Estimates for discrete spectrum of a certain system (Q1299839)

Can you hear the shape of a drum? Professor \textit{M. H. Protter} asked in [SIAM Rev. 29, 185-197 (1987; Zbl 0645.35074)]. Usually, this kind of problems can mathematically be solved by estimating the bounds of the eigenvalues to polyharmonic operators. If one strikes simultaneously \(N\) drums made by the same material and possessing the same size, we should ask that can you hear the shape of the drums? To answer this question, one might consider an eigenvalue problem for a certain system as follows: \[ \begin{cases} \sum^N_{l=1} a_{kl}\Delta^{k+1} u^l=\lambda u^k,\quad & x\in\Omega,\\ u^k= {\partial u^k\over\partial\nu}=\dots= {\partial^{N+k-1} u^k\over\partial\nu^{N+ k-1}}= 0,\quad &k= 1,2,\dots, N,\quad x\in\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^m\) \((m\geq 2)\), \(\overline\nu= (\nu_1,\nu_2,\dots, \nu_m)\) is the outward unit normal to the boundary of \(\Omega\), \(a_{kl}= a_{lk}\), \(l,k= 1,2,\dots, N\), are constants with \(a_{Nl}\neq 0\) for \(l= 1,2,\dots, N\), which guarantees the reasonability of boundary conditions in (1) since the order of derivatives for each \(u^k\) is of \(2(N+ k)\). Suppose \(a_{kl}\), \(k,l= 1,2,\dots, N\), satisfy the following conditions: (\(\text{A}_1\)) \((-1)^{k+l} a_{kl}\geq 0\), \(k,l= 1,2,\dots, N\); (\(\text{A}_2\)) For all \(\xi= (\xi_1,\xi_2,\dots, \xi_N)\) there holds \(\sum^N_{k,l=1} a_{kl} \xi_k \xi_l\geq \mu|\xi|^2\), where \(\mu>0\) is the smallest eigenvalue of \(A= (a_{ij})_{N\times N}\). Theorem. Let \(\lambda_i\), \(i= 1,2,\dots,n+1\), be the first \(n+1\) eigenvalues of problem (1). If \(a_{kl}\), \(k,l= 1,2,\dots, N\), satisfy conditions (\(\text{A}_1\)) and \((\text{A}_2\)), then for \(m\geq 2\), \(n\geq 2\), the following estimates hold \[ \begin{multlined} \lambda_{n+1}\leq \lambda_n+{4\over m^2 n^2} \Biggl[\sum^n_{i= 1} \sum^N_{k,l= 1} (k+l)(2k+ 2l+ m-2)\cdot\\ | a_{kl}| (\lambda_i/\mu)^{1-(k+ l)/4kl}\Biggr]\Biggl[ \sum^n_{i=1} \sum^N_{k=1} (\lambda_i/\mu)^{1/2k}\Biggr],\end{multlined} \] \[ \sum^n_{i= 1} \sum^N_{k=1} {(\lambda_i/\mu)^{1/2k}\over \lambda_{n+1}- \lambda_i}\geq {m^2n^2\over \sum^n_{i=1} \sum^N_{k,l=1} 4(k+ l)(2k+ 2l+ m-2)| a_{kl}|(\lambda/\mu)^{1- (k+l)/4kl}}. \]