发表机构
Waseda University; Shanghai Center for Mathematical Sciences, Fudan University; School of Mathematical Sciences, Fudan University; Sorbonne Université, Université de Paris Cité, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche（早稻田大学; 复旦大学上海数学中心; 复旦大学数学科学学院; 索邦大学，巴黎西岱大学，法国国家科学研究中心，儒勒·让苏-巴黎左岸数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究非单调Hamilton-Jacobi方程的消失折扣问题，在Mather商小性假设下证明最大粘性解收敛，并利用Peierls势垒刻画极限，首次实现多个临界解的选取。
AI 中文摘要
我们研究了一类广义的Hamilton-Jacobi方程的消失折扣问题，不假设关于所有Mather测度在逐点意义或平均意义上的单调性。具体地，我们考虑\\[ \lambda a(x)u(x)+H\big(x,Du(x)\big)=c_0, \\] 并假设对于每个充分小的$\lambda>0$，该方程存在一个关于$\lambda$一致下有界的严格次解。我们证明，在Mather商的小性假设下，当$\lambda\to 0^+$时，最大粘性解在由Aubry集的指定部分确定的折扣系数$a(x)$的符号条件下一致收敛。极限由Peierls势垒和所选取的Mather测度显式刻画。作为应用，与任何孤立静态类相关的临界Hamilton-Jacobi方程\\[ H\big(x,Du(x)\big)=c_0 \\] 的基本解可以实现为消失折扣极限。这提供了在经典单调性框架之外，在消失折扣问题中选取多个临界解的第一个机制，表明不同的Mather测度族产生不同的极限临界解。最后，我们提出了折扣问题的一个变体，对于该变体，严格次解假设自动成立，因此适用于更一般的Hamiltonian $H$和函数$a$。
英文摘要
We study a generalized vanishing discount problem for Hamilton--Jacobi equations without assuming monotonicity either pointwise or in the averaged sense with respect to all Mather measures. Specifically, we consider \[ λa(x)u(x)+H\big(x,Du(x)\big)=c_0, \] and assume that, for every sufficiently small $λ>0$, this equation admits a strict subsolution bounded from below uniformly with respect to $λ$. We prove that, under a smallness assumption on the Mather quotient, the maximal viscosity solution converges uniformly as $λ\to 0^+$ under a sign condition on the discount coefficient $a(x)$ determined by a prescribed part of the Aubry set. The limit is characterized explicitly in terms of the Peierls barrier and the selected Mather measures. As an application, the elementary solution of the critical Hamilton--Jacobi equation \[ H\big(x,Du(x)\big)=c_0 \] associated with any isolated static class can be realized as the vanishing discount limit. This provides the first mechanism for selecting multiple critical solutions in vanishing discount problems beyond the classical monotonicity framework, showing that different families of Mather measures yield different limiting critical solutions. We finally propose a variant of the discounted problems for which the strict subsolution hypothesis is automatically verified and hence applies to more general Hamiltonians $H$ and functions $a$.
Comments35 pages, 1 figure. This paper substantially extends and supersedes arXiv:2602.09697
