12.4E: Ecuación de Laplace en Coordenadas Polares (Ejercicios)
- Page ID
- 114805
Q12.4.1
1. Definir la solución formal de
\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad \rho_0<r<\rho,\quad -\pi\le\theta<\pi,\\[4pt] u(\rho_0,\theta)=f(\theta),\quad u(\rho,\theta)=0,\quad -\pi\le\theta<\pi, \end{array}\nonumber \]
donde\(0<\rho_0<\rho\).
2. Definir la solución formal de
\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad \rho_0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u(\rho_0,\theta)=0,\quad u(\rho,\theta)=f(\theta),\quad 0\le\theta\le\gamma,\\[4pt] u(r,0)=0,\quad u(r,\gamma)=0,\quad \rho_0<r<\rho, \end{array}\nonumber \]
dónde\(0<\gamma<2\pi\) y\(0<\rho_0<\rho\).
3. Definir la solución formal de
\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad \rho_0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u(\rho_0,\theta)=0,\quad u_r(\rho,\theta)=g(\theta),\quad 0\le\theta\le\gamma,\\[4pt] u_\theta(r,0)=0,\quad u_\theta(r,\gamma)=0,\quad \rho_0<r<\rho, \end{array}\nonumber \]
dónde\(0<\gamma<2\pi\) y\(0<\rho_0<\rho\).
4. Definir la solución formal acotada de
\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad 0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u(\rho,\theta)=f(\theta),\quad 0\le\theta\le\gamma,\\[4pt] u_\theta(r,0)=0,\quad u(r,\gamma)=0,\quad 0<r<\rho, \end{array}\nonumber \]
donde\(0<\gamma<2\pi\).
5. Definir la solución formal de
\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad \rho_0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u_r(\rho_0,\theta)=g(\theta),\quad u_r(\rho,\theta)=0,\quad 0\le\theta\le\gamma,\\[4pt] u(r,0)=0,\quad u_\theta(r,\gamma)=0,\quad \rho_0<r<\rho, \end{array}\nonumber \]
dónde\(0<\gamma<2\pi\) y\(0<\rho_0<\rho\).
6. Definir la solución formal acotada de
\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad 0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u(\rho,\theta)=f(\theta),\quad 0\le\theta\le\gamma,\\[4pt] u_\theta(r,0)=0,\quad u_\theta(r,\gamma)=0,\quad 0<r<\rho, \end{array}\nonumber \]
donde\(0<\gamma<2\pi\).
7. Demostrar que el problema de Neumann
\[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad 0<r<\rho,\quad -\pi\le\theta<\pi,\\[4pt] u_r(\rho,\theta)=f(\theta),\quad -\pi\le\theta<\pi \end{array}\nonumber \]
no tiene una solución formal acotada a menos que\(\int_{-\pi}^\pi f(\theta)\,d\theta=0\). En este caso tiene infinitamente muchas soluciones. Encuentra esas soluciones.