GLOBAL GENERALIZED CHARACTERISTICS FOR THE DIRICHLET PROBLEM FOR HAMILTON-JACOBI EQUATIONS AT A SUPERCRITICAL ENERGY LEVEL
GLOBAL GENERALIZED CHARACTERISTICS FOR THE DIRICHLET PROBLEM FOR HAMILTON-JACOBI EQUATIONS AT A SUPERCRITICAL ENERGY LEVEL
复制标题
超临界能级汉密尔顿-雅可比方程狄利克雷问题的全局广义特征
DOI：
10.1137/18m1203547
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发表时间：
2019-01-01
影响因子：
2
通讯作者：
Wang, Kaizhi
中科院分区：
文献类型：
--
作者：
Cannarsa, Piermarco;Cheng, Wei;Wang, Kaizhi
本文研究了在R ~ n的有界区域Ω上的一阶Hamilton-Jacobi方程的非齐次Dirichlet问题，其中Tonelli哈密顿算子是在超临界能级下的.首先，我们证明了这样一个问题的粘性（弱KAM）解是Lipschitz连续的，并且在Omega中局部收敛。然后，我们分析了奇异集的解决方案，表明奇异性传播沿着合适的曲线，所谓的广义特征，并且这些曲线保持奇异，除非他们达到边界的欧米茄。此外，我们证明了后者是从来没有的情况下，力学系统和奇异广义特征收敛到一个临界点的解决方案在有限或无限的时间。最后，在更强的假设下的域和Dirichlet数据，我们能够得出结论，解决方案是全球性的，并在边界附近。
We study the nonhomogeneous Dirichlet problem for first-order Hamilton-Jacobi equations associated with Tonelli Hamiltonians on a bounded domain Omega of R-n assuming the energy level to be supercritical. First, we show that the viscosity (weak KAM) solution of such a problem is Lipschitz continuous and locally semiconcave in Omega. Then, we analyze the singular set of a solution showing that singularities propagate along suitable curves, the so-called generalized characteristics, and that such curves stay singular unless they reach the boundary of Omega. Moreover, we prove that the latter is never the case for mechanical systems and that singular generalized characteristics converge to a critical point of the solution in finite or infinite time. Finally, under stronger assumptions for the domain and Dirichlet data, we are able to conclude that solutions are globally semiconcave and semiconvex near the boundary.