Abstract

We study nondifferentiability points for a class of continuous functions f:RN→Rf:\mathbb R^N\to\mathbb R whose epigraph satisfies a kind of external sphere condition with uniform radius (called φ\varphi-convexity or proximal smoothness). The functions belonging to this class are not necessarily Lipschitz. However, they enjoy some properties analogous to semiconvex functions; in particular they are twice LN\mathcal L^{N}-a.e. differentiable (see the authors in Calc. Var. 25 (2006) 1–31). In partial analogy with the study of singularities of semiconcave functions (see P. Cannarsa, C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control", Birkhäuser, Boston (2004)), under suitable conditions we give estimates from below of the nondifferentiability set, which consists of points where the subdifferential is not a singleton, as well as (differently from semiconvex functions) of points where it is empty. Furthermore, we show that if a function in this class is an a.e. solution of a Hamilton-Jacobi equation, then under suitable assumptions it is actually a viscosity solution. Methods of nonsmooth analysis and geometric measure theory are used, including a representation of Clarke's generalized gradient as the closed convex hull of limits of Fréchet derivatives.

Author information

Contact details are reproduced from the original publication and may be historical.

Giovanni Colombo

Dip. di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy

colombo@math.unipd.it

Antonio Marigonda

Dipartimento di Matematica Pura e Applicata, Università di Padova, via Belzoni 7, 35131 Padova, Italy

amarigo@math.unipd.it

Keywords

  • Nonsmooth analysis
  • proximal smoothness
  • semiconvex functions

Mathematics Subject Classification

49J52; 49L25

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Suggested citation

G. Colombo, A. Marigonda. “Singularities for a Class of Non-Convex Sets and Functions, and Viscosity Solutions of some Hamilton-Jacobi Equations.” Journal of Convex Analysis 15 (2008), No. 1, 105–129. https://doi.org/10.68381/jca15008

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