On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions
On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions
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DOI：
10.1007/s00205-011-0484-1
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发表时间：
2011-09
影响因子：
2.5
通讯作者：
G. Barles;Hitoshi Ishii;Hiroyoshi Mitake
中科院分区：
文献类型：
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作者：
G. Barles;Hitoshi Ishii;Hiroyoshi Mitake
本文研究了有界区域上具有非线性Neumann边界条件的一阶Hamilton-Jacobi方程组解的大时间性态，包括动力学边界条件的情形.我们通过使用两种完全不同的方法建立了这些Cauchy-Neumann问题粘性解的一般收敛结果：第一种方法仅依赖于偏微分方程方法，其即使在Hamilton不是凸的情况下也提供结果，第二种方法是最优控制/动力系统方法，称为“弱KAM方法”，它要求Hamilton算子的凸性，并给出了基于Aubry-Mather集的渐近解的公式。
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.