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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...
We study the Dirichlet boundary value problem for eikonal type equations of ray
light propagation in an inhomogeneous medium with discontinuous
refraction index. We prove a comparison principle
that allows us to obtain existence and uniqueness of a continuous
viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander
type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value...
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