Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations.
Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations.
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DOI：
10.1090/s0002-9947-1984-0732102-x
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发表时间：
1984-02
影响因子：
1.3
通讯作者：
M. Crandall;L. Evans;P. Lions
中科院分区：
文献类型：
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作者：
M. Crandall;L. Evans;P. Lions
摘要：M. G. Crandall和P. L. Lions最近引入了标量非线性一阶偏微分方程的“粘性解”的概念。黏度解不必处处可微，因此对经典的特征交叉问题不敏感。应用领域中出现的许多问题的粘度解都具有非常普遍的存在性、唯一性和连续依赖性，这一事实证明了这一概念的价值。“粘度解”的概念允许几种等价的公式。在这里，我们更仔细地考察这两个等效的标准，并通过证明几个新的事实和以一种更简单的方式反驳各种已知的结果来展示它们的优点。此外，通过放弃技术上的一般性，我们在此提供了一个比原论文更适合的介绍。(作者)
Abstract : Recently M. G. Crandall and P. L. Lions introduced the notion of 'viscosity solutions' of scalar nonlinear first order partial differential equations. Viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics. The value of this concept is established by the fact that very general existence, uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application. The notion of a 'viscosity solution' admits several equivalent formulations. Here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner. Moreover, by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper. (Author)