Differentiability properties for a class of non-convex functions

Colombo, Giovanni; Marigonda, Antonio
January 2006
Calculus of Variations & Partial Differential Equations;Jan2006, Vol. 25 Issue 1, p1
Academic Journal
Closed sets K ? $$\mathbb R^{n}$$ satisfying an external sphere condition with uniform radius (called ?-convexity or proximal smoothness) are considered. It is shown that for $$\mathcal H^{n-1}$$ -a.e. x ? ? K the proximal normal cone to K at x has dimension one. Moreover if K is the closure of an open set satisfying a (sharp) nondegeneracy condition, then the De Giorgi reduced boundary is equivalent to ? K and the unit proximal normal equals $$\mathcal H^{n-1}$$ -a.e. the (De Giorgi) external normal. Then lower semicontinuous functions f : $$\mathbb R^{n}\rightarrow \mathbb R\cup\{ +\infty\}$$ with ?-convex epigraph are shown, among other results, to be locally BV and twice $$\mathcal L^{n}$$ -a.e. differentiable; furthermore, the lower dimensional rectifiability of the singular set where f is not differentiable is studied. Finally we show that for $$\mathcal L^{n}$$ -a.e. x there exists d ( x) > 0 such that f is semiconvex on B( x,d( x)). We remark that such functions are neither convex nor locally Lipschitz, in general. Methods of nonsmooth analysis and of geometric measure theory are used.


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