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

Abstract

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 ) {H(x,p,u)} be a continuous Hamiltonian which is strictly increasing in u, and is convex and coercive in p. For each parameter λ > 0 {\lambda>0} , we denote by u λ {u^{\lambda}} the unique viscosity solution of the Hamilton–Jacobi equation H ⁢ ( x , D ⁢ u ⁢ ( x ) , λ ⁢ u ⁢ ( x ) ) = c . H\big{(}x,Du(x),\lambda u(x)\big{)}=c. Under quite general assumptions, we prove that u λ {u^{\lambda}} converges uniformly, as λ tends to zero, to a specific solution of the critical Hamilton–Jacobi equation H ⁢ ( x , D ⁢ u ⁢ ( x ) , 0 ) = c {H(x,Du(x),0)=c} . We also characterize the limit solution in terms of Peierls barrier and Mather measures.