Inverse iteration for the Monge–Ampère eigenvalue problem
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- by Farhan Abedin and Jun Kitagawa
- Proc. Amer. Math. Soc. 148 (2020), 4875-4886
- DOI: https://doi.org/10.1090/proc/15157
- Published electronically: August 11, 2020
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Abstract:
We present an iterative method based on repeatedly inverting the Monge–Ampère operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain $\Omega \subset \mathbb {R}^{n}$. We prove that the iterates $u_k$ generated by this method converge as $k \to \infty$ to a solution of the Monge–Ampère eigenvalue problem \begin{equation*} \begin {cases} \mathrm {det} D^2u = \lambda _{MA} (-u)^n & \quad \text {in } \Omega ,\\ u = 0 & \quad \text {on } \partial \Omega . \end{cases} \end{equation*} Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates $\hat {u}_k \coloneq \frac {u_k}{||u_k||_{L^{\infty }(\Omega )}}$ converge to the eigenfunction of unit height. In addition, we show that $\lim _{k \to \infty } R(u_k) = \lim _{k \to \infty } R(\hat {u}_k) = \lambda _{MA}$, where the Rayleigh quotient $R(u)$ is defined as \begin{equation*} R(u) \coloneq \frac {\int _{\Omega } (-u) \ \mathrm {det}D^2u}{\int _{\Omega } (-u)^{n+1}}. \end{equation*} Our method converges for a wide class of initial choices $u_0$ that can be constructed explicitly, and does not rely on prior knowledge of the Monge–Ampère eigenvalue $\lambda _{MA}$.References
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Bibliographic Information
- Farhan Abedin
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- MR Author ID: 884070
- Email: abedinf1@msu.edu
- Jun Kitagawa
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- MR Author ID: 1000616
- ORCID: 0000-0002-6145-6664
- Email: kitagawa@math.msu.edu
- Received by editor(s): January 15, 2020
- Received by editor(s) in revised form: April 6, 2020, and April 10, 2020
- Published electronically: August 11, 2020
- Additional Notes: The first author is the corresponding author.
The second author’s research was supported in part by National Science Foundation grant DMS-1700094. - Communicated by: Ryan Hynd
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 4875-4886
- MSC (2010): Primary 35J96, 35P30
- DOI: https://doi.org/10.1090/proc/15157
- MathSciNet review: 4143401