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{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 $\lambda _{MA}$.