【online】On the vanishing discount problem from the negative direction

(*)λuλ+H(x,dxuλ)=c(H)in M,

uniformly converges, for λ→0+, to a specific solution u0 of the critical equation

H(x,dxu)=c(H)in M,

where M is a closed and connected Riemannian manifold and c(H) is the critical value.

In this seminar, we will consider the same problem for λ→0−. In this case, viscosity solutions of equation (*) are not unique, in general, so we focus on the asymptotics of the minimal solution uλ− of (*). Under the assumption that constant functions are subsolutions of the critical equation, we prove that the uλ−also converges to u0 as λ→0−. Furthermore, we exhibit an example of H for which equation (*) admits a unique solution for λ<0 as well. The talk is based on a joint work with Lin Wang (Tsinghua University).