Inverse Iteration for the Monge-Ampère Eigenvalue Problem

2020/01/05 by Abedin, Farhan, Kitagawa, Jun · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2001.01291

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 Ω⊂ ℝⁿ. We prove that the iterates u_k generated by this method converge as k → ∞ to a solution of the Monge-Ampère eigenvalue problem \begincases det D²u = λ_MA (-u)ⁿ amp; in Ω,
u = 0 amp; on ∂ Ω. \endcases Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates u_k := \fracu_k||u_k||_L^∞(Ω) converge to the eigenfunction of unit height. In addition, we show that lim_{k → ∞} R(u_k) = lim_{k → ∞} R(u_k) = λ_MA, where the Rayleigh quotient R(u) is defined as R(u) := \frac∫_Ω (-u) det D²u∫_Ω (-u)ⁿ⁺¹. Our method converges for a wide class of initial choices u₀ that can be constructed explicitly, and does not rely on prior knowledge of the Monge-Ampère eigenvalue λ_MA.

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• A new approach to the Monge-Ampère eigenvalue problem

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