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Boundary-variation solution of eigenvalue problems for elliptic operators

Bruno, Oscar P. and Reitich, Fernando (2001) Boundary-variation solution of eigenvalue problems for elliptic operators. Journal of Fourier Analysis and Applications, 7 (2). pp. 169-187. ISSN 1069-5869. doi:10.1007/BF02510422.

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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.

Item Type:Article
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Bruno, Oscar P.0000-0001-8369-3014
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.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)F49620-96-1-0008
Air Force Office of Scientific Research (AFOSR)F49620-99-1-0010
Charles Lee Powell FoundationUNSPECIFIED
Air Force Office of Scientific Research (AFOSR)F49620-99-1-0193
Subject Keywords:boundary perturbations; eigenvalues and eigenfunctions; elliptic operators; analytic continuation
Issue or Number:2
Classification Code:Math Subject Classification: 35P99; 65N25; 35B20; 41A58
Record Number:CaltechAUTHORS:20181106-075027317
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Official Citation:Bruno, O.P. & Reitich, F. The Journal of Fourier Analysis and Applications (2001) 7: 169.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:90656
Deposited By: Tony Diaz
Deposited On:06 Nov 2018 18:32
Last Modified:16 Nov 2021 03:34

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