Generalized Hessians of C1, 1-Functions and Second-Order Viscosity Subjets

Given a C1,1–function f : U → R (where U ⊂ Rn open) we deal with the question of whether or not at a given point x0 ∈ U there exists a local minorant φ of f of class C2 that satisfies φ(x0) = f(x0), Dφ(x0) = Df(x0) and Dφ(x0) ∈ Hf(x0) (the generalized Hessian of f at x0). This question is motivated by the second-order viscosity theory of the PDE, since for nonsmooth functions, an analogous result between subgradients and first-order viscosity subjets is known to hold in every separable Asplund space. In this work we show that the aforementioned second–order result holds true whenever Hf(x0) has a minimum with respect to the semidefinite cone (thus in particular, in one dimension), but it fails in two dimensions even for piecewise polynomial functions. We extend this result by introducing a new notion of directional minimum of Hf(x0).