# Advanced PDE I

## Content

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## Description

This is the first of two Advanced PDE modules taught as part of the MIGSAA CDT programme. It is open to everyone, including non-MIGSAA students, as an SMSTC supplementary module.

Registration is essential!

If you want to take this course, please register straight away otherwise we will not know which video conference sites are needed. Staff and students should log into the SMSTC site, click on My Account and then My SMSTC and follow the instructions there.

**Syllabus:**

1) The model equation $\Delta u-\lambda u=f$ for constants $\lambda>0$

2) Sobolev spaces, generalised and Sobolev derivatives. Motivation and basic properties.

3) Solvability of the model equation when $f\in L_2$.

4) Solvability of elliptic PDEs on the whole space in Sobolev spaces.

5) Equations in divergence form

6) Elliptic Equations on domains. Interior regularity, Boundary regularity

7) Parabolic equations, main examples, maximum principle

8) Parabolic setting and Sobolev spaces

9) Global in time solutions for nonlinear parabolic problems with small initial data

10) Energy estimates

**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.)

### Suggested text book:

N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev spaces, AMS Graduate Studies in Mathematics.

L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics.

### Assessment:

We propose 4 (2+2) sets of homework.