We consider the Dirichlet problem for a class of Hamilton-Jacobi equations and we show that if the viscosity solution is semiconcave near the boundary then it is globally semiconcave. As an application, we discuss the particular case of eikonal-type equations. Finally, we describe a sort of stability property for the singular set of the semiconcave viscosity solution of Hamilton-Jacobi equations.
"On the regularity of the solution of the Dirichlet problem for Hamilton-Jacobi equations." Differential Integral Equations 18 (6) 601 - 610, 2005.