Section B.3 Problem Sheet 3
Due 27 February at 4pm, either in class or in the pigeon hole in 4W.
Exercise 2.5.1 exists to get you thinking about the interior and exterior ball properties, and to try your hand at least once at verifying them. The second part is tricky, but comes with a serious hint. These conditions will appear as hypotheses for many of our results.
Connectedness is another important hypothesis in some of our most important results later on. How to actually use this hypothesis is often non-obvious, and Exercise 2.5.2 gives you some practice with this, again with a pretty serious hint.
Exercise 2.6.2 is an important fact about the Laplacian which we will use several times, and which can be proved by two applications of the chain rule and some careful bookkeeping. This symmetry property of the Laplacian is one of the reasons it is so ubiquitous in applications. A related fact is Exercise 2.6.3, which will definitely not be assigned and is a bit harder.
Exercise 2.7.1 is a very concrete application of the basic ideas in Section 2.6 and Section 2.7. If you are short on time, this is the exercise I would recommend skipping.
We will constantly use Exercise 2.8.1 to interchange suprema over \(\Omega\) with maxima over \(\overline\Omega\text{.}\) This exercise also exists to give you some practice making arguments about maxima and minima using analysis techniques.
Please feel free to email me, or drop by office hours (Wednesdays 1:15–2:05 in 4W 1.12). I am also more than happy to meet one-on-one or in a small group.
