Vanishing discount problem and the additive eigenvalues on changing domains

Abstract

Let Ω be a bounded open subset of Rn and H(x,p):Ω×Rn→R be a continuous Hamiltonian that is convex in the second argument. We study the asymptotic behavior, as λ→0+, of the state-constraint Hamilton–Jacobi equation(Sλ){ϕ(λ)uλ(x)+H(x,Duλ(x))⩽0in(1+r(λ))Ω,ϕ(λ)uλ(x)+H(x,Duλ(x))⩾0on(1+r(λ))Ω‾, and the corresponding additive eigenvalues, or ergodic constant(Eλ){H(x,Dvλ(x))⩽c(λ)in(1+r(λ))Ω,H(x,Dvλ(x))⩾c(λ)on(1+r(λ))Ω‾. Here, ϕ(λ),r(λ):(0,∞)→R are continuous functions such that ϕ is nonnegative and limλ→0+⁡ϕ(λ)=limλ→0+⁡r(λ)=0. We obtain both convergence and non-convergence results for the convex Hamilton–Jacobi equations. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue c(λ) as λ→0+. The main tool we use is a duality representation of solution with viscosity Mather measures.