# Math 547: Partial Differential Equations 1

### Title

Partial Differential Equations 1.

3

### Prerequisite

Math 334, 342; or equivalents.

### Recommended(?)

Math 314, 341; or equivalents.

### Description

Methods of analysis for hyperbolic, elliptic, and parabolic equations, including characteristic manifolds, distributions, Green's functions, maximum principles and Fourier analysis.

## Desired Learning Outcomes

### Minimal learning outcomes

1. General Cauchy problem
• Cauchy-Kowalevski Theorem
• Lewy Example
2. Method of characteristics for first-order equations
• Semilinear case
• Quasilinear case
• General case
3. Quasilinear systems of conservation laws on a line
• Riemann problem
• Rankine-Hugoniot jump condition
• Entropy condition
• Shocks
• Rarefaction waves
4. Classification of general second-order equations
5. Canonical forms for semilinear second-order equations
6. Hyperbolic equations
7. Elliptic equations
8. Parabolic equations
9. Classical theory for the canonical second-order linear equations on Rn
• Laplace's equation
• Green's first and second identities
• Mean Value Principle and its converse
• Weak and strong maximum principles
• Uniqueness for the Dirichlet problem
• Poisson integral formula
• Existence for the Dirichlet Problem on a ball
• Fundamental solutions
• Green's functions
• Harnack inequality
• Liouville's Theorem
• Harnack's Convergence Theorem
• Existence for the Dirichlet Problem on domains with regular boundaries and for continuous boundary data
• Interior and exterior sphere conditions
• Wave equation
• Method of spherical means
• Huygen’s Principle
• Conservation of Energy
• Domain of Dependence
• Heat equation
• Fourier transforms
• The heat kernel
• Existence for the IVP
• Weak and strong maximum principles
• Uniqueness for the IBVP

### Textbooks

Possible textbooks for this course include (but are not limited to):