Description
Book SynopsisThis book is devoted to the theory of infinite-order linear and nonlinear differential operators with several real arguments and their applications to problems of partial differential equations and numerical analysis.Part I develops the theory of pseudodifferential operators with real analytic symbols, the local representatives of which are linear differential operators of infinite order acting in the spaces of basic and generalized functions based on the duality of the spaces of real analytic functions and functionals. Applications to a variety of problems of PDEs and numerical analysis are given. Part II is devoted to the theory of Sobolev-Orlicz spaces of infinite order and the solvability of nonlinear partial differential equations with arbitrary nonlinearities.
Table of ContentsPart 1 Preliminaries: Convolution of Distributions; Fourier Transforms of Distributions; Entire Functions of Exponential Type that are Bounded on Rn; The Dirichlet Kernels; Markov Type Theorems; The Second Interpolation Method of Bernstein; Orlicz Classes and Orlicz Spaces. Part 2 Pseudodifferential Operators with Real Analytic Symbols: The Space of Test Functions in a Neighbourhood of Zero; The Differential Operators of Infinite Order. Part 3 Applications to Pseudodifferential Equations: Boundary-Value Problems; Multi-Dimensional Integral Equations of the First Kind with Entire Kernel. Part 4 Approximation Methods: Approximating the Symbols by Algebraic Polynomials; Approximating the Symbols by Trigonometric Polynomials; Approximate the Data by Sinc Functions. Part 5 Mollification Methods: The Heat Equation Backwards in Time; The Cauchy Problem for the Laplace Equation; The Noncharacteristic Cauchy Problem for Parabolic Equations. Part 6 Sobolev-Orlicz Spaces of Infinite Order: Limits of the Monotone Sequence of Banach Spaces; The Nontriviality of Sobolov-Orlicz Spaces of Infinite Order; Embeddings of Sobolev-Orlicz Spaces of Infinite Order. Part 7 Nonlinear Differential Equations of Infinite Order: Elliptic Boundary Value Problems of Infinite Order. (Part contents).
