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On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00623000
Contributor : Guy Barles <>
Submitted on : Thursday, January 12, 2012 - 7:54:22 AM
Last modification on : Friday, October 25, 2019 - 12:18:28 PM
Document(s) archivé(s) le : Friday, April 13, 2012 - 2:22:49 AM

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  • HAL Id : hal-00623000, version 2
  • ARXIV : 1109.2762

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Guy Barles, Hiroyoshi Mitake, Hitoshi Ishii. On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions. Archive for Rational Mechanics and Analysis, Springer Verlag, 2012, 204 (2), pp.515-558. ⟨hal-00623000v2⟩

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