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### Topics - Yiyun Liu

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##### HA8 / question 1 (a)-(b)
« on: March 19, 2015, 09:10:40 PM »
1.  Find the solutions that depend only on $r$ of the equation
\begin{equation*}
\Delta u:=u_{xx}+u_{yy}=0.
\end{equation*}
2.  Find the solutions that depend only on $\rho$ of the equation
\begin{equation*}
\Delta u:=u_{xx}+u_{yy}+u_{zz}=0.
\end{equation*}
3.  (bonus) In $n$-dimensional case prove that if $u=u(r)$ with  $r=(x_1^2+x_2^2+\ldots+x_n^2)^{\frac{1}{2}}$ then
\begin{equation}
\Delta u = u_{rr}+ \frac{n-1}{r}u_r=0.
\label{equ-H8.1} \end{equation}
4.  (bonus) In $n$-dimensional case prove ($n\ne 2$) that  $u=u(r)$ satisfies Laplace equation as $x\ne 0$ iff  $u=Ar^{2-n}+B$.

$\begin{array}{l} part(a):\\ \Delta u = {u_{rr}} + \frac{1}{r}{u_r} + \frac{1}{{{r^2}}}{u_{\theta \theta }} = 0\\ \Delta u = {u_{rr}} + \frac{1}{r}{u_r} = 0\\ \frac{\partial }{{\partial r}}(r{u_r}) = {u_r} + r{u_{rr}} = r({u_{rr}} + \frac{1}{r}{u_r}) = 0\\ \frac{\partial }{{\partial r}}(r{u_r}) = C\\ u = D\ln (r) + E\\ \\ part(b):\\ \Delta u = {u_{\rho \rho }} + \frac{2}{\rho }{u_\rho } + \frac{1}{{{\rho ^2}}}({u_{\theta \theta }} + \cot (\theta ){u_\theta } + \frac{1}{{{{\sin }^2}(\theta )}}{u_{\theta \theta }}) = 0\\ \Delta u = {u_{\rho \rho }} + \frac{2}{\rho }{u_\rho } = 0\\ \frac{\partial }{{\partial \rho }}({\rho ^2}{u_\rho }) = 2\rho {u_\rho } + {\rho ^2}{u_{\rho \rho }} = {\rho ^2}({u_{\rho \rho }} + \frac{2}{\rho }{u_\rho }) = 0\\ {\rho ^2}{u_\rho } = C,cons\tan ts\\ u = D\frac{1}{\rho } + E \end{array}$

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