Generalized Hessians of $C^{1,1}$-Functions and Second-Order Viscosity Subjets

Abstract

Given a $C^{1,1}$-function $f:U\rightarrow R$ (where $U\subset R^{n}$ open), we deal with the question of whether or not at a given point $x_{0}\in U$ there exists a local minorant $\varphi$ of $f$ of class $C^{2}$ that satisfies $\varphi(x_{0})=f(x_{0})$, $D\varphi(x_{0})=Df(x_{0})$, and $D^{2}\varphi(x_{0})\in\mathcal{H}f(x_{0})$ (the generalized Hessian of $f$ at $x_{0}$). This question is motivated by the second-order viscosity theory of the PDEs, 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 $\mathcal{H}f(x_{0})$ has a minimum with respect to the positive 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 $\mathcal{H}f(x_{0})$.