Bhattacharyya Distributions
1. Auflage 2012
ISBN: 978-3-11-026929-1
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
Generalized Functions with Applications in Sobolev Spaces
E-Book, Englisch, 872 Seiten
Reihe: De Gruyter Textbook
ISBN: 978-3-11-026929-1
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
This book grew out of a course taught in the Department of Mathematics, Indian Institute of Technology, Delhi, which was tailored to the needs of the applied community of mathematicians, engineers, physicists etc., who were interested in studying the problems of mathematical physics in general and their approximate solutions on computer in particular. Almost all topics which will be essential for the study of Sobolev spaces and their applications in the elliptic boundary value problems and their finite element approximations are presented. Also many additional topics of interests for specific applied disciplines and engineering, for example, elementary solutions, derivatives of discontinuous functions of several variables, delta-convergent sequences of functions, Fourier series of distributions, convolution system of equations etc. have been included along with many interesting examples.
Zielgruppe
Graduate Students and Young Researchers in Mathematics, Physical Sciences and Engineering; Academic Libraries
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
1;Preface;7
2;How to use this book in courses;21
3;Acknowledgment;25
4;Notation;27
5;1 Schwartz distributions;39
5.1;1.1 Introduction: Dirac’s delta function d(x) and its properties
;39
5.2;1.2 Test space D (O) of Schwartz ;44
5.2.1;1.2.1 Support of a continuous function;44
5.2.2;1.2.2 Space D (O)
;47
5.2.3;1.2.3 Space Dm(O
);51
5.2.4;1.2.4 Space DK (O)
;51
5.2.5;1.2.5 Properties of D (O)
;52
5.3;1.3 Space D'(O) of (Schwartz) distributions;63
5.3.1;1.3.1 Algebraic dual space D*(O);63
5.3.2;1.3.2 Distributions and the space D'(O) of distributions on O;64
5.3.3;1.3.3 Characterization, order and extension of a distribution;65
5.3.4;1.3.4 Examples of distributions;67
5.3.5;1.3.5 Distribution defined on test space D(O) of complex-valued functions ;78
5.4;1.4 Some more examples of interesting distributions;79
5.5;1.5 Multiplication of distributions by C8-functions
;89
5.6;1.6 Problem of division of distributions;92
5.7;1.7 Even, odd and positive distributions;95
5.8;1.8 Convergence of sequences of distributions in D'(O);97
5.9;1.9 Convergence of series of distributions in D'(O)
;105
5.10;1.10 Images of distributions due to change of variables, homogeneous, invariant, spherically symmetric, constant distributions;106
5.10.1;1.10.1 Periodic distributions;113
5.11;1.11 Physical distributions versus mathematical distributions;122
5.11.1;1.11.1 Physical interpretation of mathematical distributions;122
5.11.2;1.11.2 Load intensity;123
5.11.3;1.11.3 Electrical charge distribution;126
5.11.4;1.11.4 Simple layer and double layer distributions;128
5.11.5;1.11.5 Relation with probability distribution [7];132
6;2 Differentiation of distributions and application of distributional derivatives;134
6.1;2.1 Introduction: an integral definition of derivatives of C1-functions;134
6.2;2.2 Derivatives of distributions;138
6.2.1;2.2.1 Higher-order derivatives of distributions T;139
6.3;2.3 Derivatives of functions in the sense of distribution;140
6.4;2.4 Conditions under which the two notions of derivatives of functions coincide;157
6.5;2.5 Derivative of product aT with T . D'(O) and a . C8(O) ;159
6.6;2.6 Problem of division of distribution revisited;163
6.7;2.7 Primitives of a distribution and differential equations;169
6.8;2.8 Properties of distributions whose distributional derivatives are known;179
6.9;2.9 Continuity of differential operator .a : D'(O) . D'(O);180
6.10;2.10 Delta-convergent sequences of functions in D'(Rn);187
6.11;2.11 Term-by-term differentiation of series of distributions;192
6.12;2.12 Convergence of sequences of Ck(O¯) (resp. Ck,.(O¯)) in D'(O);211
6.13;2.13 Convergence of sequences of Lp (O), 1 = p = 8, in D'(O);211
6.14;2.14 Transpose (or formal adjoint) of a linear partial differential operator;213
6.15;2.15 Applications: Sobolev spaces Hm(O),Wm,p(O);215
6.15.1;2.15.1 Sobolev Spaces;215
6.15.2;2.15.2 Space Hm(O);216
6.15.3;2.15.3 Examples of functions belonging to or not belonging to Hm(O);220
6.15.4;2.15.4 Separability of Hm(O);222
6.15.5;2.15.5 Generalized Poincaré inequality in Hm(O);224
6.15.6;2.15.6 Space H0m(O);225
6.15.7;2.15.7 Space H–m(O);229
6.15.8;2.15.8 Quotient space Hm(O)/M;229
6.15.9;2.15.9 Quotient space Hm(O)/Pm-1;231
6.15.10;2.15.10 Other equivalent norms in Hm(O);232
6.15.11;2.15.11 Density results;233
6.15.12;2.15.12 Algebraic inclusions (.) and imbedding (.) results;233
6.15.13;2.15.13 Space Wm,p(O) with m . N, 1 = p < 8;234
6.15.14;2.15.14 Space W0m,p(O), 1 = p < 8;238
6.15.15;2.15.15 Space W-m,q (O);241
6.15.16;2.15.16 Quotient space Wm,p (O)/M for m . N, 1 = p < 8;241
6.15.17;2.15.17 Density results;245
6.15.18;2.15.18 A non-density result;246
6.15.19;2.15.19 Algebraic inclusion . and imbedding (.) results;247
6.15.20;2.15.20 Space Ws,p (O) for arbitrary s . R;247
7;3 Derivatives of piecewise smooth functions, Green’s formula, elementary solutions, applications to Sobolev spaces;249
7.1;3.1 Distributional derivatives of piecewise smooth functions;249
7.1.1;3.1.1 Case of single variable (n = 1);249
7.1.2;3.1.2 Case of two variables (n = 2);253
7.1.3;3.1.3 Case of three variables (n = 3);268
7.2;3.2 Unbounded domain O . Rn, Green’s formula;273
7.3;3.3 Elementary solutions;276
7.4;3.4 Applications;295
8;4 Additional properties of D'(O);301
8.1;4.1 Reflexivity of D(O) and density of D(O) in D'(O);301
8.2;4.2 Continuous imbedding of dual spaces of Banach spaces in D'(O);303
8.3;4.3 Applications: Sobolev spaces H-m(O), W-m,q (O);307
8.3.1;4.3.1 Space W-m,q (O), 1 < q = 8, m . N;311
9;5 Local properties, restrictions, unification principle, space E'(Rn) of distributions with compact support;318
9.1;5.1 Null distribution in an open set;318
9.2;5.2 Equality of distributions in an open set;318
9.3;5.3 Restriction of a distribution to an open set;318
9.4;5.4 Unification principle;321
9.5;5.5 Support of a distribution;323
9.6;5.6 Distributions with compact support;324
9.7;5.7 Space E'(Rn) of distributions with compact support;325
9.7.1;5.7.1 Space E'(Rn);325
9.7.2;5.7.2 Space E'(Rn);326
9.8;5.8 Definition of for f . C8 (Rn) and T . D'(Rn) with non-compact support;334
10;6 Convolution of distributions;336
10.1;6.1 Tensor product;336
10.2;6.2 Convolution of functions;341
10.3;6.3 Convolution of two distributions;353
10.4;6.4 Regularization of distributions by convolution;365
10.5;6.5 Approximation of distributions by C8-functions;367
10.6;6.6 Convolution of several distributions;369
10.7;6.7 Derivatives of convolutions, convolution of distributions on a circle G and their Fourier series representations on G;371
10.8;6.8 Applications;387
10.9;6.9 Convolution equations (see also Section 8.7, Chapter 8);402
10.10;6.10 Application of convolutions in electrical circuit analysis and heat flow problems;413
10.10.1;6.10.1 Electric circuit analysis problem [7];413
10.10.2;6.10.2 Excitations and responses defined by several functions or distributions [7];418
11;7 Fourier transforms of functions of L1 (Rn) and S(Rn);421
11.1;7.1 Fourier transforms of integrable functions in L1 (Rn);421
11.2;7.2 Space S(Rn) of infinitely differentiable functions with rapid decay at infinity;443
11.2.1;7.2.1 Space S(Rn);445
11.3;7.3 Continuity of linear mapping from S(Rn) into S(Rn);450
11.4;7.4 Imbedding results;451
11.5;7.5 Density results;453
11.6;7.6 Fourier transform of functions of S(Rn);455
11.7;7.7 Fourier inversion theorem in S(Rn);456
12;8 Fourier transforms of distributions and Sobolev spaces of arbitrary order HS (Rn);461
12.1;8.1 Motivation for a possible definition of the Fourier transform of a distribution;461
12.2;8.2 Space S'(Rn) of tempered distributions;462
12.2.1;8.2.1 Tempered distributions;462
12.2.2;8.2.2 Space S'(Rn);464
12.2.3;8.2.3 Examples of tempered distributions of S'(Rn);464
12.2.4;8.2.4 Convergence of sequences in S'(Rn);467
12.2.5;8.2.5 Derivatives of tempered distributions;470
12.3;8.3 Fourier transform of tempered distributions;473
12.3.1;8.3.1 Fourier transforms of Dirac distributions and their derivatives;476
12.3.2;8.3.2 Inversion theorem for Fourier transforms on S'(Rn);478
12.3.3;8.3.3 Fourier transform of even and odd tempered distributions;479
12.4;8.4 Fourier transform of distributions with compact support;483
12.5;8.5 Fourier transform of convolution of distributions;488
12.5.1;8.5.1 Fourier transforms of convolutions;489
12.6;8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives of tempered distributions;496
12.7;8.7 Fourier transform methods for differential equations and elementary solutions in S'(Rn);514
12.8;8.8 Laplace transform of distributions on R;530
12.8.1;8.8.1 Space D'+;530
12.8.2;8.8.2 Distribution T-1 . D'+ (see also convolution algebra A = D'+ (6.9.15b));534
12.8.3;8.8.3 Inverse L-1 of Laplace transform L;535
12.9;8.9 Applications;540
12.9.1;8.9.1 Sobolev spaces Hs (Rn);540
12.9.2;8.9.2 Imbedding result;541
12.9.3;8.9.3 Sobolev spaces Hm(Rn) of integral order m on Rn;545
12.9.4;8.9.4 Sobolev’s Imbedding Theorem (see also imbedding results in Section 8.12);550
12.9.5;8.9.5 Imbedding result: S(Rn) . HS (Rn);559
12.9.6;8.9.6 Density results HS (Rn);560
12.9.7;8.9.7 Dual space (Hs (Rn))';561
12.9.8;8.9.8 Trace properties of elements of Hs (Rn);564
12.10;8.10 Sobolev spaces on O . Rn revisited;584
12.10.1;8.10.1 Space Hs (O¯) with s . R, O . Rn;584
12.10.2;8.10.2 m-extension property of O;588
12.10.3;8.10.3 m-extension property of R+n;596
12.10.4;8.10.4 m-extension property of Cm -regular domains O;607
12.10.5;8.10.5 Space Hs (O) with s . R+, O . Rn;611
12.10.6;8.10.6 Density results in Hs (O);616
12.10.7;8.10.7 Dual space H-s (O);617
12.10.8;8.10.8 Space H0s (O) with s > 0;617
12.10.9;8.10.9 Space H-s (O) with s > 0;618
12.10.10;8.10.10 Space Ws, p (O) for real s > 0 and 1 = p < 8;618
12.10.11;8.10.11 Space Hs00 (O) with s > 0;623
12.10.12;8.10.12 Dual space (H00s(O))' for s > 0;629
12.10.13;8.10.13 Space W00s,p (O) for s > 0, 1 < p < 8;629
12.10.14;8.10.14 Restrictions of distributions in Sobolev spaces;631
12.10.15;8.10.15 Differentiation of distributions in Hs (O) with s . R;636
12.10.16;8.10.16 Differentiation of distributions u . Hs (O¯) with s > 0;639
12.11;8.11 Compactness results in Sobolev spaces;643
12.11.1;8.11.1 Compact imbedding results in Hs(O), Hs0(O) and Hs00(O);654
12.12;8.12 Sobolev’s imbedding results;655
12.12.1;8.12.1 Compact imbedding results;670
12.13;8.13 Sobolev spaces Hs (G), Ws,p (G) on a manifold boundary G;672
12.13.1;8.13.1 Surface integrals on boundary G of bounded O . Rn;672
12.13.2;8.13.2 Alternative definition of Hs(G) with G . Cm-class (resp. C8-class);675
12.13.3;8.13.3 Space Hs (G) (s > 0) with G in Cm-class (resp. C8-class);676
12.13.4;8.13.4 Sobolev spaces on boundary curves G in R2;679
12.13.5;8.13.5 Spaces H0s (Gi), HS00(Gi) for polygonal sides Gi . C8-class, 1 = i = N;689
12.14;8.14 Trace results in Sobolev spaces on O . Rn;689
12.14.1;8.14.1 Trace results in Hm(Rn+);690
12.14.2;8.14.2 Trace results in Hm(O) with bounded domain O . Rn;692
12.14.3;8.14.3 Trace results in Ws,p-spaces;708
12.14.4;8.14.4 Trace results for polygonal domains O . R2;710
12.14.5;8.14.5 Trace results for bounded domains with curvilinear polygonal boundary G in Rn;723
12.14.6;8.14.6 Traces of normal components in Lp (div; O);724
12.14.7;8.14.7 Trace theorems based on Green’s formula;729
12.14.8;8.14.8 Traces on G0 . G;748
13;9 Vector-valued distributions;750
13.1;9.1 Motivation;750
13.2;9.2 Vector-valued functions;750
13.3;9.3 Spaces of vector-valued functions;753
13.4;9.4 Vector-valued distributions;756
13.5;9.5 Derivatives of vector-valued distributions;761
13.6;9.6 Applications;762
13.6.1;9.6.1 Space E(0, T; V, W);763
13.6.2;9.6.2 Hilbert space W1 (0, T; V);763
13.6.3;9.6.3 Hilbert space W2 (0, T; V);766
13.6.4;9.6.4 Green’s formula;767
14;A Functional analysis (basic results);769
14.1;A.0 Preliminary results;769
14.1.1;A.0.1 An important result on logical implication (.) and non-implication (.);769
14.1.2;A.0.2 Supremum (l.u.b.) and infimum (g.l.b.);770
14.1.3;A.0.3 Metric spaces and important results therein;770
14.1.4;A.0.4 Important subsets of a metric space X = (X, d);773
14.1.5;A.0.5 Compact sets in Rn with the usual metric d2;775
14.1.6;A.0.6 Elementary properties of functions of real variables;776
14.1.7;A.0.7 Limit of a function at a cluster point x0 . Rn;776
14.1.8;A.0.8 Limit superior and limit inferior of a sequence in R;777
14.1.9;A.0.9 Pointwise and uniform convergence of sequences of functions;778
14.1.10;A.0.10 Continuity and uniform continuity of f . F (O);778
14.2;A.1 Important properties of continuous functions;779
14.2.1;A.1.1 Some remarkable properties on compact sets in Rn;779
14.2.2;A.1.2 C80(O)-partition of unity on compact set K .. O . Rn;779
14.2.3;A.1.3 Continuous extension theorems;779
14.3;A.2 Finite and infinite dimensional linear spaces;781
14.3.1;A.2.1 Linear spaces;781
14.3.2;A.2.2 Linear functionals;784
14.3.3;A.2.3 Linear operators;785
14.4;A.3 Normed linear spaces;786
14.4.1;A.3.1 Semi-norm and norm;786
14.4.2;A.3.2 Closed subspace, dense subspace, Banach space and its separability;788
14.5;A.4 Banach spaces of continuous functions;788
14.5.1;A.4.1 Banach spaces C0(O¯), Ck(O¯);788
14.6;A.5 Banach spaces C0,. (O¯), 0 < . < 1, of Hölder continuous functions;791
14.6.1;A.5.1 Hölder continuity and Lipschitz continuity;791
14.6.2;A.5.2 Hölder space C0,. (O¯);792
14.6.3;A.5.3 Space Ck,. (O¯), 0 < . < 1;792
14.7;A.6 Quotient space V/M;794
14.8;A.7 Continuous linear functionals on normed linear spaces;794
14.8.1;A.7.1 Space V';794
14.8.2;A.7.2 Hahn-Banach extension of linear functionals in analytic form;795
14.8.3;A.7.3 Consequences of the Hahn-Banach theorem in normed linear spaces;796
14.9;A.8 Continuous linear operators on normed linear spaces;798
14.9.1;A.8.1 Space L (V; W);798
14.9.2;A.8.2 Continuous extension of continuous linear operators by density;799
14.9.3;A.8.3 Isomorphisms and isometric isomorphisms;800
14.9.4;A.8.4 Graph of an operator A . L (V; W) and graph norm;800
14.10;A.9 Reflexivity of Banach spaces;801
14.11;A.10 Strong, weak and weak-* convergence in Banach space V;801
14.11.1;A.10.1 Strong convergence .;801
14.11.2;A.10.2 Weak convergence .;802
14.11.3;A.10.3 Weak-* convergence.* in Banach space V';802
14.12;A.11 Compact linear operators in Banach spaces;802
14.13;A.12 Hilbert space V;803
14.14;A.13 Dual space V' of a Hilbert space V, reflexivity of V;806
14.15;A.14 Strong, weak and weak-* convergences in a Hilbert space;807
14.16;A.15 Self-adjoint and unitary operators in Hilbert space V;807
14.17;A.16 Compact linear operators in Hilbert spaces;807
15;B Lp -spaces;809
15.1;B.1 Lebesgue measure µ on Rn;809
15.1.1;B.1.1 Lebesgue-measurable sets in Rn;809
15.1.2;B.1.2 Sets with zero (Lebesgue) measure in Rn;810
15.1.3;B.1.3 Property P holds almost everywhere (a.e.) on O;813
15.2;B.2 Space M(O) of Lebesgue-measurable functions on O;814
15.2.1;B.2.1 Measurable functions and space M(O);814
15.2.2;B.2.2 Pointwise convergence a.e. on O;816
15.3;B.3 Lebesgue integrals and their important properties;816
15.3.1;B.3.1 Lebesgue integral of a bounded function on bounded domain O;816
15.3.2;B.3.2 Important properties of Lebesgue integrals (Kolmogorov and Fomin [20]);818
15.3.3;B.3.3 Some important approximation and density results in L1(O)822;822
15.4;B.4 Spaces Lp(O), 1 = p = 8;826
15.4.1;B.4.1 Basic properties;826
15.4.2;B.4.2 Dual space (Lp(O))' of Lp(O) for 1 = p = 8;832
15.4.3;B.4.3 Space L2(O);835
15.4.4;B.4.4 Some negative properties of L8(O);836
15.4.5;B.4.5 Some nice properties of L8(O);837
15.4.6;B.4.6 Space Lp loc(O) inclusion results;837
16;C Open cover and partition of unity;841
16.1;C.1 C80(O)-partition of unity theorem for compact sets;841
17;D Boundary geometry;846
17.1;D.1 Boundary geometry;846
17.1.1;D.1.1 Locally one-sided and two-sided bounded domains O;846
17.1.2;D.1.2 Star-shaped domain O;846
17.1.3;D.1.3 Cone property and uniform cone property;847
17.1.4;D.1.4 Segment property;849
17.2;D.2 Continuity and differential properties of a boundary;850
17.2.1;D.2.1 Continuity and differential properties;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;851
17.2.3;D.2.3 Properties of the mappings .r: Rn-1 . R, 1 = r = N;852
17.3;D.3 Alternative definition of locally one-sided domain;854
17.4;D.4 Alternative definition of continuity and differential properties of O as a manifold in Rn;855
17.5;D.5 Atlas/local charts of G;856
18;Bibliography;857
19;Index;861
