I'm taking a course of particial diferencial equation, I think it was a mistake xD, but now here I'm and I have some troubles with the following problems:
Let $u(x_0,y_0)$ be a point of the boundary of a domain $\Omega$ contained in a circle of radius $R$ with center at $(x_0,y_0)$. Let $u$ be an harmonic function in $\Omega$, and continuous in $\Omega \cup \partial \Omega$ except in $(x_0,y_0)$.
we define $A(x,y)= log \left( { {2R^2}\over{(x-x_0)^2+(y-y_0)^2} } \right)$
Then we write $u$ as :
$u(x,y) = \phi(x,y)A(x,y)+M$
Doing some computacion it follows that:
$0= (∇^2\phi)A+2(\phi_xA_x+\phi_yA_y)+(∇^2A)\phi$
(since $∇^2u=0 { }$ by definition of u)
and since $A$ is harmonic too:
$0= (∇^2\phi)A+2(\phi_xA_x+\phi_yA_y)$
I need to show that $\phi$ satisfies the maximum principle, so I can prove some stuff of the harmonic function. But I dont know if $\phi$ does really satisfies the maximum principle.
(Some one gave me a hint: Green's identities)
Thanks alot for your help!