Geodesic
Viscosity and Almost Everywhere Solutions of First-Order Carnot-Carathèodory Hamilton-Jacobi Equations
Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 391-406.

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We consider viscosity and distributional derivatives of functions in the directions of a family of vector fields, generators of a Carnot-Carathèodory (C-C in brief) metric. In the framework of convex and non coercive Hamilton-Jacobi equations of C-C type we show that viscosity and a.e. subsolutions are equivalent concepts. The latter is a concept related to Lipschitz continuity with respect to the metric generated by the family of vector fields. Under more restrictive assumptions that include Carnot groups, we prove that viscosity solutions of C-C HJ equations are Lipschitz continuous with respect to the corresponding Carnot-Carathèodory metric and satisfy the equation a.e. 
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Soravia, Pierpaolo. Viscosity and Almost Everywhere Solutions of First-Order Carnot-Carathèodory Hamilton-Jacobi Equations. Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 391-406. http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_2_a9/

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