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<p>In this paper, we introduce a discrete version of the nonlinear implicit Lax-Oleinik operator, that provides solutions to the so-called contact type Hamilton–Jacobi equation, as studied for instance in Wang et al. (J Math Pures Appl 123:167–200, 2019). We consider the associated vanishing discount problem with a non-degenerate condition and prove convergence of the discounted solutions as the discount factor goes to 0. We also discuss the uniqueness of the discounted solution. The proof relies on discrete Mather measures and tools from discrete weak KAM theory.</p>

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Nonlinear and degenerate discounted approximation in discrete weak KAM theory

  • 期刊论文

  • 发表日期：

    2025年5月21日

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  • Panrui Ni,
  • Maxime Zavidovique

摘要

In this paper, we introduce a discrete version of the nonlinear implicit Lax-Oleinik operator, that provides solutions to the so-called contact type Hamilton–Jacobi equation, as studied for instance in Wang et al. (J Math Pures Appl 123:167–200, 2019). We consider the associated vanishing discount problem with a non-degenerate condition and prove convergence of the discounted solutions as the discount factor goes to 0. We also discuss the uniqueness of the discounted solution. The proof relies on discrete Mather measures and tools from discrete weak KAM theory.