Comparison theorems for eigenvalues of elliptic operators and the generalized Pólya conjecture (Q430017)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n, n \geq 2\) with \(\partial \Omega\) being smooth. For the Dirichlet eigenvalue problem \(\triangle u + \lambda u = 0\) in \(\Omega\), \(u = 0\) on \(\partial \Omega\), denote the eigenvalues as \[ 0 < \lambda_1 < \lambda_2 \leq \cdots \rightarrow \infty, \] the Pólya conjecture states that \(\lambda_k \geq C(n) \bigl(\frac{k}{|\Omega|} \bigr )^{2/n}\) for all \(k\). On the other hand, let the elliptic operator of order \(2t\) defined by \[ Lu = \sum_{m=r+1}^t a_{m-r}(-\triangle)^m u\, \text{ for}\, u \in C^\infty (\Omega) \] with \(r \geq 0\) being an integer, \(a_{m-r} \geq 0\) are constants, \(r+1 \leq m \leq t\), \(a_{t-r}=1\) and \(t\) a fixed integer. For the boundary eigenvalue problem \[ \begin{cases} Lu = \lambda (-\triangle)^r u, u \in C^\infty(\Omega),\\ (\frac{\partial}{\partial \nu})^j u = 0\, \text{ on}\, \partial \Omega, j = 0, 1, \cdots, t-1,\end{cases} \] where \(\nu\) is the outward unit normal vector field of \(\partial \Omega\), denote the eigenvalues as \(0 < \lambda_{1,r} \leq \lambda_{2,r} \leq \cdots \rightarrow \infty\), the generalized Pólya conjecture states that \[ \lambda_{k,r} \geq \sum_{m=1}^{t-r} a_m C^m \bigl(\frac{k}{|\Omega|}\bigr )^{2m/n} \] for \(k = 1, 2, \cdots\) with \(C = C(n)\) as in Pólya conjecture. In this paper, the authors establish comparison theorems for eigenvalues between higher order elliptic equations on compact manifolds with boundary. With this result, they prove that if the Pólya conjecture is true, then so is the generalized Pólya conjecture.