Ergodic problems for contact Hamilton-Jacobi equations

TL;DR

This paper investigates the ergodic problem for contact Hamilton-Jacobi equations, characterizing the set of constants for which viscosity solutions exist and revealing the structure of this set without requiring monotonicity assumptions.

Contribution

It provides necessary and sufficient conditions for existence of solutions and characterizes the solution set interval using min-max and max-min formulas, without monotonicity assumptions.

Findings

The set of constants admitting solutions is an interval.

Endpoints of this interval are characterized by min-max and max-min formulas.

The structure of the solution set is determined without monotonicity assumptions.

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 $u$.