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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

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Published Paper

Inserted: 13 jul 2023
Last Updated: 13 jul 2023

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

Abstract:

We derive a discrete version of the results of 2. If M is a compact metric space, c:M×M\R a continuous cost function and λ(0,1), the unique solution to the discrete λ-discounted equation is the only function uλ:M\R such that xM,uλ(x)=min 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.


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