Measure upper bounds for nodal sets of eigenfunctions of the bi‐harmonic operator

Abstract

In this paper, we investigate the measure estimate of nodal set of eigenfunction u $u$ of the bi-harmonic operator, that is, ▵ 2 u = λ 2 u $\hbox{{$\triangle$}} ^2u=\lambda ^2u$ in Ω $\Omega$ with some homogeneous linear boundary conditions. We assume that Ω ⊆ R n $\Omega \subseteq \mathbb {R}^n$ ( n ⩾ 2 $n\geqslant 2$ ) is a C ( n + 1 ) / 2 $C^{(n+1)/2}$ -bounded domain, ∂ Ω $\partial \Omega$ is piecewise analytic, and analytic except a set Γ ⊆ ∂ Ω $\Gamma \subseteq \partial \Omega$ which is a finite union of some compact ( n − 2 ) $(n-2)$ -dimensional submanifolds of ∂ Ω $\partial \Omega$ . The main result of this paper is that the measure upper bound of nodal set of the eigenfunction u $u$ is controlled by λ $\sqrt {\lambda }$ . We first define a frequency function and a doubling index related to the eigenfunction. With the help of establishing the monotonicity formula, doubling conditions and various a priori estimates, we obtain that the ( n − 1 ) $(n-1)$ -dimensional Hausdorff measure of nodal set of the eigenfunction in a ball is controlled by the frequency function and λ $\sqrt {\lambda }$ . In order to further control the frequency function by λ $\sqrt {\lambda }$ , we establish the relationship between the frequency function and the doubling index, and then separate the domain Ω $\Omega$ into two parts: the domain away from Γ $\Gamma$ and the domain near Γ $\Gamma$ , and develop a new iteration argument to deal with the two cases, respectively.