Differential Operators And Related Topics

Presenting research from more than 30 international authorities, this reference provides a complete arsenal of tools and theorems to analyze systems of hyperbolic partial differential equations. The authors investigate a wide variety of problems in areas such as thermodynamics, electromagnetics, fluid dynamics, differential geometry, and topology. Renewing thought in the field of mathematical physics, Hyperbolic Differential Operators defines the notion of pseudosymmetry for matrix symbols of order zero as well as the notion of time function. Surpassing previously published material on the topic, this text is key for researchers and mathematicians specializing in hyperbolic, Schrödinger, Einstein, and partial differential equations; complex analysis; and mathematical physics.

A self-contained and systematic development of an aspect of analysis which deals with the theory of fundamental solutions for differential operators, and their applications to boundary value problems of mathematical physics, applied mathematics, and engineering, with the related computational aspects.

The aim of this third edition is to give an accessible and essentially self-contained account of pseudo-differential operators based on the previous edition. New chapters notwithstanding, the elementary and detailed style of earlier editions is maintained in order to appeal to the largest possible group of readers. The focus of this book is on the global theory of elliptic pseudo-differential operators on Lp(Rn). The main prerequisite for a complete understanding of the book is a basic course in functional analysis up to the level of compact operators. It is an ideal introduction for graduate students in mathematics and mathematicians who aspire to do research in pseudo-differential operators and related topics.

This collection of counter-examples highlights the theory of differential equations and related topics which is now playing an enormously important role in the area of science, engineering and mathematics.

About the Mark Krein International Conference.- Mark Grigorevich Krein (A short biography).- The Seminar on Ship Hydrodynamics, Organized by M.G. Krein.- Review Papers: The Works of M.G. Krein on Eigenfunction Expansion for Selfadjoint Operators and their Applications and Development.- M.G. Krein and the Extension Theory of Symmetric Operators. Theory of Entire Operators.- Works by M.G. Krein on Inverse Problems.- Research Papers: The Spectrum of Periodic Point Perturbations and the Krein Resolvent Formula.- The Periodic Choquard Equation.- On the Best Constant in a Poincare-Sobolev Inequality.- On Solutions of Parabolic Equations from Families of Banach Spaces Dependent on Time.- Canonical Systems on the Line with Rational Spectral Densities: Explicit Formulas.- Oscillations in Systems with Periodic Coefficients and Sector-restricted Nonlinearities.- Differential Operator Matrices of Mixed Order with Periodic Coefficients.- Asymptotics of Generalized Eigenvectors for Unbounded Jacobi Matrices with Power-like Weights, Pauli Matrices Commutation Relations and Cesaro Averaging.- Functional Means, Convolution Operators and Semigroups.- The Inverse Spectral Problem for First Order Systems on the Half Line.- Exact Solution of the Marchenko Equation Relevant to Inverse Scattering on the Line.- An Arbitrary Oriented Crack in the Box Shell.- Homogeneity of a String having Three Unperturbed Spectra.- On the Integro-differential Equation of a Torsion of an Elastic Medium Including a Cylindrical Crack.- Green's Formula and Theorems on Isomorphisms for General Elliptic Problems for Douglis-Nirenberg Elliptic Systems.- Sobolev's Problem in Complete Scale of Banach Spaces.- On the Simple Waves with Profiles in the Form of some Special Functions-Chebyshov-Hermite, Mathiev, Whittaker-in Two-phase Media.- Inverse Spectral Problem Related to the N-wave Equation.- Degenerated Hyperbolic Approximations of the Wave Theory of Elastic Plates.- Elliptic Problems with a Shift in Complete Scales of Sobolev-type Spaces.- On the Extremal Regularization of the Variational Inequalities with Multivalued Operators.- Poly-Fock Spaces.- Diffraction of Longitudinal Shear Waves by a Hollow Thick Circular Cylinder which is Situated in the Elastic Halfspace.- On M.G. Krein's Spectral Shift Function for Canonical Systems of Differential Equations.- Table of Contents of Volume II.

This volume consists of eighteen peer-reviewed papers related to lectures on pseudo-differential operators presented at the meeting of the ISAAC Group in Pseudo-Differential Operators (IGPDO) held at Imperial College London on July 13-18, 2009. Featured in this volume are the analysis, applications and computations of pseudo-differential operators in mathematics, physics and signal analysis. This volume is a useful complement to the volumes “Advances in Pseudo-Differential Operators”, “Pseudo-Differential Operators and Related Topics”, “Modern Trends in Pseudo-Differential Operators”, “New Developments in Pseudo-Differential Operators” and “Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations” published in the same series in, respectively, 2004, 2006, 2007, 2009 and 2010.