Vanishing contact structure problem and convergence of the viscosity solutions

Abstract

This paper is devoted to study the vanishing contact structure problem which is a generalization of the vanishing discount problem. Let $H^{\hat{I}}»(x,p,u)$ be a family of Hamiltonians of contact type with parameter $\hat{I}»>0$ and converges to $G(x,p)$. For the contact type Hamilton-Jacobi equation with respect to $H^{\hat{I}}»$, we prove that, under mild assumptions, the associated viscosity solution $u^{\hat{I}}»$ converges to a specific viscosity solution $u^0$ of the vanished contact equation. As applications, we give some convergence results for the nonlinear vanishing discount problem.