Syllabus

1) Examples of elliptic equations, maximum principles (strong weak), Hopf's Lemma, comparison principle.

2) Classical solutions, Bernstein's gradient estimate, applications.

3) Schauder estimates.

4) Approximation by smooth functions, Sobolev spaces, embeddings, traces.

5) Weak solutions, Lax-Milgram.

6) Interior regularity, Boundary regularity.

7) Parabolic equations, main examples, maximum principle.

8) Parabolic setting and Sobolev spaces.

9) Energy estimates

10) Global in time solutions for nonlinear parabolic equations with small initial data.

Prerequisites

1) Rigorous multivariable calculus (continuity, differentiability, chain rule, integration)

2) Metric spaces, Banach spaces, Hilbert space, weak/strong convergence

3) Vector calculus, Green's formula, (normal, tangent/vectors, parametrisation of surfaces and curves.)

We propose 4 (2+2) sets of homework

The suggested text book is L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics.

Some material will be from elsewhere.

Notes-1
Notes-2
Notes-3 (Schauder estimates)

Notes-4 (Weak solutions)

Homework 1
Homework 2