| The eigenvalue problem for the negative Laplace operator in two dimensions is classical in mathematics and physics. Nevertheless, analytical methods for estimating the eigenvalues are still of much current interest. In this work, a modified perturbation method is formulated by applying perturbation method, reflection method, and the Fredholm alternative theorem. The method provides the asymptotic expansion formulas of the lowest eigenvalue to bounded doubly connected regions having the inner boundary which encloses a region with the maximum dimension of 2c, c 1. The first three order terms of the asymptotic expansion formulas are found explicitly by correcting the inner and outer boundary conditions alternatively and by applying the generalized Green's functions. The relations between the first three order terms of the asymptotic expansion formulas and geometric properties of the regions are also investigated. |
