Curvature Estimate of Nodal Sets of Harmonic Functions in the Plane

In this paper, we study curvature estimates for nodal sets of harmonic functions in the plane. We prove that at any point $p$, the curvature of each nodal curve of any harmonic function $u$ is upper bounded by $$\mid \kappa (u)(p)\mid \le \frac{4(n+1)}{nr}\cos n{\alpha }_0,$$ where $u$ has only $n$ nodal curves in $B_r(p)$ intersecting at $p$, and ${\alpha }_0=0$ for odd $n$ or ${\alpha }_0=\frac{\pi }{2n(n+1)}$ for even $n$. This result is sharp for all $n\ge 1$. In extreme cases, $u$ can be given by the Poisson extension of Dirac measure and its derivatives. Moreover, the curvature of any nodal curve is uniformly upper bounded at every point in the nodal set of $u$ in a small neighborhood $B_{cr}(p)$, where $c<1$ depends only on $n$. Furthermore, with the frequency tool, we prove that the area of the positive part and the negative part of $u$ have a uniform lower bound, which depends only on the number of nodal domains in $B_r(p)$.