Regularity Results for Solutions of a Class of Hamilton-Jacobi Equations

Abstract.

The regularity of the gradient of viscosity solutions of first‐order Hamilton‐Jacobi equations \begin{eqnarray*} \begin{array}{rll} \partial_t u(t,x) + H( t, x, D_x u(t,x))=0,&\quad & t\in\real_+,\es x \in \real^n\,, \\[3pt] u(0,x) = \uzero (x), &\quad & x \in \real^n\,, \end{array} \end{eqnarray*} is studied under a strict convexity assumption on H(t,x,⋅). Estimates on the discontinuity set of Du are derived. Such estimates imply that solutions of the above problem are smooth in the complement of a closed ℋⁿ‐rectifiable set. In particular, it follows that Du belongs to the classSBV, i.e., D ² u$ is a measure with no Cantor part.