Abstract linear theory
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Notations and Conventions.- 1 Topological Spaces.- 2 Locally Convex Spaces.- 3 Complexifications.- 4 Unbounded Linear Operators.- 5 General Conventions.- I Generators and Interpolation.- 1 Generators of Analytic Semigroups.- 1.1 Properties of Linear Operators.- 1.2 The Class H(E1E0).- 1.3 Perturbation Theorems.- 1.4 Spectral Estimates.- 1.5 Compact Perturbations.- 1.6 Matrix Generators.- 2 Interpolation Functors.- 2.1 Definitions.- 2.2 Interpolation Inequalities.- 2.3 Retractions.- 2.4 Standard Interpolation Functors.- 2.5 Continuous Injections.- 2.6 Duality Properties.- 2.7 Compactness.- 2.8 Reiteration Theorems.- 2.9 Fractional Powers and Interpolation.- 2.10 Semigroups and Interpolation.- 2.11 Admissible Interpolation Functors.- II Cauchy Problems and Evolution Operators.- 1 Linear Cauchy Problems.- 1.1 Holder Spaces.- 1.2 Existence and Regularity Theorems.- 2 Parabolic Evolution Operators.- 2.1 Basic Properties.- 2.2 Determining Integral Equations.- 3 Linear Volterra Integral Equations.- 3.1 Weakly Singular Kernels.- 3.2 Resolvent Kernels.- 3.3 Singular Gronwall Inequalities.- 4 Existence of Evolution Operators.- 4.1 A Class of Parameter Integrals.- 4.2 Semigroup Estimates.- 4.3 Construction of Evolution Operators.- 4.4 The Main Result.- 4.5 Solvability of the Cauchy Problem.- 5 Stability Estimates.- 5.1 Estimates for Evolution Operators.- 5.2 Continuity Properties of Mild Solutions.- 5.3 Holder Estimates.- 5.4 Boundedness of Mild Solutions.- 6 Invariance and Positivity.- 6.1 Yosida Approximations.- 6.2 Approximations of Evolution Operators.- 6.3 Invariance.- 6.4 Orderings and Positivity.- III Maximal Regularity.- 1 General Principles.- 1.1 Sobolev Spaces.- 1.2 Absolutely Continuous Functions.- 1.3 Generalized Solutions.- 1.4 Trace Spaces.- 1.5 Pairs of Maximal Regularity.- 1.6 Stability.- 2 Maximal Holder Regularity.- 2.1 Singular Holder Spaces.- 2.2 Semigroup Estimates.- 2.3 Trace Spaces.- 2.4 Estimates for KA.- 2.5 Maximal Regularity.- 2.6 Nonautonomous Problems.- 3 Maximal Continuous Regularity.- 3.1 Necessary Conditions.- 3.2 Higher Order Interpolation Spaces.- 3.3 Estimates for KA.- 3.4 Maximal Regularity.- 4 Maximal Sobolev Regularity.- 4.1 Temperate Distributions.- 4.2 Fourier Transforms and Convolutions.- 4.3 The Hilbert Transform.- 4.4 UMD Spaces and Fourier Multipliers.- 4.5 Properties of UMD Spaces.- 4.6 Fractional Powers.- 4.7 Bounded Imaginary Powers.- 4.8 Perturbation Theorems.- 4.9 Sums of Closed Operators.- 4.10 Maximal Regularity.- IV Variable Domains.- 1 Higher Regularity.- 1.1 Properties of Differentiable Functions.- 1.2 General Solvability Results for Cauchy Problems.- 1.3 Estimates for Evolution Operators.- 1.4 Evolution Operators on Interpolation Spaces.- 1.5 The Cauchy Problem.- 2 Constant Interpolation Spaces.- 2.1 Semigroup and Convergence Estimates.- 2.2 Assumptions and Consequences.- 2.3 Construction of Evolution Operators.- 2.4 Estimates for Evolution Operators.- 2.5 The Cauchy Problem.- 2.6 Abstract Boundary Value Problems.- 3 Maximal Regularity.- 3.1 Abstract Initial Boundary Value Problems.- 3.2 Isomorphism Theorems.- V Scales of Banach Spaces.- 1 Banach Scales.- 1.1 General Concepts.- 1.2 Power Scales.- 1.3 Extrapolation Spaces.- 1.4 Dual Scales.- 1.5 Interpolation-Extrapolation Scales.- 2 Evolution Equations in Banach Scales.- 2.1 Semigroups in Interpolation-Extrapolation Scales.- 2.2 Parabolic Evolution Equations in Banach Scales.- 2.3 Duality.- 2.4 Approximation Theorems.- 2.5 Final Value Problems.- 2.6 Weak Solutions and Duality.- 2.7 Positivity.- 2.8 General Evolution Equations.- List of Symbols.