Calculus of Variations and Geometric Measure Theory

A. Davini - A. Fathi - R. Iturriaga - M. Zavidovique

Convergence of the solutions of the discounted equation: the discrete case

created by davini on 13 Jul 2023


Published Paper

Inserted: 13 jul 2023
Last Updated: 13 jul 2023

Journal: Math. Z.
Volume: 284
Number: 3-4
Pages: 1021–1034
Year: 2016


We derive a discrete version of the results of 2. If $M$ is a compact metric space, $c : M\times M \to \R$ a continuous cost function and $\lambda \in (0,1)$, the unique solution to the discrete $\lambda$-discounted equation is the only function $u_\lambda : M\to \R$ such that \[ \forall x\in M, \quad u_\lambda(x) = \min_{y\in M} \lambda u_\lambda (y) + c(y,x). \] We prove that there exists a unique constant $\alpha\in \R$ such that the family of $u_\lambda+\alpha/(1-\lambda)$ is bounded as $\lambda \to 1$ and that for this $\alpha$, the family uniformly converges to a function $u_0 : M\to \R$ which then verifies \[ \forall x\in X, \quad u_0(x) = \min_{y\in X}u_0(y) + c(y,x)+\alpha. \] The proofs make use of Discrete Weak KAM theory. We also characterize $u_0$ in terms of Peierls barrier and projected Mather measures.