(mM)∞算子及Hamilton-Jacobi方程粘性解

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 ex-pressed. The (mM)^∞ type Solutions is proved to be the viscosity solutions ofHamilton-Jacobi equations. In fact our ( (mM)^∞ ) formula is a nonconvex generalization of the convex Lax-Oleinik-Hopf’s formula.