Appendix

Literature

This script derives from the script of Prof Martin Schmidt and I am very thankful to have had such a strong base from which to work.

In PDEs there are two gospels:

• Lawrence C. Evans (1998). Partial Differential Equations. isbn: 978-0-8218-0772-9
• David Gilbarg and Neil S. Trudinger (2001). Elliptic Partial Differential Equations of Second Order. isbn: 978-3-540-41160-4

The later is focused on general elliptic theory, which is not the focus in this course. So for the student who want to dig deeper I would recommend Evans.

Overall, our course is most similar to:

• Qing Han (2011). A Basic Course in Partial Differential Equations. isbn: 978-0-8218-5255-2

There are also a number of other sources that might be a useful supplement for particular sections of the script. We have mentioned with respect to distributions:

• Lars Hörmander (1964). Linear Partial Differential Operators. isbn: 978-3-662-30724-3

For the material on Fourier analysis I also drew from

• G. I. Eskin (2011). Lectures on Linear Partial Differential Equations. isbn: 978-0-8218-5284-2

As you might have deduced, there are a great many textbooks that cover the material and many of them are good; find one that speaks to you.

Changes for 2024

Here is a list of significant changes to the script in 2024.

• The example of a PDE with no solution is new. It is more direct, but requires tools from Chapter 3, so has been moved from Chapter 2.
• I once again changed the definition of a submanifold. The independence of the integral on parameterisation is now only proved for submanifolds, which simplifies the proof.
• Co-area formula as a consequence of the divergence theorem.
• There is a definition of the spherical mean of a distribution, which is used to explain the weak mean value theorem.
• Green’s functions and heat kernels are only defined for bounded domains $\mathrm{\Omega }$.

Changes for 2023

Here is a list of significant changes to the script in 2023.

• Split out the concept of non-characteristic from the proof of the method of characteristics.
• Defined integrals for regular parameterisations. Changed the definition of submanifolds.
• New approach to the heat equation using Fourier transforms first.
• Prove the maximum principle for the heat equation using local methods (not heat balls).
• Cut wave equation in dimensions $>3$.
• Moved all material on energy methods to the end. Added Weierstrass’ counterexample to Dirichlet principle and limits of weak solutions of the Laplace equation.