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

Pierpaolo Soravia — 2010

Bollettino dell'Unione Matematica Italiana

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...

Degenerate Eikonal equations with discontinuous refraction index

Pierpaolo Soravia — 2006

ESAIM: Control, Optimisation and Calculus of Variations

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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