Advanced Partial Differential Equations (Linear Equations)

Syllabus

M645 Introduction

Boundary Value Problems

1 Function Spaces

1a The Mollifier Theorem

2 Basic Facts About Hilbert Space

3 Fourier-Sobolev Spaces

4 Advanced Properties of Sobolev Spaces

5 Intro to Distributions

Problems

6 Existence Proofs for Elliptic BVP's

7 H-Space Duality

8 Weak formulation of Elliptic BVP's

Problems on Weak Formulations

9 Spectral properties of Elliptic Operators

Evolution Equations

H valued Functions of t

Parabolic Equations

Hyperbolic Equations

Linear Operators

Intro to Semigroups

The Hille-Yosida theorem

The Lumer-Phillips theorem

Alternative Development of S/G's

Summary of Sg results

Analytic SemiGroups

Nonlinear IVP's