12.4E: Laplace's Equation in Polar Coordinates (Exercises)

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Q12.4.1

1. Define the formal solution of

\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad \rho_0<r<\rho,\quad -\pi\le\theta<\pi,\\[4pt] u(\rho_0,\theta)=f(\theta),\quad u(\rho,\theta)=0,\quad -\pi\le\theta<\pi, \end{array}\nonumber \]

where \(0<\rho_0<\rho\).

2. Define the formal solution of

\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad \rho_0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u(\rho_0,\theta)=0,\quad u(\rho,\theta)=f(\theta),\quad 0\le\theta\le\gamma,\\[4pt] u(r,0)=0,\quad u(r,\gamma)=0,\quad \rho_0<r<\rho, \end{array}\nonumber \]

where \(0<\gamma<2\pi\) and \(0<\rho_0<\rho\).

3. Define the formal solution of

\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad \rho_0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u(\rho_0,\theta)=0,\quad u_r(\rho,\theta)=g(\theta),\quad 0\le\theta\le\gamma,\\[4pt] u_\theta(r,0)=0,\quad u_\theta(r,\gamma)=0,\quad \rho_0<r<\rho, \end{array}\nonumber \]

where \(0<\gamma<2\pi\) and \(0<\rho_0<\rho\).

4. Define the bounded formal solution of

\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad 0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u(\rho,\theta)=f(\theta),\quad 0\le\theta\le\gamma,\\[4pt] u_\theta(r,0)=0,\quad u(r,\gamma)=0,\quad 0<r<\rho, \end{array}\nonumber \]

where \(0<\gamma<2\pi\).

5. Define the formal solution of

\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad \rho_0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u_r(\rho_0,\theta)=g(\theta),\quad u_r(\rho,\theta)=0,\quad 0\le\theta\le\gamma,\\[4pt] u(r,0)=0,\quad u_\theta(r,\gamma)=0,\quad \rho_0<r<\rho, \end{array}\nonumber \]

where \(0<\gamma<2\pi\) and \(0<\rho_0<\rho\).

6. Define the bounded formal solution of

\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad 0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u(\rho,\theta)=f(\theta),\quad 0\le\theta\le\gamma,\\[4pt] u_\theta(r,0)=0,\quad u_\theta(r,\gamma)=0,\quad 0<r<\rho, \end{array}\nonumber \]

where \(0<\gamma<2\pi\).

7. Show that the Neumann problem

\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad 0<r<\rho,\quad -\pi\le\theta<\pi,\\[4pt] u_r(\rho,\theta)=f(\theta),\quad -\pi\le\theta<\pi \end{array}\nonumber \]

has no bounded formal solution unless \(\int_{-\pi}^\pi f(\theta)\,d\theta=0\). In this case it has infinitely many solutions. Find those solutions.