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{ \large \textbf{\textsc{Math 5588 -- Homework 2 (Due Thursday January 26)}}}
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In all problems $x=(x_1,x_2,\dots,x_n)\in \R^n$ and $u$ is a real-valued function on $\R^n$, $u:\R^n \to \R$. This aim of this homework is to give you practice with multi-variable calculus.
\begin{enumerate}
\item Let $|x| = \sqrt{x_1^2 + \cdots + x_n^2}.$
\begin{enumerate}
\item Show that for $x\neq 0$
\[\frac{\partial }{\partial x_i} |x| = \frac{x_i}{|x|}.\]
\item Show that for $x\neq 0$
\[\frac{\partial^2 }{\partial x_i\partial x_j} |x| = \frac{\delta_{ij}}{|x|} - \frac{x_ix_j}{|x|^3},\]
where $\delta_{ij}$ is the Kronecker delta defined by
\[\delta_{ij} = \begin{cases}
1,&\text{if } i=j\\
0,&\text{if } i\neq j.
\end{cases}\]
\item Show that for $x\neq 0$
\[\Delta |x| = \frac{n-1}{|x|}.\]
\end{enumerate}
\item Find all real numbers $\alpha$ for which $u(x) = |x|^\alpha$ is a solution of Laplace's equation
\[\Delta u(x) = 0 \ \ \text{ for } x\neq 0.\]
\item Let $1 \leq p \leq \infty$. The $p$-Laplacian is defined by
\[\Delta_p u := \div\left(|\nabla u|^{p-2}\nabla u\right)\]
for $1 \leq p < \infty$, and 
\[\Delta_\infty u := \frac{1}{|\nabla u|^2}\sum_{i=1}^n\sum_{j=1}^n u_{x_ix_j}u_{x_i}u_{x_j}.\]
Notice that $\Delta_2 u = \Delta u$. A function $u$ is called $p$-\emph{harmonic} if $\Delta_p u=0$.  
\begin{enumerate}
\item Show that
\[\Delta_p u =|\nabla u|^{p-2} \left( \Delta u + (p-2)\Delta_\infty u\right). \]
\item Show that
\[\Delta_\infty u = \lim_{p\to\infty} \frac{1}{p}|\nabla u|^{2-p}\Delta_p u.\]
\end{enumerate}
\item Let $1 \leq p \leq \infty$. Find all real numbers $\alpha$ for which the function $u(x) = |x|^\alpha$ is $p$-harmonic away from $x=0$.
\item Let $u,w \in C^2(\bar{U})$ where $U \subset \R^n$ is open and bounded. Show that for $1 \leq p < \infty$
\[\int_U u\,\Delta_p w \, dx = \int_{\partial U} u\,|\nabla w|^{p-2}\frac{\partial w}{\partial \nu} \, dS - \int_U |\nabla w|^{p-2}\nabla u\cdot \nabla w\, dx.\]
[Hint: Use one of the integration by parts formulas in the appendix of the course notes.]
\end{enumerate}

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