Regularity and Global Structure of Solutions to Hamilton-Jacobi equations II. Convex initial data

The paper is concerned with the Hamilton-Jacobi (HJ) equations of multidimensional space variables with convex initial data and general Hamiltonians. Using Hopf’s formula (II), we will study the differentiability of the HJ solutions. For any given point,we give a sufficient and necessary condition such that the solutions are Ck smooth in some neighborhood of this point. We also study the characteristics of the equations which play important roles in our analysis. It is shown that there are only two kinds of characteristics, one never touches the singularity point, but the other one touches the singularity point in a finite time. Based on these results, we study the global structure of the set of singularity points for the solutions. It is shown that there exists a one-to-one correspondence between the path connected components of the set of singularity points and path connected component of the set {(Dg(y), H(Dg(y)))| y ∈ Rn} \ {(Dg(y), convH(Dg(y)))| y ∈ Rn}, where convH is the convex hull of H. A path connected component of the set of singularity points never terminates as t increases. Moreover, our results depend only on H and its domain of definition.