An introduction to partial differential equations / Michael Renardy, Robert C. Rogers.

1.Introduction -- 1.1.Basic Mathematical Questions -- 1.2.Elementary Partial Differential Equations -- 2.Characteristics -- 2.1.Classification and Characteristics -- 2.2.The Cauchy-Kovalevskaya Theorem -- 2.3.Holmgren's Uniqueness Theorem -- 3.Conservation Laws and Shocks -- 3.1.Systems in One Space Dimension -- 3.2.Basic Definitions and Hypotheses -- 3.3.Blowup of Smooth Solutions -- 3.4.Weak Solutions -- 3.5.Riemann Problems -- 3.6.Other Selection Criteria -- 4.Maximum Principles -- 4.1.Maximum Principles of Elliptic Problems -- 4.2.An Existence Proof for the Dirichlet Problem -- 4.3.Radial Symmetry -- 4.4.Maximum Principles for Parabolic Equations -- 5.Distributions -- 5.1.Test Functions and Distributions -- 5.2.Derivatives and Integrals -- 5.3.Convolutions and Fundamental Solutions -- 5.4.The Fourier Transform -- 5.5.Green's Functions -- 6.Function Spaces -- 6.1.Banach Spaces and Hilbert Spaces -- 6.2.Bases in Hilbert Spaces -- 6.3.Duality and Weak Convergence -- 6.4.Sobolev Spaces -- 7.Operator Theory -- 7.1.Basic Definitions and Examples -- 7.2.The Open Mapping Theorem -- 7.3.Spectrum and Resolvent -- 7.4.Symmetry and Self-adjointness -- 7.5.Compact Operators -- 7.6.Sturm-Liouville Boundary-Value Problems -- 7.7.The Fredholm Index -- 8.Linear Elliptic Equations -- 8.1.Definitions -- 8.2.Existence and Uniqueness of the Solutions of the Dirichlet Problem -- 8.3.Eigenfunction Expansions -- 8.4.General Linear Elliptic Problems -- 8.5.Interior Regularity -- 8.6.Boundary Regularity -- 9.Nonlinear Elliptic Equations -- 9.1.Perturbation Results -- 9.2.Nonlinear Variational Problems -- 9.3.Nonlinear Operator Theory Methods -- 10.Energy Methods for Evolution Problems -- 10.1.Parabolic Equations -- 10.2.Hyperbolic Evolution Problems -- 11.Semigroup Methods -- 11.1.Semigroups and Infinitesimal Generators -- 11.2.The Hille-Yosida Theorem -- 11.3.Applications to PDEs -- 11.4.Analytic Semigroups.