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

Pierpaolo Soravia

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 2, page 391-406
  • ISSN: 0392-4041

Abstract

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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.

How to cite

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Soravia, Pierpaolo. "Viscosity and Almost Everywhere Solutions of First-Order Carnot-Carathèodory Hamilton-Jacobi Equations." Bollettino dell'Unione Matematica Italiana 3.2 (2010): 391-406. <http://eudml.org/doc/290645>.

@article{Soravia2010,
abstract = {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.},
author = {Soravia, Pierpaolo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {391-406},
publisher = {Unione Matematica Italiana},
title = {Viscosity and Almost Everywhere Solutions of First-Order Carnot-Carathèodory Hamilton-Jacobi Equations},
url = {http://eudml.org/doc/290645},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Soravia, Pierpaolo
TI - Viscosity and Almost Everywhere Solutions of First-Order Carnot-Carathèodory Hamilton-Jacobi Equations
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/6//
PB - Unione Matematica Italiana
VL - 3
IS - 2
SP - 391
EP - 406
AB - 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.
LA - eng
UR - http://eudml.org/doc/290645
ER -

References

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