A maximum principle for a fourth-order Dirichlet problem on smooth domains
Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain with a smooth boundary, \(\lambda \in \mathbb{R}\) an \(f\in L^2(\Omega)\). In [\textit{I. Schnieders} and \textit{G. Sweers}, Positivity 24, No. 3, 677--710 (2020; Zbl 1441.35082)], the following converse of the Krein-Rutman Theorem for the problem \[(\Delta^2 u-\lambda u)=f \ \ \text{in} \ \ \Omega, \ \ \ u=\frac{\partial u}{\partial \nu}=0 \ \ \text{on} \ \ \partial \Omega \tag{1}\] was proved: if there exists a simple eigenvalue \(\lambda_j>0\) to the biharmonic eigenvalue problem \[(\Delta^2 u-\lambda u)=0 \ \ \text{in} \ \ \Omega, \ \ \ u=\frac{\partial u}{\partial \nu}=0 \ \ \text{on} \ \ \partial \Omega,\] with eigenfunction \(\varphi_j\) satisfying \[\inf_{x\in \Omega}\frac{\varphi_j(x)}{d(x,\partial \Omega)}>0, \tag{2}\] then there exists a left neighborhood \(I\) of \(\lambda_j\) such that for each \(\lambda \in I\), the maximum principle \[f\geq 0, f\neq 0 \ \ \Longrightarrow \ \ u > 0,\] holds for the solution \(u\) of \((1)\). Eigenfunctions satisfying \((2)\) exist for some domain (for instance when \(\Omega\) is a ball), but this fact is not true for general domains. In this paper, the authors show that restrictions on domains can be removed provided to introduce a suitable weight function. To be precise, they show that for any domain \(\Omega\) of class \(C^{4,\gamma}\) (\(0<\gamma<1)\), there exists a weight function \(w\in C^{0,\gamma}(\overline{\Omega})\), with \(\inf_{\overline{\Omega}}w>0\), such that the eigenvalue problem \[(\Delta^2 u-\lambda w u)=0 \ \ \text{in} \ \ \Omega, \ \ \ u=\frac{\partial u}{\partial \nu}=0 \ \ \text{on} \ \ \partial \Omega\] admits a simple eigenvalue \(\lambda_{j,w}>0\) whose corresponding eigenfunction satisfies \((2)\). From this, the authors derive a maximum principle for the problem \[(\Delta^2 u-\lambda w u)=f \ \ \text{in} \ \ \Omega, \ \ \ u=\frac{\partial u}{\partial \nu}=0 \ \ \text{on} \ \ \partial \Omega.\] for each \(\lambda\) in a left neighborhood of \(\lambda_{j,w}\).
scientific article; zbMATH DE number 7337025
scientific article