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., ²u=λ²u on $Ω$ with some homogeneous linear boundary conditions. We assume that Ωⁿ (n≥2) is a C∞ bounded domain, ∂Ω is piecewise analytic and ∂Ω is analytic except a set Γ⊆∂Ω which is a finite union of some compact $(n-2)$ dimensional submanifolds of ∂Ω. The main result of this paper is that the measure upper bounds of the nodal sets of the eigenfunctions is controlled by √λ. 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 √λ. In order to further control the frequency function with √λ, we first establish the relationship between the frequency function and the doubling index, and then separate the domain $Ω$ into two parts: a domain away from $Γ$ and a domain near $Γ$, and develop iteration arguments to deal with the two cases respectively.