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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.

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  • $\begingroup$ Any differential operator with smooth coefficients acts on distributions, $\endgroup$ Apr 23, 2014 at 9:54
  • $\begingroup$ @AlexanderShamov:I add a condition M is $C^0$ Riemannian maniold. And would you please write it in detail? $\endgroup$
    – wang mu
    Apr 23, 2014 at 11:15
  • $\begingroup$ My comment would only apply if the manifold and the metric were smooth, in which case $f \mapsto \nabla^2 f$ would be a smooth differential operator. In your nonsmooth case I don't know how to make sense of this. $\endgroup$ Apr 23, 2014 at 21:42

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