On the Variational Behaviour of the Subhessians of the Lasry-Lions Envelope

A full analysis is made of the effect that the Lasry-Lions double-envelope has on the subjet and associated subhessians of a prox-bounded, lower-semicontinuous function. This enables the (basic) subdifferential of nonsmooth analysis and the limiting subjet of viscosity-solution theory to be characterized via accumulation points of the first- and second-order derivative information of approximating functions provided by the Lasry-Lions double-envelope