Third (delayed) Problem set for 18.155
Due October 17 in class or 2-174.

These are Problems 24 - 30 of the notes on the web.

Problem 1   Prove the continuity in the mean of $L^2$ functions on $\mathbb{R}^n,$ that

$$\sup_{\vert t\vert<\epsilon}\int_{\mathbb{R}^n}\vert u(x+t)-u(x)\vert^2dx\to0 \epsilon \to0.$$

You will probably have to go back to first principles to do this. Show that it is enough to assume $u \geq 0$ has compact support. Then show it is enough to assume that $u$ is a simple, and integrable, function. Finally look at the definition of Lebesgue measure and show that if $E \subset \mathbb{R}^n$ is Borel and has finite Lebesgue measure then

$$\lim_{\left\vert t \right\vert \to \infty} \mu (E \backslash (E+t)) =0$$

where $\mu =$ Lebesgue measure and

$$E+t = \left\{ p \in \mathbb{R}^n ; p' +t , p' \in E \right\} .$$

Problem 2   Prove Leibniz' formula

$$D^{\alpha}_x (\varphi \psi) = \sum_{\beta \leq \alpha} \binom{\alpha}{\beta} D^{\beta}_x \varphi \cdot D^{\alpha -\beta}_x \psi$$

for any $\mathcal{C}^\infty$ functions and $\varphi$ and $\psi$. Here $\alpha$ and $\beta$ are multiindices, $\beta \leq \alpha$ means $\beta_j \leq \alpha_j$ for each $j$ and

$$\binom{\alpha}{\beta} = \prod_j \binom{\alpha_j}{\beta_j}.$$

I suggest induction!

Problem 3   Prove the generalization of a result from class that $u\in\mathcal{S}'(\mathbb{R}^n)$, $\operatorname{supp} (u) \subset \left\{ 0 \right\}$ (which means that $u(\phi)=0$ for all $\phi \in\mathcal{S}(\mathbb{R}^n)$ such that $\phi =0$ in $\vert x\vert<\epsilon$ for some $\epsilon >0)$ implies there are constants $c_\alpha$ for $\vert\alpha\vert \leq m$, for some $m$, such that

$$u= \sum_{\left\vert \alpha \right\vert \leq m} c_{\alpha} {D^{\alpha}} \delta.$$

Hint This is not so easy! I would be happy if you can show that $u \in M(\mathbb{R}^n)$, $\operatorname{supp} u \subset\{ 0\}$ implies $u= c \delta.$ To see this, you can show that

$$\begin{multline*} \varphi \in \mathcal{S} (\mathbb{R}^n), \varphi (0) = 0 \\... ...hi_j - \varphi \right\vert \to 0 \hbox{ as } j \to \infty . \end{multline*}$$

To prove the general case you need something similar -- that given $m,$ if $\varphi \in \mathcal{S} (\mathbb{R}^n)$ and $D^{\alpha}_x \varphi (0) =0$ for $\left\vert \alpha \right\vert \leq m$ then $\exists \varphi_j \in \mathcal{S} (\mathbb{R}^n)$, $\varphi_j =0$ in $\left\vert x \right\vert \leq \epsilon_j , \epsilon_j \downarrow 0$ such that $\varphi_j \to \varphi$ in the $\mathcal{C}^m$ norm.

Problem 4   If $m\in\mathbb{N}$, $m' >0$ show that $u \in H^m (\mathbb{R}^n)$ and ${D^{\alpha}} u \in H^{m'} (\mathbb{R}^n)$ for all $\left\vert \alpha \right\vert \leq m$ implies $u \in H^{m+m'} (\mathbb{R}^n)$. Is the converse true?

Problem 5   Show that every element $u \in L^2 (\mathbb{R}^n)$ can be written as a sum

$$u= u_0 + \sum^n_{j=1} D_j u_j , u_j \in H^1 (\mathbb{R}^n) , j=0 , \ldots , n .$$

Problem 6   Consider for $n=1$, the locally integrable function (the Heaviside function),

$$H(x) = \left\{ \begin{array}{cl} 0 & x \leq 0 \ 1 & x>0 . \end{array} \right.$$

Show that $D_x H (x) = c \delta$; what is the constant $c$?

Problem 7   For what range of orders $m$ is it true that $\delta \in H^m (\mathbb{R}^n) , \delta ( \varphi ) = \varphi (0)$?