The upper bounds of arbitrary eigenvalues of uniformly elliptic operators with higher orders
Let \(\Omega \subset \mathbb R^m (m \geq 2)\) be a bounded domain with piecewisely smooth and Lipschitz boundary \(\partial \Omega\) and \(t\) and \(r\) two nonnegative integers \(t\geq r+1\). The paper deals with the following eigenvalue problem \[ (-1)^t\sum_{i_1,i_2,\dots,i_t=1}^m D_{i_1,i_2,\dots,i_t} \left(a_{i_1,i_2,\dots, i_t}(x)D_{i_1,i_2,\dots,i_t}u\right) = \lambda(-\Delta)^ru, \quad x \in \Omega, \] \[ u=\frac{\partial u}{\partial \nu}=\cdots= \frac{\partial^{t-1}u}{\partial \nu^{t-1}}=0, \quad x \in \partial \Omega, \] where \(\nu\) is the unit outward normal to \(\partial \Omega\), \(a_{i_1,i_2,\dots,i_t} \in C^t({\overline{\Omega}})\) with \[ 0 < \mu \leq a_{i_1,i_2,\dots,i_t}(x) < \infty, \] for \( i_1,i_2,\dots,i_t=1,2,\dots,t,\) and \(\mu\) is a positive constant. Using the Rayleigh principle the authors obtain inequalities for the eigenvalue \(\lambda_{n+1}\) in terms of \(\lambda_1,\lambda_2,\dots,\lambda_n\). A typical result in a special case is the following. For \(a_{i_i,i_2,\dots,i_t}\equiv 1\), \(r=0\) it holds \[ \lambda_{n+1} \leq \lambda_n + \frac{4}{m^2n^2}t(2t+m-2)\bigg(\sum_{i=1}^n\lambda_i^{1/t}\bigg) \bigg(\sum_{i=1}^n\lambda_i^{1-1/t}\bigg) \] and \[ \lambda_{n+1} \leq \lambda_n + \frac{4}{m^2}t(2t+m-2)\lambda_n. \]
scientific article; zbMATH DE number 5119465