# Math 647: Theory of Partial Differential Equations 1

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## Contents

## Catalog Information

### Title

Theory of Partial Differential Equations 1.

### Credit Hours

3

### Prerequisite

### Description

## Chris Grant's Proposed Core Topics for Math 647/648

- Linear elliptic operators of order
*n*- Classification
- Strong and weak solutions
- Gårding's inequality
- Existence of weak solutions for the Dirichlet and Neumann problems
- Agmon-Douglis-Nirenberg regularity
- Green's formula

- Fundamental solutions for general linear differential operators
- Green's functions for general linear BVPs
- Dirichlet's Principle for Laplace’s equation in
**R**^{n} - Poisson's Equation
- Newtonian Potential
- Local existence for the Dirichlet Problem with locally Hölder boundary data
- Interior Hölder estimates
- Kellogg's Theorem

- Second-order linear elliptic operators
- Weak Maximum Principle
- Perron's Method
- Uniqueness for the Dirichlet Problem
- Hopf's bondary-point lemma
- Hopf's Strong Maximum Principle
- Alexandroff Maximum Principle
- Gidas-Ni-Nirenberg
- Uniqueness for the Neumann Problem
- Harnack inequality
- Finite difference methods
- Interior regularity
- Schauder estimates
- Moser iteration
- De Giorgi's theorem
- Boundary/Global regularity

- Second-order quasilinear equations in divergence form
- Existence of weak solutions for the Dirichlet problem via the Browder-Minty theorem
- Local-in-time existence for reaction-diffusion IBVPs and systems using the contraction mapping principle

- Abstract evolution equations
- General theory
- Existence and reqularity for parabolic IVPs
- Existence for hyperbolic IVPs

- Viscosity solutions

## Desired Learning Outcomes

### Prerequisites

### Minimal learning outcomes

### Textbooks

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