Ebook: Wave Factorization of Elliptic Symbols: Theory and Applications: Introduction to the Theory of Boundary Value Problems in Non-Smooth Domains

To summarize briefly, this book is devoted to an exposition of the foundations of pseudo differential equations theory in non-smooth domains. The elements of such a theory already exist in the literature and can be found in such papers and monographs as [90,95,96,109,115,131,132,134,135,136,146, 163,165,169,170,182,184,214-218]. In this book, we will employ a theory that is based on quite different principles than those used previously. However, precisely one of the standard principles is left without change, the "freezing of coefficients" principle. The first main difference in our exposition begins at the point when the "model problem" appears. Such a model problem for differential equations and differential boundary conditions was first studied in a fundamental paper of V. A. Kondrat'ev [134]. Here also the second main difference appears, in that we consider an already given boundary value problem. In some transformations this boundary value problem was reduced to a boundary value problem with a parameter . - in a domain with smooth boundary, followed by application of the earlier results of M. S. Agranovich and M. I. Vishik. In this context some operator-function R('-) appears, and its poles prevent invertibility; iffor differential operators the function is a polynomial on A, then for pseudo differential operators this dependence on . - cannot be defined. Ongoing investigations of different model problems are being carried out with approximately this plan, both for differential and pseudodifferential boundary value problems.

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This monograph is devoted to the development of a new approach to studying elliptic differential and integro-differential (pseudodifferential) equations and their boundary problems in non-smooth domains. This approach is based on a special representation of symbols of elliptic operators called wave factorization. In canonical domains, for example, the angle on a plane or a wedge in space, this yields a general solution, and then leads to the statement of a boundary problem. Wave factorization has also been used to obtain explicit formulas for solving some problems in diffraction and elasticity theory.

Audience: This volume will be of interest to mathematicians, engineers, and physicists whose work involves partial differential equations, integral equations, operator theory, elasticity and viscoelasticity, and electromagnetic theory. It can also be recommended as a text for graduate and postgraduate students for courses in singular integral and pseudodifferential equations.

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This monograph is devoted to the development of a new approach to studying elliptic differential and integro-differential (pseudodifferential) equations and their boundary problems in non-smooth domains. This approach is based on a special representation of symbols of elliptic operators called wave factorization. In canonical domains, for example, the angle on a plane or a wedge in space, this yields a general solution, and then leads to the statement of a boundary problem. Wave factorization has also been used to obtain explicit formulas for solving some problems in diffraction and elasticity theory.

Audience: This volume will be of interest to mathematicians, engineers, and physicists whose work involves partial differential equations, integral equations, operator theory, elasticity and viscoelasticity, and electromagnetic theory. It can also be recommended as a text for graduate and postgraduate students for courses in singular integral and pseudodifferential equations.

Content:
Front Matter....Pages i-ix
Distributions and Their Fourier transforms....Pages 1-7
Multidimensional complex analysis....Pages 7-13
Sobolev-Slobodetskii spaces....Pages 13-17
Pseudodifferential operators and equations in a half-space....Pages 17-26
Wave factorization....Pages 27-36
Diffraction on a quadrant....Pages 36-42
The problem of indentation of a wedge-shaped punch....Pages 42-50
Equations in an infinite plane angle....Pages 51-67
General boundary value problems....Pages 67-83
The Laplacian in a plane infinite angle....Pages 84-105
Problems with potentials....Pages 105-113
Back Matter....Pages 114-176

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This monograph is devoted to the development of a new approach to studying elliptic differential and integro-differential (pseudodifferential) equations and their boundary problems in non-smooth domains. This approach is based on a special representation of symbols of elliptic operators called wave factorization. In canonical domains, for example, the angle on a plane or a wedge in space, this yields a general solution, and then leads to the statement of a boundary problem. Wave factorization has also been used to obtain explicit formulas for solving some problems in diffraction and elasticity theory.

Audience: This volume will be of interest to mathematicians, engineers, and physicists whose work involves partial differential equations, integral equations, operator theory, elasticity and viscoelasticity, and electromagnetic theory. It can also be recommended as a text for graduate and postgraduate students for courses in singular integral and pseudodifferential equations.

Content:
Front Matter....Pages i-ix
Distributions and Their Fourier transforms....Pages 1-7
Multidimensional complex analysis....Pages 7-13
Sobolev-Slobodetskii spaces....Pages 13-17
Pseudodifferential operators and equations in a half-space....Pages 17-26
Wave factorization....Pages 27-36
Diffraction on a quadrant....Pages 36-42
The problem of indentation of a wedge-shaped punch....Pages 42-50
Equations in an infinite plane angle....Pages 51-67
General boundary value problems....Pages 67-83
The Laplacian in a plane infinite angle....Pages 84-105
Problems with potentials....Pages 105-113
Back Matter....Pages 114-176
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