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On the vanishing discount problem from the negative direction

Discrete and Continuous Dynamical Systems- Series A, ISSN: 1553-5231, Vol: 41, Issue: 5, Page: 2377-2389
2021
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It has been proved in [7] that the unique viscosity solution of λu + H(x, dxu) = c(H) in M, (*) uniformly converges, for λ → 0, to a specific solution u 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 note, we 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 u of (*). Under the assumption that constant functions are subsolutions of the critical equation, we prove that the u 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.

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