Vanishing contact structure problem and convergence of the viscosity solutions

TL;DR

This paper investigates the vanishing contact structure problem, a generalization of the vanishing discount problem, proving convergence of viscosity solutions of Hamilton-Jacobi equations with contact Hamiltonians as the parameter tends to zero.

Contribution

It establishes the convergence of viscosity solutions for a family of contact Hamilton-Jacobi equations, extending the understanding of vanishing contact structure problems.

Findings

Viscosity solutions $u^{mbda}$ converge to a specific solution $u^0$ as $mbda o 0$.

Provides convergence results for nonlinear vanishing discount problems.

Extends the theory of Hamilton-Jacobi equations to contact type Hamiltonians.

Abstract

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