Consider the open unit disk $\mathbb{D}$ in $\mathbb{R}^2$. Given a continuous function $f : \partial\mathbb{D} \to \mathbb{R}$ we define
$$
u(x) \overset{\texttt{def}}{=} \int_{\partial \mathbb{D}} f(y)\ln\left\vert x-y\right\vert\,\mathrm{d}S(y)
$$
for any point $x \in \overline{\mathbb{D}}$.
I have proven that this integral exists for all $x$ belonging to the closure of $\mathbb{D}$.
Using the dominated convergence theorem, I have also shown that $u$ is harmonic in $\mathbb{D}$. In particular, $u \in C(\mathbb{D})$. However, I have been unable to show that $u$ is continuous up to the boundary of $\mathbb{D}$. Dominated convergence does not seem to apply here since the logarithm blows up as $x$ approaches a point on the boundary. I suspect that I will have to use the fact that $u$ is harmonic in the open disk to prove this.
Any help would be greatly appreciated.
Best Answer
Consider $x_0 \in \partial \mathbb{D}$ and an open disk $B_r(x_0)$ of radius r where $|x -x_0| <r$.
We have
$$\tag{*}|u(x) - u(x_0)| \\ \leqslant \left|\int_{\partial \mathbb{D} - B_r(x_0)}f(y) [\ln |x-y|-\ln|x_0-y|] \, dS(y) \right| + \left|\int_{\partial \mathbb{D} \cap B_r(x_0)}f(y) \ln |x-y| \, dS(y) \right|+ \left|\int_{\partial \mathbb{D} \cap B_r(x_0)}f(y) \ln |x_0-y| \, dS(y) \right|$$
Choosing $|x - x_0|$ (and $r$) sufficiently small we can make each term on the RHS of (*) smaller than $\epsilon/3$.
For the second and third terms we can use the same argument that proves the integral exists.
For the first term note that $||x-y| - |x_0-y|| \leqslant |x - x_0|$. Since $x,x_0 \neq y$ for $y \in \partial \mathbb{D} - B_r(x_0)$ and the logarithm function is continuous, we have for sufficiently small $|x-x_0|$ and $M = \sup_{y \in \partial \mathbb{D}} f(y),$
$$| \ln |x-y| - \ln |x_0-y|| < \frac{\epsilon}{6\pi M}, $$
which implies that the first term on the RHS of (*) is less than $\epsilon/3$.
To be more precise, the argument rests on first choosing $r$ such that the contribution from the second and third terms on the RHS of (*) is smaller than $2\epsilon/3$. Then we would show that $|\ln|x -y| - \ln|x_0 - y|| \to 0$ as $x \to x_0$ uniformly for $y \in \partial \mathbb{D} - B_r(x_0)$