SBV Regularity for Hamilton–Jacobi Equations in \({{\mathbb R}^n}\)

TLDR: In this article, the regularity of viscosity solutions to the Hamilton-Jacobi equation was studied under the assumption that the Hamiltonian is uniformly convex, and it was shown that the class SBV isEnabled loc                 (Ω) belongs to the class of SBV▬▬▬▬▬▬▬ ǫ.

Abstract: In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations $$\partial_{t}u+H(D_{x}u)=0\quad\hbox{in }\Omega\subset{\mathbb R}\times{\mathbb R}^{n}.$$ In particular, under the assumption that the Hamiltonian $${H\in C^2({\mathbb R}^n)}$$ is uniformly convex, we prove that D x u and ∂ t u belong to the class SBV loc (Ω).