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Let $u: U \mapsto \mathbb{R}$ be given by \begin{gather} u(y) = -\frac{1}{\pi} \int_{\partial U} g(x) \log |x-y|\ \mathrm{d}x \quad \text{with} \quad U := \{ y \in \mathbb{R}^2 : |y| < 1\} \end{gather} which is the Poisson formula for the Neumann problem on the unit disk solving \begin{align} \Delta u &= 0 \quad \text{in} \ \ U\\ \partial_{\nu}u &= g \quad \text{on} \ \ \partial U \end{align} and $g$ is continuous on the boundary $\partial U$.

Now I want to prove that $u$ is harmonic in $U$, continuous in $\overline{U}$ and that $\int_{\partial U} u = 0$ must hold.

I have a hard time understanding how to approach the proof for continuity, because I don't know how to take the value of $u$ at the boundary (for a Dirichlet problem $u=g$ on $\partial U$ and we can take advantage of the continuity of $g$). Also, I understand that according to Green's identity there is the consistency condition \begin{gather} \int_U \Delta u = \int_{\partial U} \partial_{\nu}u = \int_{\partial U} g = 0 \end{gather} but how can I deduce from this that $\int_{\partial U} u = 0$ ? Is it that $\int_{\partial U} \log |x-y|\ \mathrm{d} x= 1$? If yes, why? Finally, I want to show that $\partial_{\nu} u(y) \to g(x)$ as $x\to y$, for $x \in \partial U$ and $y \in U$, but if I understand all of the above I believe I can proceed with it.

Any hint?

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