I am to find a function U, harmonic on the disk [tex] x^2 + y^2 < 6 [/tex] and satisfying [tex] u(x, y) = y + y^2 [/tex] on the disk's boundary. I am not sure where to start. Hints, help, anything?
I would think Cauchy's integral formual would be useful here: you have the value of a function on a boudry and want the value in the interior.
You are trying to solve the Laplace equation on a disk. Try seperation of variables, then break it down to 2 ODE's. Here is a start for you.. You will probably need to solve the PDE in polar coordinates. - harsh
Then is [tex] u(\sqrt{6}, \theta) = \sqrt{6} \sin(\theta) + 6\sin^2(\theta) [/tex] a boundary condition?
Looks right. Make sure you solve the correct PDE, the laplacian in r,theta is not as simple as U_rr and U_theta*theta - harsh
I know. In an earlier problem I had to compute the laplacian in polar. Oh and one more thing, is there anything else I need to know about [tex] \theta [/tex]? Other than [tex] 0 < \theta < 2\pi [/tex] ?
The theta condition that you are going to use, I believe, will be that theta is 2pi periodic. - harsh