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Problem 1

Let $u\in C^2$ be a solution of

$\begin{cases} \Delta u = 0 & \text{in } \R^n\setminus B_R\\ u=0 & \text{on } \partial B_R.\end{cases}$

Then $u\equiv 0$ if

$\displaystyle \lim_{|x|\to\infty}\frac{u(x)}{\log |x|} = 0,\quad n=2;\qquad \lim_{|x|\to\infty} u(x) = 0,\quad n=3.$

Problem 2

Let $\Omega\subset\R^n$ be a bounded $C^1$-domain that satisfies the inner sphere condition, and $f\in C(\Omega)$ bounded. If $u\in C^2(\Omega)\cap C^1(\bar \Omega)$ is a solution of

$\begin{cases} \Delta u = f & \text{in } \Omega\\ u=0 & \text{on } \partial\Omega,\end{cases}$

then

$\displaystyle \sup_{\partial\Omega}\Big|\frac{\partial u}{\partial\nu}\Big| \le C \sup_\Omega|f|,$

where $C>0$ depends only on $n,\Omega$.

Problem 3

1. A polynomial $p\in\mathscr P_k$ if and only if $\sum x_j \partial_j p = kp.$

2. Let $p\in\mathscr P_k$, and consider its orthogonal projection $|x|^2q$ onto the space $|x|^2\mathscr P_{k-2}$ with respect to the inner product seen in class. Then $p - |x|^2q$ is harmonic.

3. If $p\in\mathscr H_k$, then $\dfrac{p(x)}{|x|^{2k+n-2}}$ is harmonic on $\R^n\setminus\{0\}$.