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Published March 2001 | public
Journal Article

Boundary-variation solution of eigenvalue problems for elliptic operators

Abstract

We present an algorithm which, based on certain properties of analytic dependence, constructs boundary perturbation expansions of arbitrary order for eigenfunctions of elliptic PDEs. The resulting Taylor series can be evaluated 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 becomes apparent as one considers the issues raised by multiplicity: domain perturbations may remove existing multiple eigenvalues and criteria must therefore be provided to obtain Taylor series expansions for all branches stemming from a given multiple point. The derivation of our algorithm depends on certain properties of joint analyticity (with respect to spatial variables and perturbations) which had not been established before this work. While our proofs, constructions and numerical examples are given for eigenvalue problems for the Laplacian operator in the plane, other elliptic operators can be treated similarly.

Additional Information

© 2001 Birkhäuser Boston. Received August 31, 1999; Revision received December 30, 1999. OB gratefully acknowledges support from NSF (through an NYI award and through contracts No. DMS-9523292 and DMS-9816802), from the AFOSR (through contracts No. F49620-96-1-0008 and F49620-99-1-0010), and from the Powell Research Foundation. FR gratefully acknowledges support from AFOSR through contract No. F49620-99-1-0193 and from NSF through contracts No. DMS-9622555 and DMS-9971379. Effort sponsored by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under grant numbers F49620-96-1-0008, F49620-99-1-0010, and F49620-99-1-0193. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon, The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government.

Additional details

Created:
August 21, 2023
Modified:
October 19, 2023