Soravia: Viscosity and Almost Everywhere Solutions of First-Order Carnot-Carathèodory Hamilton-Jacobi Equations

Soravia, Pierpaolo:
Viscosity and Almost Everywhere Solutions of First-Order Carnot-Carathèodory Hamilton-Jacobi Equations
Bollettino dell'Unione Matematica Italiana Serie 9 3 (2010), fasc. n.2, p. 391-406, (English)
pdf (302 Kb), djvu (153 Kb). | MR 2666366 | Zbl 1195.49034

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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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