Harmonic inside with zero average

Question

Assume $\Omega\in\mathbb{C}$ is a domain with nice enough boundary,say smooth boundary. What can be said about $f\in C(\bar\Omega)$, harmonic in $\Omega$ and $\int_{\partial\Omega}f(z)|dz|=0$, where $|dz|$ is arc-length measure?

Answer 1

We can say that $f$ is either identically $0$, or it attains both positive and negative values in $\Omega$. This follows from the maximum principle.

When $\Omega$ is a disk, we can say that $f$ is zero at the center of the disk. But for any other shapes, no such conclusion is possible: i.e., there is not a point $z_0\in \Omega$ such that $f(z_0)=0$ for all $f$ satisfying your assumption. This follows from the (nontrivial) fact that the equality of harmonic measure and arclength measure characterizes disks.

Other than that, there isn't much. The space of such functions $f$ has codimension $1$ in the space of all harmonic functions; for any harmonic $g\in C(\overline{\Omega})$ we have a constant $c$ such that $g+c$ integrates to $0$ over $\partial \Omega$. So, such harmonic functions cannot be "nicer" than general harmonic functions in the sense of their analytic properties.