Convergence of Viscosity Solutions of Generalized Contact Hamilton–Jacobi Equations

For any compact connected manifold $M$, we consider the generalized contact Hamiltonian $H(x,p,u)$ defined on $T^*M\times\mathbb R$ which is conex in $p$ and monotonically increasing $u$. Let $u_\epsilon^-:M\rightarrow\mathbb be viscosity solution of parametrized Hamilton-Jacobi equation \[ H(x,\partial_x u_\epsilon^-(x),\epsilon u_\epsilon^-(x))=c(H) \] with $c(H)$ being Ma\~n\'e Critical Value. We prove that $u_\epsilon^-$ converges uniformly, as $\epsilon\rightarrow 0_+$, to a specfic $u_0^-$ critical u_0^-(x),0)=c(H) can characterized minimal combination associated Peierls barrier functions.