MA40203: Theory of Partial Differential Equations

During the semester last year, I added a paragraph was added on treating formulas as functions. This is perhaps the main place where the conventions in this unite differ from those in MA30252. The paragraph is now officially numbered as Notation 2.15.

There is a new sentence in Notation 2.11 making it more explicit how the zeroth-order partial \(\partial^0\) is to be interpreted. I have also added an additional displayed equation below (3.1) to further emphasize the meaning of the zeroth-order term \(c\text{.}\)

I have added a pair of exercises Exercise 2.3.1 and Exercise 3.1.2 on shifting and scaling variables, which are then referenced e.g. at the start of the proof of Lemma 3.7.

For ease of reference, Theorem 4.10 now has two separate equation numbers, (4.11) and (4.12).

The conclusion of Exercise 4.2.2 has been but more firmly into the main narrative as Lemma 4.4.

Part of Theorem 4.6 has been split off into Lemma 4.5 so that it can be more easily reused (e.g. in the proof of Theorem 4.8, and some details have been added to the proof.

As usual, several additional hints were added to exercises during the semester last year, some typos were caught and fixed, and various other small tweaks and improvements were made.

Solutions which have been assigned in problem sheet X will now have (PSX) added to their title rather than an asterisk.

Notation 2.14 now makes explicit the order of operations in expressions like \(\nabla f(2x)\) which involve both a differential operator and a composition. A related remark has been added to Notation 2.55.

The proof of Lemma 2.31 has been rewritten to match the proof given in lectures last year. The hope is that the new argument is a bit more straightforward.

During the semester last year, Figure 2.1 was added. This is a classic method for visualising the terms which appear in the multivariate chain rule.

During the proof of Lemma 5.8 last year, students requested additional details on a compactness argument, leading to the creation of Section D.1.

A new exercise Exercise 2.6.1 has been added. This gives an opportunity to think more about the proof of Lemma 2.31 on inner products between symmetric matrices.

Another new exercise Exercise 2.3.3 has been added on the Kelvin transform. While we do not make use of this tool in the text, it is quite useful, and also offers a more substantial chain rule exercise than Exercise 2.3.2.

Further changes made during the semester will be recorded in Appendix F.

I’ve rephrased the conclusion the weak maximum principle. The version involving ‘positive parts’ has not been particularly popular in previous years. Similarly for parabolic equations in Section 6.2.

Corollary 3.5 on uniqueness is now presented as a corollary of the comparison principle rather than the weak maximum principle.

Lemma 2.48, which used to be an unnumbered remark, has been upgraded to a lemma so that it can be more easily referenced. The distinction between this lemma and Theorem 3.9 is important but can be easy to miss on a first reading.

Over the course of the previous year, several exercises were added, expanded or split up into multiple parts, including Exercise 2.6.3, Exercise 3.1.1, Exercise 3.2.1 and especially Exercise 3.2.4.

During the previous year Section 3.4 and Section 4.6 were also added. These are extended problems at the end of Chapter 3 and Chapter 4, based on the open-ended ‘Question 5’ from the 2021 exam, which was scrubbed from the version available on the Library website.

Further changes made during the semester will be recorded in Appendix F.

To emphasise the conceptual distinction between a differential operator \(L\) and its application to a particular function, such operators are now introduced as, e.g., \(L=a_{ij}\partial_{ij}+b_i\partial_i + c\) whenever possible.

Chapter 1 now opens by defining some notation which is used for the rest of the chapter, resulting in briefer statements of the results. An extended concrete application has been added in Exercise 1.1.2.

Exercise 2.1.1 has been added to Section 2.1 to offer practice with the summation convention, Exercise 2.2.1 has been significantly expanded to offer more practice with the product rule. Last year many students had trouble with the multivariable chain rule, and so this is now recalled more explicitly in the new section Section 2.3, which includes a significantly expanded Exercise 2.3.2.

A series of exercises has been added to Section 2.9. These are unlikely to be officially assigned, but offer some practice with the vector calculus concepts being reviewed and also highlight some connections to elliptic partial differential equations.

Definition 3.6 now defines subsolutions of boundary value problems, and comes with an associated Exercise 3.2.2. This opens the door to stating (but not proving) non-examinable existence results based on the method of sub- and supersolutions.

The hypotheses of Theorem 4.11 have been weakened to that \(u \in C^0(\Omega)\) is required, allowing for a more satisfying statement of Exercise 4.3.1.

All references to strong maximum principles for parabolic equations have been removed.

In addition to the index, there is now a separate Appendix J with a list of notation. Appendix E has been added with automatically generated lists of exercises, definitions, and results. These are most useful in the html output. To make them easier to navigate, I have added ‘titles’ to all of the exercises and to many more of the definitions and results. In the pdf output, solutions have been moved to Appendix I to keep the main text from getting too busy.

I have switched to another standard notation for averages over balls and spheres in Notation 4.9. This is motivated purely by the sub-par html output for the previously used notation.

Many other small changes, including some new exercises.

Further changes made during the semester will be recorded in Appendix F.

I have greatly expanded Chapter 2 on preliminaries to include necessary facts about symmetric matrices, vector calculus, and compactness and diagonalization. I have removed the discussion of \(C^k\) domains, since in the end we only need the interior and exterior ball properties.

Chapter 4 has been partially reworked. In general, I have de-emphasised techniques involving integrals, such as the mean value property, mollifications, and the Poisson formula. Instead, the basic interior gradient estimate is established using the so-called Bernstein technique, and the basic existence result in balls is proved first for polynomials and then extended to general continuous boundary data using the Weierstrass approximation theorem.

The strong maximum principle for parabolic equations has been made non-examinable and relegated to an appendix. Linear and semilinear parabolic equations are now combined in a single chapter.

A large number of additional exercises have been added. These are listed at the end of each section, and include exercises not assigned on any problem sheet.

The lecture notes have been translated to PreTeXt, because I prefer its html output to the alternatives.