Boundary-variation solution of eigenvalue problems for elliptic operators

We present an algorithm which, based on certain properties of analytic depe ndence, constructs boundary perturbation expansions of arbitrary order for eigenfunctions of elliptic PDEs. The resulting Taylor series can be evaluat ed far outside their radii of convergence - by means of appropriate methods of analytic continuation in the domain of complex perturbation parameters. A difficulty associated with calculation of the Taylor coefficients become s apparent as one considers the issues raised by multiplicity: domain pertu rbations may remove existing multiple eigenvalues and criteria must therefo re be provided to obtain Taylor series expansions for all branches stemming from a given multiple point. The derivation of our algorithm depends on ce rtain properties of joint analyticity (with respect to spatial variables an d perturbations) which had not been established before this work. While our proofs, constructions and numerical examples are given for eigenvalue prob lems for the Laplacian operator in the plane, other elliptic operators can be treated similarly.