Inverse Iteration for the Monge-Ampère Eigenvalue Problem

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 $$\{\begin{array}{ll}\text{det}{D}^{2}u={\lambda }_{MA}(-u{)}^n& \text{in }\Omega ,\\ u=0& \text{on }\partial \Omega .\end{array}$$ Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates ${\hat{u}}_k:=\frac{u_k}{||u_k|{|}_{L^{\infty }(\Omega )}}$ converge to the eigenfunction of unit height. In addition, we show that $\underset{k\to \infty }{\lim }R(u_k)=\underset{k\to \infty }{\lim }R({\hat{u}}_k)={\lambda }_{MA}$, where the Rayleigh quotient $R(u)$ is defined as $$R(u):=\frac{{\int }_{\Omega }(-u)\text{det}{D}^{2}u}{{\int }_{\Omega }(-u{)}^{n+1}}.$$ 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 ${\lambda }_{MA}$.

To appear in Proc. Amer. Math. Soc