静态类引导的非单调消失折扣问题中的基本解选择
Static class-guided selection of elementary solutions in non-monotone vanishing discount problems
AI总结：
本文通过静态类引导的折扣系数，首次在非单调消失折扣问题中实现了多个粘性解的选择，突破了传统单调性和积分假设的限制。
摘要：
我们研究了Hamilton-Jacobi方程的广义消失折扣问题，去除了标准单调性假设，无论是全局还是当与所有Mather测度积分时。具体来说，我们考虑方程$\lambda a(x)u(x)+H(x,Du(x))-A\lambda=c_0$，其中$A$是一个合适选择的常数$A>0$。通过适当改变函数$a(x)$在不同静态类上的符号，我们证明了最大粘性解在$\lambda\to 0^+$时一致收敛，并且所有稳态方程$H(x,Du(x))=c_0$的基本解都可以被选择为极限。这为在通常单调性和积分假设之外的消失折扣问题中选择多个粘性解提供了首个结果，只要$a(x)$在某一静态类上为正。我们的结果强调了静态类在控制粘性解渐进行为中的关键作用。以前，在通常单调性假设下，只能选择单一解（如文献[GL]所述），而我们的方法通过静态类引导的折扣系数允许受控地选择多个解。
英文摘要：
We study a generalized vanishing discount problem for Hamilton--Jacobi equations, removing the standard monotonicity assumption, either in a global sense or when integrated against all Mather measures. Specifically, we consider \[ λa(x)u(x)+H(x,Du(x))-Aλ=c_0, \] with a suitably chosen constant $A>0$. By appropriately changing the signs of the function $a(x)$ on different static classes associated with $H$, we show that the maximal viscosity solution converges uniformly as $λ\to 0^+$ and that all elementary solutions of the stationary equation \[ H(x,Du(x))=c_0 \] can be selected as limits. This provides the first result for selecting multiple viscosity solutions in vanishing discount problems beyond the usual monotonicity and integral assumptions, as long as $a(x)$ is positive on one static class. Our results highlight the crucial role of static classes in controlling the asymptotic behavior of viscosity solutions. Previously, under usual monotonicity assumptions, only a single solution could be selected (as discussed in \cite{GL}), whereas our approach allows controlled selection of multiple solutions via static class-guided discount coefficients.
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