trace properties of elements of Hs (Rn) [Seite 564]
12.10 - 8.10 Sobolev spaces on O . Rn revisited [Seite 584]
12.10.1 - 8.10.1 Space Hs (O¯) with s . R, O . Rn [Seite 584]
12.10.2 - 8.10.2 m-extension property of O [Seite 588]
12.10.3 - 8.10.3 m-extension property of R+n [Seite 596]
12.10.4 - 8.10.4 m-extension property of Cm -regular domains O [Seite 607]
12.10.5 - 8.10.5 Space Hs (O) with s . R+, O . Rn [Seite 611]
12.10.6 - 8.10.6 Density results in Hs (O) [Seite 616]
12.10.7 - 8.10.7 Dual space H-s (O) [Seite 617]
12.10.8 - 8.10.8 Space H0s (O) with s > 0 [Seite 617]
12.10.9 - 8.10.9 Space H-s (O) with s > 0 [Seite 618]
12.10.10 - 8.10.10 Space Ws, p (O) for real s > 0 and 1 = p < 8 [Seite 618]
12.10.11 - 8.10.11 Space Hs00 (O) with s > 0 [Seite 623]
12.10.12 - 8.10.12 Dual space (H00s(O))' for s > 0 [Seite 629]
12.10.13 - 8.10.13 Space W00s,p (O) for s > 0, 1 < p < 8 [Seite 629]
12.10.14 - 8.10.14 Restrictions of distributions in Sobolev spaces [Seite 631]
12.10.15 - 8.10.15 Differentiation of distributions in Hs (O) with s . R [Seite 636]
12.10.16 - 8.10.16 Differentiation of distributions u . Hs (O¯) with s > 0 [Seite 639]
12.11 - 8.11 Compactness results in Sobolev spaces [Seite 643]
12.11.1 - 8.11.1 Compact imbedding results in Hs(O), Hs0(O) and Hs00(O) [Seite 654]
12.12 - 8.12 Sobolev's imbedding results [Seite 655]
12.12.1 - 8.12.1 Compact imbedding results [Seite 670]
12.13 - 8.13 Sobolev spaces Hs (G), Ws,p (G) on a manifold boundary G [Seite 672]
12.13.1 - 8.13.1 Surface integrals on boundary G of bounded O . Rn [Seite 672]
12.13.2 - 8.13.2 Alternative definition of Hs(G) with G . Cm-class (resp. C8-class) [Seite 675]
12.13.3 - 8.13.3 Space Hs (G) (s > 0) with G in Cm-class (resp. C8-class) [Seite 676]
12.13.4 - 8.13.4 Sobolev spaces on boundary curves G in R2 [Seite 679]
12.13.5 - 8.13.5 Spaces H0s (Gi), HS00(Gi) for polygonal sides Gi . C8-class, 1 = i = N [Seite 689]
12.14 - 8.14 Trace results in Sobolev spaces on O . Rn [Seite 689]
12.14.1 - 8.14.1 Trace results in Hm(Rn+) [Seite 690]
12.14.2 - 8.14.2 Trace results in Hm(O) with bounded domain O . Rn [Seite 692]
12.14.3 - 8.14.3 Trace results in Ws,p-spaces [Seite 708]
12.14.4 - 8.14.4 Trace results for polygonal domains O . R2 [Seite 710]
12.14.5 - 8.14.5 Trace results for bounded domains with curvilinear polygonal boundary G in Rn [Seite 723]
12.14.6 - 8.14.6 Traces of normal components in Lp (div [Seite 724]
12.14.7 - 8.14.7 Trace theorems based on Green's formula [Seite 729]
12.14.8 - 8.14.8 Traces on G0 . G [Seite 748]
13 - 9 Vector-valued distributions [Seite 750]
13.1 - 9.1 Motivation [Seite 750]
13.2 - 9.2 Vector-valued functions [Seite 750]
13.3 - 9.3 Spaces of vector-valued functions [Seite 753]
13.4 - 9.4 Vector-valued distributions [Seite 756]
13.5 - 9.5 Derivatives of vector-valued distributions [Seite 761]
13.6 - 9.6 Applications [Seite 762]
13.6.1 - 9.6.1 Space E(0, T [Seite 763]
13.6.2 - 9.6.2 Hilbert space W1 (0, T [Seite 763]
13.6.3 - 9.6.3 Hilbert space W2 (0, T [Seite 766]
13.6.4 - 9.6.4 Green's formula [Seite 767]
14 - A Functional analysis (basic results) [Seite 769]
14.1 - A.0 Preliminary results [Seite 769]
14.1.1 - A.0.1 An important result on logical implication (.) and non-implication (.) [Seite 769]
14.1.2 - A.0.2 Supremum (l.u.b.) and infimum (g.l.b.) [Seite 770]
14.1.3 - A.0.3 Metric spaces and important results therein [Seite 770]
14.1.4 - A.0.4 Important subsets of a metric space X = (X, d) [Seite 773]
14.1.5 - A.0.5 Compact sets in Rn with the usual metric d2 [Seite 775]
14.1.6 - A.0.6 Elementary properties of functions of real variables [Seite 776]
14.1.7 - A.0.7 Limit of a function at a cluster point x0 . Rn [Seite 776]
14.1.8 - A.0.8 Limit superior and limit inferior of a sequence in R [Seite 777]
14.1.9 - A.0.9 Pointwise and uniform convergence of sequences of functions [Seite 778]
14.1.10 - A.0.10 Continuity and uniform continuity of f . F (O) [Seite 778]
14.2 - A.1 Important properties of continuous functions [Seite 779]
14.2.1 - A.1.1 Some remarkable properties on compact sets in Rn [Seite 779]
14.2.2 - A.1.2 C80(O)-partition of unity on compact set K .. O . Rn [Seite 779]
14.2.3 - A.1.3 Continuous extension theorems [Seite 779]
14.3 - A.2 Finite and infinite dimensional linear spaces [Seite 781]
14.3.1 - A.2.1 Linear spaces [Seite 781]
14.3.2 - A.2.2 Linear functionals [Seite 784]
14.3.3 - A.2.3 Linear operators [Seite 785]
14.4 - A.3 Normed linear spaces [Seite 786]
14.4.1 - A.3.1 Semi-norm and norm [Seite 786]
14.4.2 - A.3.2 Closed subspace, dense subspace, Banach space and its separability [Seite 788]
14.5 - A.4 Banach spaces of continuous functions [Seite 788]
14.5.1 - A.4.1 Banach spaces C0(O¯), Ck(O¯) [Seite 788]
14.6 - A.5 Banach spaces C0,. (O¯), 0 < . < 1, of Hölder continuous functions [Seite 791]
14.6.1 - A.5.1 Hölder continuity and Lipschitz continuity [Seite 791]
14.6.2 - A.5.2 Hölder space C0,. (O¯) [Seite 792]
14.6.3 - A.5.3 Space Ck,. (O¯), 0 < . < 1 [Seite 792]
14.7 - A.6 Quotient space V/M [Seite 794]
14.8 - A.7 Continuous linear functionals on normed linear spaces [Seite 794]
14.8.1 - A.7.1 Space V' [Seite 794]
14.8.2 - A.7.2 Hahn-Banach extension of linear functionals in analytic form [Seite 795]
14.8.3 - A.7.3 Consequences of the Hahn-Banach theorem in normed linear spaces [Seite 796]
14.9 - A.8 Continuous linear operators on normed linear spaces [Seite 798]
14.9.1 - A.8.1 Space L (V [Seite 798]
14.9.2 - A.8.2 Continuous extension of continuous linear operators by density [Seite 799]
14.9.3 - A.8.3 Isomorphisms and isometric isomorphisms [Seite 800]
14.9.4 - A.8.4 Graph of an operator A . L (V [Seite 800]
14.10 - A.9 Reflexivity of Banach spaces [Seite 801]
14.11 - A.10 Strong, weak and weak-* convergence in Banach space V [Seite 801]
14.11.1 - A.10.1 Strong convergence . [Seite 801]
14.11.2 - A.10.2 Weak convergence . [Seite 802]
14.11.3 - A.10.3 Weak-* convergence.* in Banach space V' [Seite 802]
14.12 - A.11 Compact linear operators in Banach spaces [Seite 802]
14.13 - A.12 Hilbert space V [Seite 803]
14.14 - A.13 Dual space V' of a Hilbert space V, reflexivity of V [Seite 806]
14.15 - A.14 Strong, weak and weak-* convergences in a Hilbert space [Seite 807]
14.16 - A.15 Self-adjoint and unitary operators in Hilbert space V [Seite 807]
14.17 - A.16 Compact linear operators in Hilbert spaces [Seite 807]
15 - B Lp -spaces [Seite 809]
15.1 - B.1 Lebesgue measure µ on Rn [Seite 809]
15.1.1 - B.1.1 Lebesgue-measurable sets in Rn [Seite 809]
15.1.2 - B.1.2 Sets with zero (Lebesgue) measure in Rn [Seite 810]
15.1.3 - B.1.3 Property P holds almost everywhere (a.e.) on O [Seite 813]
15.2 - B.2 Space M(O) of Lebesgue-measurable functions on O [Seite 814]
15.2.1 - B.2.1 Measurable functions and space M(O) [Seite 814]
15.2.2 - B.2.2 Pointwise convergence a.e. on O [Seite 816]
15.3 - B.3 Lebesgue integrals and their important properties [Seite 816]
15.3.1 - B.3.1 Lebesgue integral of a bounded function on bounded domain O [Seite 816]
15.3.2 - B.3.2 Important properties of Lebesgue integrals (Kolmogorov and Fomin [20]) [Seite 818]
15.3.3 - B.3.3 Some important approximation and density results in L1(O)822 [Seite 822]
15.4 - B.4 Spaces Lp(O), 1 = p = 8 [Seite 826]
15.4.1 - B.4.1 Basic properties [Seite 826]
15.4.2 - B.4.2 Dual space (Lp(O))' of Lp(O) for 1 = p = 8 [Seite 832]
15.4.3 - B.4.3 Space L2(O) [Seite 835]
15.4.4 - B.4.4 Some negative properties of L8(O) [Seite 836]
15.4.5 - B.4.5 Some nice properties of L8(O) [Seite 837]
15.4.6 - B.4.6 Space Lp loc(O) inclusion results [Seite 837]
16 - C Open cover and partition of unity [Seite 841]
16.1 - C.1 C80(O)-partition of unity theorem for compact sets [Seite 841]
17 - D Boundary geometry [Seite 846]
17.1 - D.1 Boundary geometry [Seite 846]
17.1.1 - D.1.1 Locally one-sided and two-sided bounded domains O [Seite 846]
17.1.2 - D.1.2 Star-shaped domain O [Seite 846]
17.1.3 - D.1.3 Cone property and uniform cone property [Seite 847]
17.1.4 - D.1.4 Segment property [Seite 849]
17.2 - D.2 Continuity and differential properties of a boundary [Seite 850]
17.2.1 - D.2.1 Continuity and differential properties [Seite 850]
17.2.2 - D.2.2 Open cover {Gr}Nr = 1 of G, local coordinate systems {.ri}ni = 1 and mappings {.r}Nr = 1 [Seite 851]
17.2.3 - D.2.3 Properties of the mappings .r: Rn-1 . R, 1 = r = N [Seite 852]
17.3 - D.3 Alternative definition of locally one-sided domain [Seite 854]
17.4 - D.4 Alternative definition of continuity and differential properties of O as a manifold in Rn [Seite 855]
17.5 - D.5 Atlas/local charts of G [Seite 856]
18 - Bibliography [Seite 857]
19 - Index [Seite 861]