A method of estimating lower bounds for eigenvalues of differential operators by the method of fictitious domains

The Bazley-Fox method of intermediate problems and the method of fictitious domains are used in combination to obtain lower bounds for the eigenvalue of a two-dimensional second-order elliptic equation with Dirichlet boundary condition. It is shown that, with this approach, under very general assumption about the coefficients of the differential equation and the domain in which it is specified, the problem of finding lower bounds for the eigenvalues can be reduced to finding the eigenvalues of a matrix.