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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!

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Clearly i Know that $∇^2\phi A$ is harmonic so it follow that $\phi A$ satisfies the maximum principle does that implies that $\phi$ satisfies the maximum? I think it do right? – Porufes Nov 28 '12 at 22:13
Isn't $\log\left(\frac{1}{r}\right)$ harmonic ($r\neq 0$)? – Pragabhava Nov 30 '12 at 19:45

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