Problem set 9, PDE

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\}.

 

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