On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions
By Guy Barles, Hitoshi Ishii, Hiroyoshi Mitake. Archive for Rational Mechanics and Analysis, 2011
116 citations recorded by the source corpus.
In this article, we study the large time behavior of solutions of first-order Hamilton–Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy–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.
Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health
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In Applied Mathematics
- Dipmat
- Alessio Porretta Ceremade
- Pierre Cardaliaguet
- P. J. Graber
- Daniel Lacker
- Diogo A. Gomes
- Edgard A. Pimentel
- João Saúde
In Modeling and Simulation
In Public Health, Environmental and Occupational Health
Citations in the corpus
- Long time behavior of the master equation in mean field game theory (2017, methodology): “4.2. However, the convergence of the solution of the time dependent master equation (3) requires new ideas since, in contrast with the Hamilton-Jacobi setting (see Fathi [11] or Barles and Souganidis [1]), there is no obvious quantity which is monotone in time: the reason is that the master equation”