Tutorial sheet 2

Let $\rho:\mathbb{R}^{n}\rightarrow [0,\infty[$ be a continuous function such that $\supp \rho\subset \overline{B(0,1)}$ , $\rho(0)>0$ , $\rho(x)=\rho(-x)$ for all $x\in\mathbb{R}^{n}$ , and $\int_{\mathbb{R}^{n}}\rho=1$ . For fixed $\eps>0$ , define $\rho_{\eps}:\mathbb{R}^{n}\rightarrow [0,\infty[$ such that $\rho_{\eps}(x):=\eps^{-n}\rho(\frac{x}{\eps})$ .

a) Show that $\int_{\mathbb{R}^{n}}\rho_{\eps}=1$ and the following holds:

\lim_{\eps\searrow 0} \rho_{\eps}(x) = \begin{cases}\infty,&x=0\\0,&\textup{otherwise}\end{cases}

b) Suppose that $n=1$ , and set $R_{\eps}(x):=\int_{-\infty}^{x}\rho_{\eps}$ for $x\in\mathbb{R}$ . Show that the following holds:

\lim_{\eps\searrow 0}R_{\eps}(x)= \begin{cases}0,&x<0\\ \frac{1}{2},&x=0\\1,&x>0 \end{cases}

Suppose that $u\in W^{1,p}(]a,b[)$ with $p>1$ . Show that there exists a function $v\in C^{0+\alpha}([a,b])$ with $\alpha=1-\frac{1}{p}$ such that $u\equiv v$ a.e. and the inequality

[v]_{\alpha}\leq ||v'||_{p}

holds.

Let $p\in\left[1,n\right[$ , and suppose that there exist a $C>0$ and $q>0$ such that the inequality

||u||_{q}\leq C \cdot ||\D u||_{p}

holds for all $u\in C_{0}^{1}(\mathbb{R}^{n})$ . By considering the rescaled function $u_{\lambda}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ defined by $u_{\lambda}(x):=u(\lambda x)$ for fixed $\lambda>0$ , show that

q=\frac{np}{n-p}.

a) Let $A$ be an $n\times n$ matrix. Show that $|\textup{tr}\ A|\leq \sqrt{n}|A|$ , where $\textup{tr}\ A=\sum_{i=1}^{n}a_{ii}$ and $|A|=\sqrt{\sum_{i,j=1}^{n}|a_{ij}|^{2}}$ .

b) Suppose that $\Omega\subset\mathbb{R}^{n}$ is open and nonempty, and let $u\in C_{0}^{2}(\Omega)$ . Show that for all $\eps>0$ , the interpolation inequality

\int_{\Omega}|\D u|^{2} \leq \eps\int_{\Omega}|\D^{2}u|^{2} + \frac{n}{4\eps}\int_{\Omega}|u|^{2}

holds. Deduce that this inequality holds for $u\in H_{0}^{2,2}(\Omega)$ .