Let $D \subset \mathbf{R}^n$ be the unit disc, and $\alpha \in (0,1)$. Let $f \in C^{0,\alpha}(\partial D)$, and $u \in C^{2,\alpha}(D)$ be the harmonic function with $u = f$ on the boundary.
Define the 'nodal map' \begin{equation} f \in C^{0,\alpha}(\partial D) \mapsto \{ u = 0 \} \in \mathcal{Z}_{n-1}(D;\mathbf{Z}_2), \end{equation} with image in the flat cycles modulo two.
This is continuous because of elliptic estimates. It also passes to the 'projectivized' function space $C^{0,\alpha}(\partial D) / \sim$ where $f_1 \sim f_2$ if there is $\lambda \in \mathbf{R} \setminus \{ 0 \}$ so that $f_1 = \lambda f_2$.
Question. Has this map been studied in the literature?
