# Preprint 2008-030

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

## Tao Tang, Jinghua Wang and Yinchuan Zhao

**Abstract:**
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
*C*^{k} 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∈**R**^{n}} \ {(*Dg*(*y*), conv *H*(*Dg*(*y*))) | *y* ∈ **R**^{n}},
where conv *H*
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.