Tutorial sheet 1

In all of the following, $\Omega\subset\mathbb{R}^{n}$ is open and nonempty.

1. Let $\{u_{j}\}_{j=1}^{\infty}\subset C^{1}(\Omega)$ be a sequence such that for all $j$ and for some $C>0$ independent of $j$,

   $\sup_{\Omega}|u_{j}|+\sum_{i=1}^{n}\sup_{\Omega}|\partial_{i}u_{j}|\leq C.$

   a) Show that $\{u_{j}\}$ is equicontinuous on any $U\Subset\Omega$.
   b) Deduce that a subsequence of $\{u_{j}\}$ converges in $C^{0}(\overline{U})$.

2. Show that the Hölder space $(C^{k+\gamma}(\overline{\Omega}),||\cdot||_{k+\gamma})$ is a Banach space.

3. Suppose that $\Omega$ is connected and $u\in W^{1,1}(\Omega)$ satisfies the equation $\D u\equiv 0$. Show that there exists a constant $C$ such that $u\equiv C$ a.e. on $\Omega$.

4. Show that if $u\in W^{1,1}(]a,b[)$, then there is a function $v:\left]a,b\right[\rightarrow\mathbb{K}$ with $u\equiv v$ a.e. possessing the following properties:

   • $v'$ exists almost everywhere;

   • $v'\in L^{1}(]a,b[)$; and

   • the fundamental theorem of calculus holds, i.e. for all $x,y\in\left]a,b\right[$ with $x<y$,

     $v(y)-v(x)=\int_{x}^{y}v'.$

To be submitted by 10 a.m. on Friday November 8.