On the negative limit of viscosity solutions for discounted Hamilton-Jacobi equations

2021/12/09 by Wang, Ya-Nan, Yan, Jun, Zhang, Jianlu · 2 citations
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.05018

Abstract

Suppose M is a closed Riemannian manifold. For a C² generic (in the sense of Mañé) Tonelli Hamiltonian H: T^*M→ℝ, the minimal viscosity solution u_λ^-:M→ ℝ of the negative discounted equation -λu+H(x,d_xu)=c(H), x∈ M, λgt;0 with the Mañé's critical value c(H) converges to a uniquely established viscosity solution u₀^- of the critical Hamilton-Jacobi equation H(x,d_x u)=c(H), x∈ M as λ→ 0₊. We also propose a dynamical interpretation of u₀^-.

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