Abstract
Closed sets K ⊂ \(\mathbb R^{n}\) satisfying an external sphere condition with uniform radius (called ϕ-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 ϕ-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 δ (x) > 0 such that f is semiconvex on B(x,δ(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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Work partially supported by M.I.U.R., project “Viscosity, metric, and control theoretic methods for nonlinear partial differential equations.”
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Colombo, G., Marigonda, A. Differentiability properties for a class of non-convex functions. Calc. Var. 25, 1–31 (2006). https://doi.org/10.1007/s00526-005-0352-7
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DOI: https://doi.org/10.1007/s00526-005-0352-7