Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function

Abstract

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function. Let $H(x,p,u)$ be a continuous Hamiltonian which is strictly increasing in $u$, and is convex and coercive in $p$. For each parameter $\lambda >0$, we denote by $u^{\lambda }$ the unique viscosity solution of the H-J equation \[H( x,Du(x),λu(x) )=c.\] Under quite general assumptions, we prove that $u^{\lambda }$ converges uniformly, as $\lambda$ tends to zero, to a specific solution of the critical H-J equation $H(x,Du(x),0)=c.$ We also characterize the limit solution in terms of Peierls barrier and Mather measures.

References in corpus (2)

• Convergence of the solutions of the discounted equation: the discrete case
• Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample

Cited by in corpus (2)

• 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