What are your favourite examples in PDE? Which examples would you want to share with students in their first PDE course?

Question

I am going to be a TA for a course on partial differential equations in the coming semester. I recall in my undergraduate years taking a course on PDE, following Evans' book. One of the key components of the course was the theory of distributions. Unfortunately, the subject was treated very abstractly, and few examples were given, let alone treated with great detail.

I'm sure that I'm not alone in my passion for a bank of examples. As a consequence, I'd be interested to know of your favourite examples that can appear in a course of this nature.

Answer 1

I should mention that I'm not proficient in PDEs, so I can only give my perspective as a student who has taken an introductory (non rigorous) course in PDEs (but having much more real/functional analysis background).

1. The very first example of distributions is of course given by locally Lebesgue integrable functions: for any open $U\subset\Bbb{R}^n$, we have the canonical injection $L^1_{\text{loc}}(U)\to \mathcal{D}'(U)$ given by integration against test functions. These of course include all continuous functions, and much more.

2. Next, we have the dirac point masses, and also analogues for submanifolds. Given a point $p\in\Bbb{R}^n$, we can define $\delta_p$ by setting $\langle\delta_p,\phi\rangle:=\phi(p)$ (for $\phi\in C^{\infty}(\Bbb{R}^n)$). This is of course a very basic distribution to introduce because it appears very naturally when talking about fundamental solutions of PDEs. Along these lines, if $M\subset\Bbb{R}^n$ is any embedded submanifold, we can define $\delta_M$ by setting $\langle \delta_M,\phi \rangle:=\int_M\phi(x)\,dV(x)$, where $dV$ is the volume element of $M$ (induced by the pullback of Riemannian metric of $\Bbb{R}^n$ to $M$). A very simple example is of course to let $M=S_r(p)$ be a sphere in $\Bbb{R}^n$ of radius $r$, center $p$. This example might be relevant if you're discussing the fundamental solution of the wave equation in $\Bbb{R}^3$.

3. Elaborating slightly on (1), some important $L^1_{\text{loc}}$ functions are of course the fundamental solutions for the Laplace equation: \begin{align} L(x)&= \begin{cases} \frac{-1}{(n-2)A_n}\frac{1}{|x|^{n-2}}&\text{if $n\geq 3$}\\\\ \frac{1}{2\pi}\log|x|&\text{if $n=2$} \end{cases} \end{align} where $A_n$ is the surface area of the unit sphere in $\Bbb{R}^n$. Also, the fundamental solution of the heat equation: \begin{align} H(x,t)&=\begin{cases} \frac{1}{(4\pi t)^{n/2}}e^{-\frac{|x|^2}{4t}}&\quad \text{if $t>0$}\\ 0&\quad \text{if $t\leq 0$} \end{cases} \end{align} Clearly, the Laplace and Heat equations are very important, so it makes sense to talk about these functions (which are smooth at most places), and it of course be good to show that indeed their distributional derivatives satisfy $\Delta L=\delta$ and $(\partial_t-\Delta)H=\delta$ (if this isn't already done in lectures). I found it very instructive when my TA went through the calculation for showing $H$ really is the fundamental solution. These calculations made it sink in how derivatives of distributions work.

4. Of course one of the most important operations on (tempered) distributions is the Fourier transform, which is very relevant when dealing with the basic three PDEs (Laplace, Heat, and Wave), so you may want to calculate some Fourier transforms (especially of functions which are not in $L^1(\Bbb{R}^n)$) rigorously.

As a side note, the heat/Weierstrass kernel $H$ above can also be used to give a proof of Weierstrass' approximation theorem (not sure how relevant this may be for your course, but I find this interesting).

Answer 2

I think, while the theory of distributions is indeed very powerful, we do a very bad job at explaining some of its limitations and/or intricacies:

1. While differentiation works really nicely for distributions, other natural operations like multiplication behave not quite as good. For instance the product $\phi \psi$, where $\phi\in C^\infty$ and $\psi\in \mathcal{D}'$ is well-defined, but what about if $\phi\in \mathcal{D}'$? This is not an abstract question either; it comes up when trying to define weak solutions to nonlinear equations like $-\Delta u = u^2$ or similar. For example the function $f(x)=1/\sqrt{|x|}\in L^1_{loc}$ is a distribution on $\mathbb{R}$, but it's square $1/x$ is not...

2. ...or is it?! Well, obviously it's not in the usual integral-pairing sense. However, we can indeed define something that resembles it in the following sense: There exists $\phi\in \mathcal{D}'$ such that $x\cdot \phi =1$ in $\mathcal{D}'$; the distribution $\phi$ is called the principal value of $1/x$, denoted $\text{p.v.}\frac{1}{x}$, and it has other nice properties, such as $(\ln |x| )' = \text{p.v.}\frac{1}{x}$ as distributions. In fact, you can define $|x|^z$ for all complex $z$ as distributions, but it's not in the usual way (rather it's done through analytic continuation). Moral of the story is: Beware of applying nonlinear operations to distributions!

3. Distributional operations might not agree with the "classical" counterparts (in a similar spirit to the second point). For instance, if $f:[0,1]\to \mathbb{R}$ is the Cantor function, then the pointwise a.e. derivative of $f$ is zero, but the distributional derivative (after all $f\in L^1_{loc}$) is the Cantor measure $\mu$. On the other hand, not all is lost, since the fact that the a.e. derivative vanishes is simply saying that $\mu$ is singular with respect to Lebesgue measure on $[0,1]$ (indeed it's supported on the Cantor set).

4. One thing to keep in mind is that, although you'll have trouble finding something that's not a distribution in the wild, some times you have to justify why your object is indeed a distribution! This is related to the first point, when you apply algebraic operations to your distributions.

Answer 3

I just want to mention some good books that deal with distribution theory

• Leoni's A First Course in Sobolev Spaces, see chapter 10.
• Leib & Loss's Analysis, see chapter 6 for distributions. They also use them to a lot of examples through the text involving PDE.
• Grubb's Distributions and Operators, see chapters 3 and 5.

I also mention some basic examples/exercises:

• Show that if $u \in \mathcal{D}'(\mathbb{R})$ is such that $u'=0$, then $u=u_{c}$, where $c \in \mathbb{R}$ is a constant.
• Show that $u(x)=e^{-|x|}$ satisfies $$-u_{xx}+u=2\delta_{0}$$ in the distribution sense.
• Similar to the previous one, you can show that $\frac{e^{-|x|}}{4 \pi |x|}$ satisfies $$-\Delta u+u=\delta_{0} \quad \text{in }\mathbb{R}^{3},$$ in the distribution sense.
• Let $f\in C^{1}(\mathbb{R} \setminus \{x_{0}\})$. Assume that the following limits exist: $$f(x_{0}^{-}):=\lim_{x \to x_{0}^{-}}f(x) \quad \text{y} \quad f(x_{0}^{+}):=\lim_{x \to x_{0}^{+}} f(x). $$ Show that $u_{f}'=u_{f'}+(f(x_{0}^{+})-f(x_{0}^{-}))\delta_{x_{0}}$.