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Preface to second editionPreface to first edition1. Introduction 1.1. Partial differential equations 1.2. Examples 1.2.1. Single partial differential equations 1.2.2. Systems of partial differential equations 1.3. Strategies for studying PDE 1.3.1. Well-posed problems, classical solutions 1.3.2. Weak solutions and regularity 1.3.3. Typical difficulties 1.4. Overview 1.5. Problems 1.6. ReferencesPART I: REPRESENTATION FORMULAS FOR SOLUTIONS2. Four Important Linear PDE 2.1. Transport equation 2.1.1. Initial-value problem 2.1.2. Nonhomogeneous problem 2.2. Laplace's equation 2.2.1. Fundamental solution 2.2.2. Mean-value formulas 2.2.3. Properties of harmonic functions 2.2.4. Green's function 2.2.5. Energy methods 2.3. Heat equation 2.3.1. Fundamental solution 2.3.2. Mean-value formula 2.3.3. Properties of solutions 2.3.4. Energy methods 2.4. Wave equation 2.4.1. Solution by spherical means 2.4.2. Nonhomogeneous problem 2.4.3. Energy methods 2.5. Problems 2.6. References3. Nonlinear First-Order PDE 3.1. Complete integrals, envelopes 3.1.1. Complete integrals 3.1.2. New solutions from envelopes 3.2. Characteristics 3.2.1. Derivation of characteristic ODE 3.2.2. Examples 3.2.3. Boundary conditions 3.2.4. Local solution 3.2.5. Applications 3.3. Introduction to Hamilton-Jacobi equations 3.3.1. Calculus of variations, Hamilton's ODE 3.3.2. Legendre transform, Hopf-Lax formula 3.3.3. Weak solutions, uniqueness 3.4. Introduction to conservation laws 3.4.1. Shocks, entropy condition 3.4.2. Lax-Oleinik formula 3.4.3. Weak solutions, uniqueness ¡­¡­ 4. Other Ways to Represent SolutionsPART II: THEORY FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS5. Sobolev Spaces6. Second-Order Elliptic Equations7. Linear Evolution EquationsPART III: THEORY FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS8. The Calculus of Variations9. Nonvariational Techniques10. Hamilton-Jacobi Equations11. Systems of Conservation Laws12. Nonlinear Wave EquationsAPPENDICESAppendix A: NotationAppendix B: InequalitiesAppendix C: CalculusAppendix D: Functional AnalysisAppendix E: Measure TheoryBibliographyIndex