2021/07/24 by Wang, Kaizhi, Yan, Jun
#35D40 #35F21 #37J51 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2107.11554
This paper deals with the generalized ergodic problem H(x,u(x),Du(x))=c, x∈ M, where the unknown is a pair (c,u) of a constant c ∈ 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∈ T^*xM and the Lipschitz condition in the argument u∈\R. For a given c∈ \R, we first discuss necessary and sufficient conditions for the existence of viscosity solutions. Let \mathfrakC denote the set of all real numbers c's for which the above equation admits viscosity solutions. Then we show \mathfrakC 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 \mathfrakC without monotonicity assumptions on u.