Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations

Abstract

For any compact connected manifold $M$, we consider the generalized contact Hamiltonian $H(x,p,u)$ defined on $T^{\ast }M\times \mathbb{R}$ which is conex in $p$ and monotonically increasing in $u$. Let $u_{\epsilon }^{-}:M\to \mathbb{R}$ be the viscosity solution of the parametrized contact Hamilton-Jacobi equation \[ H(x,\partial_x u_ε^-(x),εu_ε^-(x))=c(H) \] with $c(H)$ being the Mañé Critical Value. We prove that $u_{\epsilon }^{-}$ converges uniformly, as $\epsilon \to 0_{+}$, to a specfic viscosity solution $u_0^{-}$ of the critical equation \[ H(x,\partial_x u_0^-(x),0)=c(H) \] which can be characterized as a minimal combination of associated Peierls barrier functions.

viscosity solution, contact Hamiltonian, action minimizer, Aubry-Mather theory, weak KAM solution, Peierls barrier

References in corpus (1)

• Convergence of the solutions of the discounted equation: the discrete case

Cited by in corpus (4)

• Reviewing the Geometric Hamilton-Jacobi Theory concerning Jacobi and Leibniz identities
• Convergence of solutions for some degenerate discounted Hamilton--Jacobi equations
• 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