发展型Hamilton-Jacobi方程的变分算子和粘性算子
Variational and viscosity operators for the evolutive Hamilton-Jacobi equation
摘要：
研究了一类具有Lipschitz初值的一阶发展型Hamilton-Jacobi方程的Cauchy问题。哈密顿量在动量变量中不一定是凸的，也不一定是先验紧支持的。我们构造并研究了一个给出该问题变分解的算子，得到了该算子的局部Lipschitz估计。迭代这个变分算子，我们得到粘性算子和扩展的估计粘性框架。我们还检查的变分算子的建设给出了Lax-Oleinik半群，如果哈密顿量是凸的或凹的动量变量。
英文摘要：
We study the Cauchy problem for the first order evolutive Hamilton-Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and study an operator giving a variational solution of this problem, and get local Lipschitz estimates on this operator. Iterating this variational operator we obtain the viscosity operator and extend the estimates to the viscosity framework. We also check that the construction of the variational operator gives the Lax-Oleinik semigroup if the Hamiltonian is convex or concave in the momentum variable.
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