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 $H^{\lambda }(x,p,u)$ 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.

References in corpus (7)

• Contact Hamiltonian Mechanics
• Convergence of the solutions of the discounted equation
• Convergence of the solutions of the discounted equation: the discrete case
• The vanishing discount problem and viscosity Mather measures. Part 1: the problem on a torus
• The vanishing discount problem and viscosity Mather measures. Part 2: boundary value problems
• Aubry-Mather Theory for Conformally Symplectic Systems
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Cited by in corpus (7)

• Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations
• Convergence of solutions for some degenerate discounted Hamilton--Jacobi equations
• Vanishing discount problem and the additive eigenvalues on changing domains
• Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function
• Global dynamics of contact Hamiltonian systems (I): monotone systems
• Limit of solutions for semilinear Hamilton-Jacobi equations with degenerate viscosity
• Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains