\((mM)^\infty\) operators and viscosity solutions for Hamilton-Jacobi equations (Q2735446)

The following initial value problems NEWLINE\[NEWLINE\begin{cases} v_t +f(v_x)= 0\text{ in }\mathbb{R}\times [0,\infty),\\ v(x,0)= v_0(x)\text{ in }\mathbb{R} \end{cases} \tag{H-J}NEWLINE\]NEWLINE are considered. The main idea of this work is to decompose the flux into convex flux plus concave flux and, with the help of a newly designed operator \((mM)^\infty\) and Legendre transform, the viscosity solutions of Hamilton-Jacobi equations can be exactly expressed. The \((mM)^\infty\) type solutions are proved to be the viscosity solutions of Hamilton-Jacobi equations. In fact, the \((mM)^\infty\) formula is a nonconvex generalization of the convex Lax-Oleinik-Hopf's formula. This method may have some relations with (though independent of) game theory where the minimax value of so-called mixed strategy may be equivalent to the \((mM)^\infty\) value.