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simon j distributions 2022

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Textbook in PDF format This book presents a simple and original theory of distributions, both real and vector, adapted to the study of partial differential equations. It deals with value distributions in a Neumann space, that is, in which any Cauchy suite converges, which encompasses the Banach and Fréchet spaces and the same “weak” spaces. Alongside the usual operations – derivation, product, variable change, variable separation, restriction, extension and regularization – Distributions presents a new operation: weighting. This operation produces properties similar to those of convolution for distributions defined in any open space. Emphasis is placed on the extraction of convergent sub-sequences, the existence and study of primitives and the representation by gradient or by derivatives of continuous functions. Constructive methods are used to make these tools accessible to students and engineers. This book is the third of seven volumes dedicated to solving partial dif- ferential equations in physics: Volume 1: Banach, Frechet, Hilbert and Neumann Spaces Volume 2: Continuous Functions Volume 3: Distributions Volume 4: Integration Volume 5: Sobolev Spaces Volume 6: Traces Volume 7: Partial Differential Equations Introduction Notations Semi-Normed Spaces and Function Spaces Semi-normed spaces Comparison of semi-normed spaces Continuous mappings Differentiable functions Spaces Cm(Ω;E), Cmb(Ω;E) and Cmb(Ω;E) Integral of a uniformly continuous function Space of Test Functions Functions with compact support Compactness in their whole of support of functions The space D(Ω) Sequential completeness of D(Ω) Comparison of D(Ω) to various spaces Convergent sequences in D(Ω) Covering by crown-shaped sets and partitions of unity Control of the CmK(Ω)-norms by the semi-norms of D(Ω) Semi-norms that are continuous on all the C∞K(Ω) Space of Distributions The space D'(Ω;E) Characterization of distributions Inclusion of C(Ω;E) into D'(Ω;E) The case where E is not a Neumann space Measures Continuous functions and measures Extraction of Convergent Subsequences Bounded subsets of D'(Ω;E) Convergence in D'(Ω;E) Sequential completeness of D'(Ω;E) Sequential compactness in D'(Ω;E) Change of the space E of values The space E-weak The space D'(Ω;E-weak) and extractability Operations on Distributions Distributions fields Derivatives of a distribution Image under a linear mapping Product with a regular function Change of variables Some particular changes of variables Positive distributions Distributions with values in a product space Restriction, Gluing and Support Restriction Additivity with respect to the domain Local character Localization-extension Gluing Annihilation domain and support Properties of the annihilation domain and support The space D'K(Ω;E) Weighting Weighting by a regular function Regularizing character of the weighting by a regular function Derivatives and support of distributions weighted by a regular weight Continuity of the weighting by a regular function Weighting by a distribution Comparison of the definitions of weighting Continuity of the weighting by a distribution Derivatives and support of a weighted distribution Miscellanous properties of weighting Regularization and Applications Local regularization Properties of local approximations Global regularization Convergence of global approximations Properties of global approximations Commutativity and associativity of weighting Uniform convergence of sequences of distributions Potentials and Singular Functions Surface integral over a sphere Distribution associated with a singular function Derivatives of a distribution associated with a singular function Elementary Newtonian potential Newtonian potential of order n Localized potential Dirac mass as derivatives of continuous functions Heaviside potential Weighting by a singular weight Line Integral of a Continuous Field Line integral along a C1 path Change of variable in a path Line integral along a piecewise C1 path The homotopy invariance theorem Connectedness and simply connectedness Primitives of Functions Primitive of a function field with a zero line integral Tubular flows and concentration theorem The orthogonality theorem for functions Poincaré’s theorem Properties of Primitives of Distributions Representation by derivatives Distribution whose derivatives are zero or continuous Uniqueness of a primitive Locally explicit primitive Continuous primitive mapping Harmonic distributions, distributions with a continuous Laplacian Existence of Primitives Peripheral gluing Reduction to the function case The orthogonality theorem Poincaré’s generalized theorem Current of an incompressible two dimensional field Global versus local primitives Comparison of the existence conditions of a primitive Limits of gradients Distributions of Distributions Characterization Bounded sets Convergent sequences Extraction of convergent subsequences Change of the space of values Distributions of distributions with values in E-weak Separation of Variables Tensor products of test functions Decomposition of test functions on a product of sets The tensorial control theorem Separation of variables The kernel theorem Regrouping of variables Permutation of variables Banach Space Valued Distributions Finite order distributions Weighting of a finite order distribution Finite order distribution as derivatives of continuous functions Finite order distribution as derivative of a single function Distributions in a Banach space as derivatives of functions Non-representability of distributions with values in a Fréchet space Extendability of distributions with values in a Banach space Cancellation of distributions with values in a Banach space Appendix Notation and numbering Semi-normed spaces Continuous mappings, duality Continuous or differentiable functions Integration of uniformly continuous functions Bibliography Index

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