Picard R., McGhee D. Partial Differential Equations: A unified Hilbert Space Approach

Amsterdam: De Gruyter, 2011. - 489p.This book presents a systematic approach to a solution theory for linear partial differential equations developed in a Hilbert space setting based on a Sobolev Lattice structure, a simple extension of the well established notion of a chain (or scale) of Hilbert spaces. The focus on a Hilbert space setting is a highly adaptable and suitable approach providing a more transparent framework for presenting the main issues in the development of a solution theory for partial differential equations. This global point of view is taken by focussing on the issues involved in determining the appropriate functional analytic setting in which a solution theory can naturally be developed. Applications to many areas of mathematical physics are presented. The book aims to be a largely self-contained. Full proofs to all but the most straightforward results are provided. It is therefore highly suitable as a resource for graduate courses and for researchers, who will find new results for particular evolutionary system from mathematical physics.NomenclatureHilber tSpace
Some Construction Principles of Hilbert Spaces
Direct Sums of Hilbert Spaces
Dual Spaces
Tensor Products of Hilbert SpacesSobolev Chains
Sobolev Lattices
Sobolev Lattices from Tensor Products of Sobolev ChainsFirst Steps Towards a Solution Theory
The Tarski–Seidenberg Theorem and some Consequences
Regularity Loss (,,)
Classification of Partial Differential Equations
The Classical Classification of Partial Differential Equations
Elliptic, Parabolic, Hyperbolic?
Evolutionary Expressions in Canonical Form
Functions of [xxx]v and Convolutions
Systems and Scalar Equations
Causality
Initial Value Problems
Transport Equation
Acoustics
Thermodynamics
Electrodynamics
Elastodynamics
Fluid Dynamics
Quantum Mechanics
Extension of the Solution Theory to H-∞(Dv)
Helmholtz Equation in R
Helmholtz Equation in R
Wave Equation in R (Method of Descent)
Plane Waves
Linearized Navier–Stokes Equations
Electro- and Magnetostatics
Force-free Magnetic Fields
Beltrami Field Expansions
Convolutions in H-∞(Dv)H, v∈Rn+
An Integral Representation of the Solution of the Transport Equation
Potentials, Single and Double Layers in R
Electro- and Magnetostatics (Biot–Savart’s Law)
Potential Theory in R
Cauchy’s Integral Formula
Integral Representations of Solutions of the Helmholtz Equation in R
Retarded Potentials
Integral Representations of Solutions of the Time-Harmonic Maxwell EquationsPolynomials of Commuting Operators
Polynomials of Commuting, Selfadjoint Operators
Classification of Operator Polynomials with Time Differentiation
Causality of Evolutionary Problems
Abstract Initial Value Problems
Systems and Scalar Equations
First-Order-in-Time Evolution Equations in Sobolev LatticesThe Selfadjoint Laplace Operator
Bounded Perturbations
Relatively Bounded Perturbations (the Coulomb Potential)
Heat Equation
TheSelfadjointOperatorCase
Prescribed Dirichlet and Neumann Boundary Data
Transmission Initial Boundary Value Problem
Stefan Boundary Condition
LowerOrderPerturbations
Acoustics
Dirichlet and Neumann Boundary Condition
Wave Equation
ReversibleHeatTransport
Electrodynamics
The Electric Boundary Condition
Some Decomposition Results
TheExtendedMaxwellSystem
The Vectorial Wave Equation for the Electromagnetic Field
Elastodynamics
The Rigid Boundary Condition
Free Boundary Condition
Shear andPressureWaves
Plate Dynamics
Thermo-ElasticityA "Royal Road" to Initial Boundary Value Problems
A Class of Evolutionary Material Laws
The Shape of Evolutionary Problems with Material Laws
Some Special Cases
Material Laws via Differential Equations
CoupledSystems
Initial Value Problems
MemoryProblems
Some Applications
ReversibleHeatTransfer
Models of Thermoelasticity
Thermo-Piezo-Electro-Magnetism