Vanishing contact structure problem and convergence of the viscosity solutions

AbstractThis article is devoted to studying the vanishing contact structure problem which is a generalization of the vanishing discount problem. Let Hλ(x,p,u) be a family of Hamiltonians of contact type with parameter λ>0 and converge to G(x, p). For the contact type Hamilton-Jacobi equation with respect to Hλ, we prove that, under mild assumptions, the associated viscosity solution uλ converges to a specific viscosity solution u0 of the vanished contact equation. As applications, we give some convergence results for the nonlinear vanishing discount problem.