On perturbation problems associated with finite boundaries

Abstract

The methods of a previous paper (9) are improved and extended. It is assumed that the eigenfunctions and eigenvalues of an eigenvalue problem given by an elliptic differential equation are known subject to given boundary conditions on a finite boundary. It is shown how the corresponding quantities can be obtained for a similar problem in which the original differential equation, boundary and boundary conditions are simultaneously perturbed. The introduction of a surface displacement vector allows of a Taylor expansion of all quantities and a subsequent separation of orders. The problem of finding the perturbed eigenfunctions for each order then reduces to the solution of an inhomogeneous differential equation subject to known boundary conditions. These equations are solved by a variational method. An application of Green's theorem at each stage enables us to find the perturbed eigenvalues. The method is applied to a problem of which an exact solution is known and good agreement is obtained.The author is greatly indebted to Dr H. Fröhlich for many interesting discussions and some valuable suggestions.