Certain estimates of the least eigenvalue for an elliptic operator in a multiply connected domain (Q1088069)

Some estimates of the lowest eigenvalue for an elliptic operator in a multiply connected domain are given. Let \(L=\sum^{2}_{i,j=1}a_{ij}(x)\partial^ 2/\partial x_ i\partial x_ j+c(x)\) with bounded coefficients \(a_{ij}\), \(c\leq 0\) and \[ \gamma_ 1\sum \xi_ i^ 2\leq \sum a_{ij}(x)\xi_ i\xi_ j\leq \gamma_ 2\sum \xi_ i^ 2. \] Assume that \(Lu+\lambda^ 2u=0\), u(x)\(\geq 0\) for \(x\in \Omega\) and \(u|_{\partial \Omega}=0\) where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ 2\) with an inner diameter \(\rho\). Let k be the number of connected components of \({\mathbb{R}}^ 2\setminus \Omega\) with diameters not exceeding \(2\rho\). It is shown that \(\lambda\geq c\rho^{-1}k^{-(\gamma +1)/4}\) if \(\gamma =\gamma_ 2/\gamma_ 1>1\). Other results of similar type are also given.