# Math 547: Partial Differential Equations 1

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## Contents

## Chris Grant's Proposed Core Topics for Math 547/548

- General Cauchy problem
- Cauchy-Kovalevskaya Theorem
- Lewy Example

- Method of characteristics for first-order equations
- Semilinear case
- Quasilinear case
- General case

- Quasilinear systems of conservation laws on a line
- Riemann problem
- Rankine-Hugoniot jump condition
- Entropy condition
- Shocks
- Rarefaction waves

- Classification of general second-order equations
- Canonical forms for semilinear second-order equations
- Classical theory for the canonical second-order linear equations on
**R**^{n}- 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
- Hadamard’s method of descent
- 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

- Laplace's equation