The method of generations for estimating the first eigenvalue of the Neumann problem (Q1975123)

The method of generations is used for estimating the first eigenvalue of the operator \(\Delta+c(x)\), whose domain is the set of functions satisfying the Neumann condition. In this paper, two estimates of a similar kind relating to the Neumann problem are given, one in differential form and the other in integral form. The estimates obtained apply to the case when \(c(x)\) is defined by the values of a bounded random variable \(\gamma(x)\) such that \(\text{E}\gamma(x)=c(x)\). Estimates are also found for the bias of the estimated eigenvalue. Some numerical examples are given.