Distributions

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.

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: Diracs 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 o
f 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 Hm(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, Greens 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, Greens 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 Sobolevs Imbedding Theorem (see also imbedding results in Section 8.12);550 12.9.5;8.9.5 Imbedding resu
lt: 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 Sobolevs 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 result
s 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 Greens 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 Greens 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 an
d 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 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