Multiple asymptotic behaviors of solutions in the generalized vanishing discount problem
Published in Proceedings of the American Mathematical Society, 2023
Recommended citation: P. Ni, Multiple asymptotic behaviors of solutions in the generalized vanishing discount problem, Proc. Amer. Math. Soc. 151 (2023), 5239-5250. https://doi.org/10.1090/proc/16420
Considered the discounted Hamilton-Jacobi equation
λu(x)+H(x,Du(x))=0,
where λ>0. We have known that there is a unique solution u_λ. The family {u_λ}(λ∈(0,1]) is equi-Lipschitz continuous. Therefore, there is a sequence λ_n→0 such that u(λ_n)-min u_(λ_n) uniformly converges to a continuous function u and -λ_n min u_(λ_n ) converges to a constant c. By the stability of viscosity solutions, (u,c) solves H(x,Du(x))=c. In [1], the authors proved that the convergence of u_λ is uniform as λ→0. In [2], the author proved that the solution u_λ of
λa(x)u(x)+H(x,Du(x))=0
is unique, where a(x)≥0 and a(x)>0 on the Aubry set of H(x,p). Moreover, the convergence of u_λ is uniform as λ→0. Following these works, I gave a counter example showing that {u_λ} may be divergent when a(x) vanishes on the Aubry set of H(x,p) or changes its signs.
[1] A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique. Convergence of the solutions of the discounted Hamilton-Jacobi equation: convergence of the discounted solutions. Invent. Math., 206 (2016), 29–55.
[2] M. Zavidovique, Convergence of solutions for some degenerate discounted Hamilton-Jacobi equations. Analysis & PDE, 15 (2022), 1287–1311.
Recommended citation: P. Ni, Multiple asymptotic behaviors of solutions in the generalized vanishing discount problem, Proc. Amer. Math. Soc. 151 (2023), 5239-5250.