I've posted this before, but I was unable to solve this...
Setting :
- U is a bounded Lipschitz domain in the complex plane
- Consider the following classical Dirichlet problem for the Laplace operator: $$\begin{align} \Delta{}u&=0\ \ in\ \ \ U\\ u&=f\ \ \partial U.\end{align} $$
- $F(x)=-\frac{1}{2\pi}\log|x|$ denotes the fundamental solution for the Laplacian in the complex plane.
- For $x\in{}U$, $B(x,\epsilon)\subseteq U$ denotes the usual ball centered at x with raidus $\epsilon>0$. $U_\epsilon{}=U\backslash B(x,\epsilon)$.
So what I want to get is the following $$\begin{align} \int\limits_{\partial{}B(x,\epsilon)}F(x-y)\frac{\partial{u}}{\partial{\nu}}(y)d\sigma(y)=\int\limits_{\partial{}B(x,\epsilon)}-\frac{1}{2\pi}\log|x|\frac{\partial{u}}{\partial{\nu}}(y)d\sigma(y) \end{align}$$ tends to zero as $\epsilon\rightarrow0$. The author says I can derive this by using the explicit formula for $F$, but I don't know how to. Please help me with this.
The book is Classical and Multilinear Harmonica Analysis Volume 2, Muscalu, Schlag.