an:06102015
Zbl 1282.70037
Barles, Guy; Ishii, Hitoshi; Mitake, Hiroyoshi
On the large time behavior of solutions of Hamilton-Jacobi equations associated with nonlinear boundary conditions
EN
Arch. Ration. Mech. Anal. 204, No. 2, 515-558 (2012).
00298814
2012
j
70H20 35Q70 49L25
Cauchy-von Neumann problem; convergence; weak KAM approach; Aubry-Mather set
Summary: We study the large time behavior of solutions of first-order Hamilton-Jacobi equations set in a bounded domain with nonlinear von Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-von Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the ``weak KAM approach'', which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry-Mather sets.