Measure Upper Bounds for Nodal Sets of Eigenfunctions of the bi-Harmonic Operator

In this article, we consider eigenfunctions $u$ of the bi-harmonic operator, i.e., $\triangle^2u=\lambda^2u$ on $\Omega$ with some homogeneous linear boundary conditions. We assume that $\Omega\subseteq\mathbb{R}^n$ ($n\geq2$) is a $C^{\infty}$ bounded domain, $\partial\Omega$ is piecewise analytic and $\partial\Omega$ is analytic except a set $\Gamma\subseteq\partial\Omega$ which is a finite union of some compact $(n-2)$ dimensional submanifolds of $\partial\Omega$. The main result of this paper is that the measure upper bounds of the nodal sets of the eigenfunctions is controlled by $\sqrt{\lambda}$. We first define a frequency function and a doubling index related to these eigenfunctions. With the help of establishing the monotonicity formula, doubling conditions and various a priori estimates, we obtain that the $(n-1)$ dimensional Hausdorff measures of nodal sets of these eigenfunctions in a ball are controlled by the frequency function and $\sqrt{\lambda}$. In order to further control the frequency function with $\sqrt{\lambda}$, we first establish the relationship between the frequency function and the doubling index, and then separate the domain $\Omega$ into two parts: a domain away from $\Gamma$ and a domain near $\Gamma$, and develop iteration arguments to deal with the two cases respectively.