Regularity and Global Structure of Solutions to Hamilton-Jacobi Equations I. Convex Hamiltonian

This paper is concerned with the Hamilton-Jacobi (HJ) equations of multidimensional space variables with convex Hamiltonian. Using Hopf’s formula (I), we will study the differentiability of the HJ solutions. For any given point, we give a sufficient and necessary condition under which the solutions are Ck smooth in some neighborhood of the point. We also study the characteristics of the HJ equations. It is shown that there are only two kinds of characteristics, one never touches the point of singularity, and the other touches the point of singularity in a finite time. The sufficient and necessary condition under which the characteristic never touches the point of singularity is given. Based on these results, we study the global structure of the set of singularity points for the HJ solutions. It is shown that there exists a one-to-one correspondence between the path connected components of the set of singularity points and the path connected components of a set on which the initial function does not attain its minimum. A path connected component of the set of singularity points never terminates at a finite time. Our results are independent of the particular forms of the equations as long as the Hamiltonian is convex.