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- W4212887834 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 $ngeqslant 2$ ) 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." @default.
- W4212887834 created "2022-02-24" @default.
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- W4212887834 date "2022-02-17" @default.
- W4212887834 modified "2023-10-16" @default.
- W4212887834 title "Measure upper bounds for nodal sets of eigenfunctions of the bi‐harmonic operator" @default.
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- W4212887834 doi "https://doi.org/10.1112/jlms.12550" @default.
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