Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T10:32:10.028Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenization of Hamilton-Jacobi equations in Carnot Groups

Published online by Cambridge University Press:  14 February 2007

Bianca Stroffolini
Bianca Stroffolini*
Affiliation:
Dipartimento di Matematica e Applicazioni Università degli studi di Napoli Federico II Complesso Monte S. Angelo Edificio “T” via Cintia, 80126 Napoli Italy; [email protected]
Get access

Abstract

We study an homogenization problem for Hamilton-Jacobi equations in the geometry of Carnot Groups. The tiling and the corresponding notion of periodicity are compatible with the dilatations of the Group and use the Lie bracket generating property.

Keywords

HomogenizationCarnot GroupsHamilton-Jacobi.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 1 , January 2007 , pp. 107 - 119
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alvarez, O., and Bardi, M., Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40 (2001) 1159–1188. CrossRef
Arisawa, M., Quasi-periodic homogenizations for second-order Hamilton-Jacobi-Bellmann equations. Adv. Sci. Appl. 11 (2001) 465–480.
M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser Boston, Boston, MA (1997).
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Math. Appl. 17 (1994).
A. Bellaïche and J.-J. Risler, ed., Sub-Riemannian Geometry. Birkhäuser, Progress. Math. 144 (1996).
Birindelli, I. and Wigniolle, J., Homogenization of Hamilton-Jacobi equations in the Heisenberg Group. Commun. Pure Appl. Anal. 2 (2003) 461–479.
Capuzzo Dolcetta, I. and Ishii, H., On the rate of convergence in homogenization of Hamilton-Jacobi equations. Indiana University Math. J. 50 (2001) 1113–1129. CrossRef
Evans, L.C., The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh 11A (1989) 359–375. CrossRef
Evans, L.C., Periodic homogenization of certain fully nonlinear PDE. Proc. Roy. Soc. Edinburgh 120 (1992) 245–265. CrossRef
G.B. Folland and E.M. Stein, Hardy spaces on homogeneous groups. Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo. Math. Notes 28 (1982)
Ishii, H., Perron's method for Hamilton-Jacobi equations. Duke Math. J. 55 (1987) 369–384. CrossRef
P. Juutinen, G. Lu, J. Manfredi and B. Stroffolini, Convex functions on Carnot Groups, to appear in Revista Mathematica Iberoamericana.
Lu, G., Manfredi, J. and Stroffolini, B., Convex functions on the Heisenberg Group. Calc. Var. Partial Differential Equations 19 (2004) 1–22. CrossRef
P.L. Lions, G. Papanicolau and R.S. Varadhan, Homogenization of Hamilton-Jacobi equations, preprint (1986).
J. Manfredi, Nonlinear subelliptic equations on Carnot Groups. Notes of a course at the School on Analysis and Geometry, Trento (2003).
J. Manfredi and B. Stroffolini A Version of the Hopf-Lax Formula in the Heisenberg Group. Comm. in Partial Differential Equations 27 (2002) 1139–1159.
R. Montgomery, A Tour of Subriemannian Geometries, their geodesics and applications. American Mathematical Society, Providence, RI. Math. Surveys Monographs 91(2002).
Monti, R. and Serra Cassano, F., Surface measures in Carnot-Carathéodory spaces. Calc. Var. 13 (2001) 339-376. CrossRef
Morbidelli, D., Fractional Sobolev norms and structure of Carnot-Caratheodory balls for Hörmander vector fields. Studia Math. 139 (2000) 213–244. CrossRef
Nagel, A., Stein, E.M. and Wainger, S., Balls and metrics defined by vector fields I: basic properties. Acta Math. 137 (1976) 247–320.