Good afternoon,
I try to understand the follwoing counterexample of Hadamard:
The classical solution of $ \begin{cases} \Delta u = 0 ~~\text{ in }\Omega\\ u=g \text{ auf }~~ \partial \Omega \end{cases} $
with $u \in C^2(\Omega) \cap C(\partial \Omega)$ not always lies in $H^1(\Omega)$.
To show this Hadamard took the function $ \begin{equation} u(r, \phi) = \sum\limits_{n=0}^{\infty}r^{n!}\frac{sin(n! \phi)}{n^2} \end{equation} $ in polar-coordinates on $\Omega =$ "the unit-disk". After computing the Laplacian in polar coordinates it was very easy to show that this function is indeed a classical solution (i.e. u harmonic) but now I am supposed to choose an appropriate boundary value function $g \in C(\partial \Omega)$, such that this $u(r,\phi)$ is not in $H^1(\Omega)$. Can someone help? By the way in our lecture we had the statement that harmonic functions have minimal seminorms, perhaps this could be useful.