Regularity and global structure for Hamilton–Jacobi equations with convex Hamiltonian

We consider the multidimensional Hamilton–Jacobi (HJ) equation ut + γ⁻¹|Du|^γ = 0 with 1 < γ < 2 being a constant and for bounded C² initial data. When γ = 2, this is the typical case of interest with a uniformly convex Hamiltonian. When γ = 1, this is the famous Eikonal equation from geometric optics, the Hamiltonian being Lipschitz continuous with homogeneity 1. We intend to fill the gap in between these two cases. When 1 < γ < 2, the Hamiltonian H(p) = γ⁻¹|p|^γ is not uniformly convex and is only C¹ in any neighborhood of 0, which causes new difficulties. In particular, points on characteristics emanating from points with vanishing gradient of the initial data could be “bad” points, so the singular set is more complicated than what is observed in the case γ = 2. We establish here the regularity of solutions and the global structure of the singular set from a topological standpoint: the solution inherits the regularity of the initial data in the complement of the singular set and there is a one-to-one correspondence between the connected components of the singular set and the path-connected components of the set {y0 |g(y0) > inf_y∈Rn g(y)}.