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Ergodic problems for contact Hamilton-Jacobi equations

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arxiv 2107.11554 v2 pith:GTKXECZX submitted 2021-07-24 math.AP math.DS

Kaizhi Wang , Jun Yan This is my paper · ORCID
classification math.APmath.DS
keywords mathfrakviscosityargumentconditionsergodicformulasolutionsabove
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abstract

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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Cited by 5 Pith papers

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