Uniqueness of semi-concave weak Solutions for Hamilton–Jacobi Equations

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.