# Book:Lawrence C. Evans/Partial Differential Equations

## Lawrence C. Evans: Partial Differential Equations

Published $\text {1998}$, American Mathematics Society

### 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. HÃ¶lder 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
5.8.2. 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
9.6.1. 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
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