Singularities Of The Viscosity Solutions And Lax-Oleinik Operators

The theory of the viscosity solutions of Hamilton-Jacobi equations has already been widely studied in the past several decades,in the fields such as PDEs,calculus of variations and optimal controls,and differential games.Now we try to combine the methods from PDEs and optimal controls with the idea from Hamilton-Lagrange dynamics.We focus on the problem on the propagation of singularities of viscosity solutions of Hamilton-Jacobi equations.Recent intrinsic approach is based on the following two points.One is the regularity properties of the fundamental solutions of the associated Hamilton-Jacobi equation,the other is the critical points of local barrier function in the procedure of sup-convolution defined by the Lax-Oleinik operators.Along this line,We obtained the following conclusions:1.For non-autonomous Tonelli-Lagrange system,under suitable conditions,the fundamental solution Ats(x,y)is locally C1,1 in the variable(t,y)on the cone S?(x,t'?):={(t,y)?R×Rn:s<t<s+t'?,|y-x|<?(t-s)}2.For time-independent Tonelli-Lagrange system,using the regularity of fundamental solution,the so-called Larsy-Lions regularization procedure is closely linked to generalized characteristics when we study the evolution of critical point of local barrier function defined in the procedure of sup-convolution defined by Lax-Oleinik operators.We also give the connection between local minimizing curve and the singularity of viscosity solution in the procedure of inf-convolution as well.