On an estimate of the first eigenvalue of a self-adjoint elliptic operator (Q802083)

The authors consider the eigenvalue problem \(Lu=\lambda Q(x)u\) in the boundary domain \(\Omega \subset R^ n\) with Dirichlet conditions. L is a differential operator of order m, \((Lu,u)\geq c_ 0\sum_{| \alpha | \leq m}(D^{\alpha}u,D^{\alpha}u)\) and Q(x)\(\geq 0\), \(\int Q^{\alpha}dx\equiv 1.\) Let \(\lambda\) be the first eigenvalue. Theorem. If \(\alpha \geq \max (1,n/(2m)),\) \(n\neq 2m\), then \(\lambda \geq \lambda_ 0>0\) and \(\lambda\) can be as great as possible. If \(\alpha <1/(2m)\), \(\alpha\) \(\neq 0\), then \(\lambda \leq \lambda_ 0\), \(\lambda_ 0>0\), and \(\lambda\) can be any small positive number. If \(1/(2m)\leq \alpha <1\) then \(\lambda\) can be any small or any great positive number.