  It is well known that when the nonlinearity is convex, the Hamilton-Jacobi
PDE admits a unique semi-convex weak solution, which is the viscosity solution.
In this paper, motivated by problems arising from spin glasses, we show that if
the Hamilton-Jacobi PDE with strictly convex nonlinearity and regular enough
initial condition admits a semi-concave weak solution, then this solution is
the viscosity solution.
