Tutorial sheet 6

Throughout this sheet, assume that $\Omega\subset\mathbb{R}^{n}$ is a bounded domain of class $C^{\infty}$. For a function $u\in L^{1}(\Omega)$, $\overline{u}:=\frac{1}{\textup{Vol}(\Omega)}\int_{\Omega}u$ denotes the average value of $u$.

1. Fix $r>0$, $x_{0}\in\mathbb{R}^{n}$ and $\lambda>0$. For each $u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$, define the rescaled function

   $\begin{aligned} u_{r}^{(x_{0})}:\Omega_{r}^{(x_{0})}&\rightarrow\mathbb{R}\\ x&\mapsto u(x_{0}+rx),\end{aligned}$

   where $\Omega_{r}^{(x_{0})}=r^{-1}\cdot (\Omega - x_{0})$. Show that the pair $(u,\lambda)$ solves the eigenvalue problem

   $\begin{aligned}-\Delta u &=\lambda u\ \textup{on }\Omega\\ \left.u\right|_{\partial\Omega}&=0 \end{aligned}$

   if and only if the pair $(u_{r},r^{2}\lambda)$ solves the eigenvalue problem

   $\begin{aligned}-\Delta u_{r} &=r^{2}\lambda u\ \textup{on }\Omega_{r}^{(x_{0})}\\ \left.u_{r}\right|_{\partial\Omega_{r}^{(x_{0})}}&=0. \end{aligned}$

   (4 marks)

2. Let $u_{0}\in C^{\infty}(\overline\Omega)$ and suppose $u\in C^{\infty}(\overline{\Omega}\cap \left[0,\infty\right[)$ solves the initial-boundary value problem

   $\begin{aligned}\partial_{t}u - \Delta u&=0\ \textup{on }\Omega\times\left]0,\infty\right[\\ u(\cdot,0)&=u_{0}\\ \frac{\partial u }{\partial \nu}&=0\ \textup{on }\partial\Omega\times\left[0,\infty\right[, \end{aligned}$

   $\nu$ denoting the outer unit normal of $\partial\Omega$. Let $\overline{u}(t)$ denote the average value of $u(\cdot,t)$ for each $t\geq 0$.

   a) Show that $\frac{d\overline{u}}{d t}\equiv 0$, i.e. the average value of $u(\cdot,t)$ is constant in $t$.

   b) Show using the Poincaré inequality $\int_{\Omega}(v-\overline{v})^{2}\leq c_{0}\cdot \int_{\Omega}|\D v|^{2}$ for $v\in W^{1,2}(\Omega)$ that the inequality

   $||u(\cdot,t)-\overline{u}_{0}||_{2} \leq e^{-c_{0}^{-1}t}||u_{0}-\overline{u}_{0}||_{2}$

   holds. Deduce that $u(\cdot,t)\xrightarrow{t\rightarrow\infty} \overline{u}_{0}$ in $L^{2}(\Omega)$.

   c) Now suppose $g\in C^{\infty}(\overline{\Omega})$ and $u$ solves the initial-boundary value problem

   $\begin{aligned}\partial_{t}u - \Delta u&=0\ \textup{on }\Omega\times\left]0,\infty\right[\\ u(\cdot,0)&=u_{0}\\ \frac{\partial u }{\partial \nu}(x,t)&=g(x)\ \textup{for }(x,t)\in\partial\Omega\times\left[0,\infty\right[ \end{aligned}$

   and let $w\in C^{\infty}(\overline{\Omega})$ be a solution to the following Neumann boundary-value problem:

   $\begin{aligned}-\Delta w&=0\ \textup{on }\Omega\\ \left.\frac{\partial w}{\partial\nu}\right|_{\partial\Omega}&=\left.g\right|_{\partial\Omega}\end{aligned}$

   Show that $u(\cdot,t)\xrightarrow{t\rightarrow\infty} w - \overline{w} + \overline{u}_{0}$ in $L^{2}(\Omega)$.

   (8 marks)

3. Let $Lu:=-\sum_{i,j=1}^{n}\partial_{i}(a_{ij}\partial_{j}u)$ be an elliptic operator with $a_{ij}\in C^{\infty}(\overline{\Omega})$ and let $u_{0},g\in C^{\infty}(\overline{\Omega})$. Suppose that $u\in C^{\infty}(\overline{\Omega}\times\left[0,\infty\right[)$ solves the initial-boundary value problem

   $\begin{aligned}\partial_{t}u +L u&=0\ \textup{on }\Omega\times\left]0,\infty\right[\\ u(\cdot,0)&=u_{0}\\ u(x,t)&=g(x)\ \textup{for }(x,t)\in\partial\Omega\times\left[0,\infty\right[. \end{aligned}$

   Show that if $w\in C^{\infty}(\overline\Omega)$ solves the boundary-value problem

   $\begin{aligned}Lw&=0\ \textup{on }\Omega\\\left.w\right|_{\partial\Omega}&=\left.g\right|_{\partial\Omega},\end{aligned}$

   then for all $t\geq 0$,

   $||u(\cdot,t)-w||_{2}\leq e^{-\lambda_{1}t}||u_{0}-w||_{2},$

   where $\lambda_{1}>0$ denotes the principal eigenvalue of $L$. Deduce that $u(\cdot,t)\xrightarrow{t\rightarrow\infty} w$ in $L^{2}(\Omega)$.

   (8 marks)

To be submitted by 10 a.m. on Friday January 31.