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Problem set 9, PDE

Problem 1

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

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

Then $latex u\equiv 0$ if

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

Problem 2

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

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


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

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

Problem 3

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

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

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