Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function
Qinbo Chen · Advances in Calculus of Variations · 2021
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 inu, and is convex and coercive inp. For each parameter λ>0 {\lambda>0} , we denote by uλ {u^{\lambda}} the unique viscosity solution of the Hamilton–Jacobi equation H(x,Du(x),λu(x))=c. H\big{(}x,Du(x),\lambda u(x)\big{)}=c. Under quite general assumptions, we prove that uλ {u^{\lambda}} converges uniformly, as λ tends to zero, to a specific solution of the critical Hamilton–Jacobi equation H(x,Du(x),0)=c {H(x,Du(x),0)=c} . We also characterize the limit solution in terms of Peierls barrier and Mather measures.