Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T14:27:37.093Z Has data issue: false hasContentIssue false
  • English
  • Français

Monge solutions for discontinuous Hamiltonians

Published online by Cambridge University Press:  15 March 2005

Ariela Briani  and
Andrea Davini
Ariela Briani
Affiliation:
Dipartimento di Matematica, Università di Pisa Lago B. Pontecorvo 5, 56127 Pisa, Italy; [email protected]; [email protected]
Andrea Davini
Affiliation:
Dipartimento di Matematica, Università di Pisa Lago B. Pontecorvo 5, 56127 Pisa, Italy; [email protected]; [email protected]
Get access

Abstract

We consider an Hamilton-Jacobi equation of the form

 $$ H(x,Du)=0\quad x\in\Omega\subset\mathbb R^N,\qquad\qquad (1) $$ 
where H(x,p) is assumed Borel measurable and quasi-convex inp. The notion of Monge solution, introduced by Newcomb and Su,is adapted to this setting making use of suitable metric devices.We establish the comparison principle for Monge sub andsupersolution, existence and uniqueness for equation ([see full text])coupled with Dirichlet boundary conditions, and a stability result. Therelation among Monge and Lipschitz subsolutions is also discussed.

Keywords

Viscosity solutionlax formulaFinsler metric.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 2 , April 2005 , pp. 229 - 251
Copyright
© EDP Sciences, SMAI, 2005

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

L. Ambrosio and P. Tilli, Selected topics on “Analysis on Metric spaces”. Scuola Normale Superiore di Pisa (2000).
M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Syst. Control Found. Appl. (1997).
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Math. Appl. 17 (1994).
Barron, E.N. and Jensen, R., Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial Diff. Equ. 15 (1990) 1713–1742. CrossRef
G. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Res. Notes Math. Ser. 207 (1989).
Buttazzo, G., De Pascale, L. and Fragalà, I., Topological equivalence of some variational problems involving distances. Discrete Contin. Dyn. Syst. 7 (2001) 247–258. CrossRef
Caffarelli, L., Crandall, M.G., Kocan, M. and Swiech, A., On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math. 49 (1996) 365–397. 3.0.CO;2-A>CrossRef
Camilli, F. and Siconolfi, A., Hamilton-Jacobi equations with measurable dependence on the state variable. Adv. Differ. Equ. 8 (2003) 733–768.
F.H. Clarke, Optimization and Nonsmooth Analysis. John Wiley & Sons, New York (1983).
A. Davini, On the relaxation of a class of functionals defined on Riemannian distances. J. Convex Anal., to appear.
A. Davini, Smooth approximation of weak Finsler metrics. Adv. Differ. Equ., to appear.
G. De Cecco and G. Palmieri, Length of curves on LIP manifolds. Rend. Accad. Naz. Lincei, Ser. 9 1 (1990) 215–221.
De Cecco, G. and Palmieri, G., Integral distance on a Lipschitz Riemannian Manifold. Math. Z. 207 (1991) 223–243. CrossRef
G. De Cecco and G. Palmieri, Distanza intrinseca su una varietà finsleriana di Lipschitz. Rend. Accad. Naz. Sci. V, XVII, XL, Mem. Mat. 1 (1993) 129–151.
De Cecco, G. and Palmieri, G., LIP manifolds: from metric to Finslerian structure. Math. Z. 218 (1995) 223–237. CrossRef
Ishii, H., A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Sc. Norm. Sup. Pisa 16 (1989) 105–135.
H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets. Bull. Facul. Sci. & Eng., Chuo Univ., Ser I 28 (1985) 33–77.
P.L. Lions, Generalized solutions of Hamilton Jacobi equations. Pitman (Advanced Publishing Program). Res. Notes Math. 69 (1982).
Newcomb II, R.T. and Eikonal, J. Su equations with discontinuities. Differ. Integral Equ. 8 (1995) 1947–1960.
Soravia, P., Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J. 51 (2002) 451–477. CrossRef