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Inverse iteration for $ p$-ground states


Authors: Ryan Hynd and Erik Lindgren
Journal: Proc. Amer. Math. Soc. 144 (2016), 2121-2131
MSC (2010): Primary 35J70, 35J60, 35P30
DOI: https://doi.org/10.1090/proc/12860
Published electronically: September 11, 2015
MathSciNet review: 3460172
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Abstract: We adapt the inverse iteration method for symmetric matrices to some nonlinear PDE eigenvalue problems. In particular, for $ p\in (1,\infty )$ and a given domain $ \Omega \subset \mathbb{R}^n$, we analyze a scheme that allows us to approximate the smallest value the ratio $ \int _\Omega \vert D\psi \vert^pdx/\int _\Omega \vert\psi \vert^pdx$ can assume for functions $ \psi $ that vanish on $ \partial \Omega $. The scheme in question also provides a natural way to approximate minimizing $ \psi $. Our analysis also extends in the limit as $ p\rightarrow \infty $ and thereby fashions a new approximation method for ground states of the infinity Laplacian.


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Additional Information

Ryan Hynd
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Email: rhynd@math.upenn.edu

Erik Lindgren
Affiliation: Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden
Email: eriklin@kth.se

DOI: https://doi.org/10.1090/proc/12860
Keywords: Nonlinear eigenvalue problem, $p$-Laplacian, inverse iteration, power method
Received by editor(s): March 5, 2015
Received by editor(s) in revised form: May 30, 2015, and June 9, 2015
Published electronically: September 11, 2015
Additional Notes: The first author was partially supported by NSF grant DMS-1301628. The second author was supported by the Swedish Research Council, grant no. 2012-3124, and partially supported by the Royal Swedish Academy of Sciences.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2015 American Mathematical Society