I'm currently working on a problem for my PDEs course. The problem states
Suppose $u(x, y)$ is an electrostatic potential in which $∆u = 0$, defined on the unit disk centered at the origin. Determine if the following statements are consistent or inconsistent. Justify each answer.
- a) $u(x,y) = x^2+y^2+5 $
- b) $u(r=1, θ) = 2 + \sin(θ) $ and $u(r = \frac{1}{2},θ = π) = −1$
- c) $ u(r = 1, θ) = 2 + 4 \sin(θ)$ and $u(r = \frac{1}{2}, θ = π) = \frac{1}{2} $
I simply do not know how to approach this problem.
My attempt at part (a) was $$ \nabla ^2 u = 4 \neq 0 $$ $$ \therefore \text{not consistent} $$
Is this process correct?
Any help would be appreciated, thank you.