(mM)^(∞)算子及 Hamilton-Jacobi方程粘性解

(mM)^(∞) operators and viscosity solutions for hamilton-Jacobi equations

Hamilton- Jacobi方程经常出现在实际应用中 ,例如控制论 ,微分对策和经济理论 .本文构造了具有非凸通量和非凸初值的 Hamilton- Jacobi方程的粘性解 .主要思想是 :将通量分解为一个凸量加一个凹通量 ,利用新设计的算子 (m M) ∞ 和 L egendre变换 ,Hamilton- Jacobi方程的粘性解可以精确地表达 .(m M) ∞ 型解证明是 Hamilton- Jacobi方程的粘性解 .实际上我们的 (m M)∞公式是凸的 L ax- Oleinik- Hopf公式的非凸推广 .

Hamilton Jacobi equations are frequently encountered in applications, e.g., in control theory, differential games, and theory of economics. construct viscosity solutions of Hamilton Jacobi equations having a nonconvex flux and a nonconvex initial value. The main idea is: decomposit flux into convex flux plus concave flux, with the help of a newly designed operator (mM) ∞ and Legendre transform, the viscosity solutions of Hamilton Jacobi equations can be exactly expressed. The (mM) ∞ type Solutions is proved to be the viscosity solutions of Hamilton Jacobi equations. In fact our ( (mM) ∞ ) formula is a nonconvex generalization of the convex Lax Oleinik Hopfs formula.