M is a $C^1$ manifold with a $C^0$ Riemannian metric, f is a convex function on M. How to define a functional on M which can represent $Hessf$?
For example: for $\Delta f$ we can define the corresponding functional: $$ Lap f(\phi)=\int_M \langle \nabla f,\nabla\phi \rangle $$ for Lipschitz function $\phi$ with compact support.