发表机构
Sapienza Università di Roma（罗马大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究周期Hamilton量的常号局部扰动的消失折扣选择问题，证明折扣解收敛到特定临界解，并给出反例说明自然公式失效。
AI 中文摘要
我们研究了定义在${\mathbb R}^d$上的临界Hamilton-Jacobi方程的消失折扣选择问题，其Hamilton量为$G=H-V$，其中$H$在空间上连续、周期、关于动量凸且超线性，$V$是具有常号的紧支撑势。对于$V\leqslant0$，我们证明了折扣解局部一致收敛到一个特殊的临界解，该解通过Mather测度和无扰动周期问题所选择的解来刻画。证明依赖于将极限折扣测度分解为保留分量和周期分量，后者解释了在无穷远处损失的质量。对于$V\geqslant0$，我们在额外假设无扰动选择的解关于每个周期Mather测度具有零平均值的情况下建立了收敛性。我们还给出了显式例子，其中折扣解收敛，但候选极限的自然公式未能识别所选择的解。
英文摘要
We study the vanishing discount selection problem for critical Hamil\-ton--Jacobi equations posed on ${\mathbb R}^d$ with Hamiltonian $G=H-V$, where $H$ is continuous, periodic in space, convex and superlinear in the momentum, and $V$ is a compactly supported potential of constant sign. For $V\leqslant0$, we prove local uniform convergence of the discounted solutions to a distinguished critical solution, which we characterize in terms of Mather measures and of the solution selected by the unperturbed periodic problem. The proof relies on a decomposition of limiting discounted measures into a retained component and a periodic component accounting for the mass lost at infinity. For $V\geqslant0$, we establish convergence under the additional assumption that the unperturbed selected solution has zero average with respect to every periodic Mather measure. We also give explicit examples in which the discounted solutions converge, but natural formulas for the candidate limit fail to identify the selected solution.
