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arXiv 2107.11554math.APmath.DS
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接触Hamilton-Jacobi方程的遍历问题

Ergodic problems for contact Hamilton-Jacobi equations

Kaizhi Wang, Jun Yan

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37 pages
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摘要：

本文讨论了广义遍历问题\[ H(x,u(x),Du(x))=c, \quad x\in M, \]，其中未知量是一个常数$c \in \mathbb{R}$和一个函数$u$在$M$上的对$(c,u)$，其中$u$是粘性解.我们假设$H=H(x,u,p)$在变元$p\in T^*_xM$中满足Tonelli条件，在变元$u\in\R$中满足Lipschitz条件。对于给定的$c\in \R$，我们首先讨论了粘性解存在的充分必要条件。令$\mathfrak{C}$表示所有实数$c$的集合，对于该集合，上述方程允许粘性解。证明了$\mathfrak{C}$是一个区间，其端点$\x$，$\y$和$\x\leqslant\y$分别可用min-max公式和max-min公式刻画.最有意义的发现是我们在不对$u$作单调性假设的情况下，得出了$\mathfrak{C}$的结构.

英文摘要：

This paper deals with the generalized ergodic problem \[ H(x,u(x),Du(x))=c, \quad x\in M, \] where the unknown is a pair $(c,u)$ of a constant $c \in \mathbb{R}$ and a function $u$ on $M$ for which $u$ is a viscosity solution. We assume $H=H(x,u,p)$ satisfies Tonelli conditions in the argument $p\in T^*_xM$ and the Lipschitz condition in the argument $u\in\R$. For a given $c\in \R$, we first discuss necessary and sufficient conditions for the existence of viscosity solutions. Let $\mathfrak{C}$ denote the set of all real numbers $c$'s for which the above equation admits viscosity solutions. Then we show $\mathfrak{C}$ is an interval, whose endpoints $\x$, $\y$ with $\x\leqslant\y$ can be characterized by a min-max formula and a max-min formula, respectively. The most significant finding is that we figure out the structure of $\mathfrak{C}$ without monotonicity assumptions on $u$.

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