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Nonlinear and degenerate discounted approximation in discrete weak KAM theory

Panrui Ni, Maxime Zavidovique

Abstract

In this paper, we introduce a discrete version of the nonlinear implicit Lax-Oleinik operator. We consider the associated vanishing discount problem with a non-degenerate condition and prove convergence of solutions as the discount factor goes to $0$. We also discuss the uniqueness of the discounted solution. The convergence result is a selection principle for fixed points of a family of nonlinear operators.

Nonlinear and degenerate discounted approximation in discrete weak KAM theory

Abstract

In this paper, we introduce a discrete version of the nonlinear implicit Lax-Oleinik operator. We consider the associated vanishing discount problem with a non-degenerate condition and prove convergence of solutions as the discount factor goes to . We also discuss the uniqueness of the discounted solution. The convergence result is a selection principle for fixed points of a family of nonlinear operators.
Paper Structure (9 sections, 21 theorems, 111 equations)

This paper contains 9 sections, 21 theorems, 111 equations.

Table of Contents

  1. Brief history of the problem
  2. The discretization
  3. Setting and statement of results
  4. Organization of the paper
  5. Classical discrete weak KAM theory
  6. Discrete version of implicit semigroup
  7. Discounted solutions
  8. Vanishing discount convergence
  9. Uniqueness of $u_\lambda$

Key Result

Proposition 1

There is a $\lambda_0>0$ such that for $0<\lambda<\lambda_0$, if $\varphi \in C^0(X,\mathbb R)$ there is a unique $T_\lambda \varphi \in C^0(X,\mathbb R)$ such that for all $x\in X$,

Theorems & Definitions (43)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Theorem 3
  • Definition 1.2
  • Proposition 1.1
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • ...and 33 more