Book:Lawrence C. Evans/Partial Differential Equations

Subject Matter

 * Partial Differential Equations

Contents

 * Preface


 * 1. 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


 * PART I: REPRESENTATION FORMULAS FOR SOLUTIONS


 * 2. Four Important Linear PDE
 * 2.1. Transport equation
 * 2.1.1. Initial-value problem
 * 2.1.2. Nonhomogeneous equation
 * 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. References


 * 3. 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
 * 3.4.4. Riemann's problem
 * 3.4.5. Long time behavior
 * 3.5. Problems
 * 3.6. References


 * 4. Other Ways to Represent Solutions
 * 4.1. Separation of variables
 * 4.2. Similarity solutions
 * 4.2.1. Plane and traveling waves, solitons
 * 4.2.2. Similarity under scaling
 * 4.3. Transform methods
 * 4.3.1. Fourier transform
 * 4.3.2. Laplace transform
 * 4.4. Converting nonlinear into linear PDE
 * 4.4.1. Hopf–Cole transformation
 * 4.4.2. Potential functions
 * 4.4.3. Hodograph and Legendre transforms
 * 4.5. Asymptotics
 * 4.5.1. Singular perturbations
 * 4.5.2. Laplace's method
 * 4.5.3. Geometric optics, stationary phase
 * 4.5.4. Homogenization
 * 4.6. Power series
 * 4.6.1. Noncharacteristic surfaces
 * 4.6.2. Real analytic functions
 * 4.6.3. Cauchy–Kovalevskaya Theorem
 * 4.7. Problems
 * 4.8. References


 * PART II: THEORY FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS


 * 5. Sobolev Spaces
 * 5.1. Hölder spaces
 * 5.2. Sobolev spaces
 * 5.2.1. Weak derivatives
 * 5.2.2. Definition of Sobolev spaces
 * 5.2.3. Elementary properties
 * 5.3. Approximation
 * 5.3.1. Interior approximation by smooth functions
 * 5.3.2. Approximation by smooth functions
 * 5.3.3. Global approximation by smooth functions
 * 5.4. Extensions
 * 5.5. Traces
 * 5.6. Sobolev inequalities
 * 5.6.1. Gagliardo–Nirenberg–Sobolev inequality
 * 5.6.2. Morrey's inequality
 * 5.6.3. General Sobolev inequalities
 * 5.7. Compactness
 * 5.8. Additional topics
 * 5.8.1. Poincaré's inequalities
 * 5.8.2. Difference quotients
 * 5.8.3. Differentiability a.e.
 * 5.8.4. Fourier transform methods
 * 5.9. Other spaces of functions
 * 5.9.1. The space $H^{-1}$
 * 5.9.2. Spaces involving time
 * 5.10. Problems
 * 5.11. References


 * 6. Second-Order Elliptic Equations
 * 6.1. Definitions
 * 6.1.1. Elliptic equations
 * 6.1.2. Weak solutions
 * 6.2. Existence of weak solutions
 * 6.2.1. Lax–Milgram Theorem
 * 6.2.2. Energy estimates
 * 6.2.3. Fredholm alternative
 * 6.3. Regularity
 * 6.3.1. Interior regularity
 * 6.3.2. Boundary regularity
 * 6.4. Maximum principles
 * 6.4.1. Weak maximum principle
 * 6.4.2. Strong maximum principle
 * 6.4.3. Harnack's inequality
 * 6.5. Eigenvalues and eigenfunctions
 * 6.5.1. Eigenvalues of symmetric elliptic operators
 * 6.5.2. Eigenvalues of nonsymmetric elliptic operators
 * 6.6. Problems
 * 6.7. References


 * 7. Linear Evolution Equations
 * 7.1. Second-order parabolic equations
 * 7.1.1. Definitions
 * 7.1.2. Existence of weak solutions
 * 7.1.3. Regularity
 * 7.1.4. Maximum principles
 * 7.2. Second-order hyperbolic equations
 * 7.2.1. Definitions
 * 7.2.2. Existence of weak solutions
 * 7.2.3. Regularity
 * 7.2.4. Propagation of disturbances
 * 7.2.5. Equations in two variables
 * 7.3. Systems of first-order hyperbolic equations
 * 7.3.1. Definitions
 * 7.3.2. Symmetric hyperbolic systems
 * 7.3.3. Systems with constant coefficients
 * 7.4. Semigroup theory
 * 7.4.1. Definitions, elementary properties
 * 7.4.2. Genearting contraction semigroups
 * 7.4.3. Applications
 * 7.5. Problems
 * 7.6. References


 * PART III: THEORY FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS


 * 8. The Calculus of Variations
 * 8.1. Introduction
 * 8.1.1. Basic ideas
 * 8.1.2. First variation, Euler–Lagrange equation
 * 8.1.3. Second variation
 * 8.1.4. Systems
 * 8.2. Existence of minimizers
 * 8.2.1. Coercivity, lower semicontinuity
 * 8.2.2. Convexity
 * 8.2.3. Weak solutions of Euler–Lagragne equation
 * 8.2.4. Systems
 * 8.3. Regularity
 * 8.3.1. Second derivative estimates
 * 8.3.2. Remarks of higher regularity
 * 8.4. Constraints
 * 8.4.1. Nonlinear eigenvalue problems
 * 8.4.2. Unilateral constraints, variational inequalities
 * 8.4.3. Harmonic maps
 * 8.4.4. Incompressibility
 * 8.5. Critical points
 * 8.5.1. Mountain Pass Theorem
 * 8.5.2. Application to semilinear elliptic PDE
 * 8.6. Problems
 * 8.7. References


 * 9. Nonvariational Techniques
 * 9.1. Monotonicity methods
 * 9.2. Fixed point methods
 * 9.2.1. Banach's Fixed Point Theorem
 * 9.2.2. Schauder's, Schaefer's Fixed Point Theorems
 * 9.3. Method of subsolutions and supersolutions
 * 9.4. Nonexistence
 * 9.4.1. Blow-up
 * 9.4.2. Derrick–Pohozaev identity
 * 9.5. Geometric properties of solutions
 * 9.5.1. Star-shaped level sets
 * 9.5.2. Radial symmetry
 * 9.6. Gradient flows
 * 9.6.1. Convex functions on Hilbert spaces
 * 9.6.2. Subdifferentials, nonlinear semigroups
 * 9.6.3. Applications
 * 9.7. Problems
 * 9.8. References


 * 10. Hamilton–Jacobi Equations
 * 10.1. Introduction, viscosity solutions
 * 10.1.1. Definitions
 * 10.1.2. Consistency
 * 10.2. Uniqueness
 * 10.3. Control theory, dynamic programming
 * 10.3.1. Introduction to control theory
 * 10.3.2. Dynamic programming
 * 10.3.3. Hamilton–Jacobi–Bellman equation
 * 10.3.4. Hopf–Lax formula revisited
 * 10.4. Problems
 * 10.5. References


 * 11. Systems of Conservation Laws
 * 11.1. Introduction
 * 11.1.1. Integral solutions
 * 11.1.2. Traveling waves, hyperbolic systems
 * 11.2. Riemann's problem
 * 11.2.1. Simple waves
 * 11.2.2. Rarefaction waves
 * 11.2.3. Shock waves, contact discontinuities
 * 11.2.4. Local solution of Riemann's problem
 * 11.3. Systems of two conservation laws
 * 11.3.1. Riemann invariants
 * 11.3.2. Nonexistence of smooth solutions
 * 11.4. Entropy criteria
 * 11.4.1. Vanishing viscosity, traveling waves
 * 11.4.2. Entropy/entropy flux pairs
 * 11.4.3. Uniqueness for a scalar conservation law
 * 11.5. Problems
 * 11.6. References


 * APPENDICES


 * Appendix A: Notation
 * A.1. Notation for matrices
 * A.2. Geometric notation
 * A.3. Notation for functions
 * A.4. Vector-valued functions
 * A.5. Notation for estimates
 * A.6. Some comments about notation


 * Appendix B: Inequalities
 * B.1. Convex functions
 * B.2. Elementary inequalities


 * Appendix C: Calculus Facts
 * C.1. Boundaries
 * C.2. Gauss–Green Theorem
 * C.3. Polar coordinates, coarea formula
 * C.4. Convolution and smoothing
 * C.5. Inverse Function Theorem
 * C.6. Implicit Function Theorem
 * C.7. Uniform convergence


 * Appendix D: Linear Functional Analysis
 * D.1. Banach spaces
 * D.2. Hilbert spaces
 * D.3. Bounded linear operators
 * D.4. Weak convergence
 * D.5. Compact operators, Fredholm theory
 * D.6. Symmetric operators


 * Appendix E: Measure Theory
 * E.1. Lebesgue measure
 * E.2. Measurable functions and integration
 * E.3. Convergence theorems for integrals
 * E.4. Differentiation
 * E.5. Banach space-valued functions


 * Bibliography
 * Index