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Hi, I need some help with the following problem:

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 $(x_0,y_0)$. Demonstrate that if $$ \frac{u(x,y)}{\log \left( { {2R^2}\over{(x-x_0)^2+(y-y_0)^2} } \right)} \longrightarrow 0 $$ when $u(x,y)$ tends to $u(x_0,y_0)$, then $u\leq M$ in omega.

Thank you for your help :)

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1 Answer 1

This looks like homework, so here is a hint: apply the maximum principle to the function $v_\epsilon(x,y)=u(x,y)-\epsilon\,\log(\frac{2R^2}{(x-x_0)^2+(y-y_0)^2})$. Then let $\epsilon\to 0$.

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