Metric character of Hamilton-Jacobi equations

Author:
Antonio Siconolfi

Journal:
Trans. Amer. Math. Soc. **355** (2003), 1987-2009

MSC (2000):
Primary 35F20, 49L25

DOI:
https://doi.org/10.1090/S0002-9947-03-03237-9

Published electronically:
January 8, 2003

Erratum:
Trans. Amer. Math. Soc. (recently posted).

MathSciNet review:
1953535

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We deal with the metrics related to Hamilton-Jacobi equations of eikonal type. If no convexity conditions are assumed on the Hamiltonian, these metrics are expressed by an - formula involving certain level sets of the Hamiltonian. In the case where these level sets are star-shaped with respect to 0, we study the induced length metric and show that it coincides with the Finsler metric related to a suitable convexification of the equation.

**1.**M. Bardi and I. Capuzzo Dolcetta,*Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations*, Birkhäuser, Boston, 1997. MR**99e:49001****2.**M. Bardi and L. C. Evans, ``On Hopf's formulas for solutions of Hamilton-Jacobi equations'',*Nonlinear Anal. TMA***8**(1984), 1373-1381. MR**85k:35043****3.**G. Barles,*Solutions de viscosité des équations de Hamilton-Jacobi*, Springer, Paris, 1994. MR**2000b:49054****4.**L. M. Blumenthal,*Theory and applications of distance geometry*, Oxford Univ. Press, London, 1953, reprinted Chelsea Publ., New York, 1979. MR**14:1009a**; MR**42:3678****5.**H. Busemann,*Metric methods in Finsler spaces and in the foundations of geometry*, Ann. Math. Study, Princeton, 1942. MR**4:109e****6.**H. Busemann,*The geometry of geodesics*, Academic Press, New York, 1955. MR**17:779a****7.**H. Busemann and W. Mayer, ``On the foundations of calculus of variations'',*Trans. Amer. Math. Soc.***49**(1941), 173-198. MR**2:225d****8.**F. Camilli and A. Siconolfi, ``Maximal subsolutions for a class of degenerate Hamilton-Jacobi equations'',*Indiana Univ. Math. J.***48**(1999), 1111-1131. MR**2001a:49028****9.**F. Camilli and A. Siconolfi, ``Nonconvex degenerate Hamilton-Jacobi equations'',*Math. Z.***242**(2002), 1-21.**10.**R. Courant and D. Hilbert,*Methods of Mathematical Physics, Volume II*, John Wiley , New York, 1962, reprinted 1989. MR**25:4216**; MR**90k:35001****11.**L. C. Evans and H. Ishii, ``Differential games and nonlinear first order PDE in bounded domains'',*Manuscripta Math.***49**(1984), 109-139. MR**86f:35048****12.**L. C. Evans and P. Souganidis, ``Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations'',*Indiana Univ. Math. J.***33**(1984), 773-797. MR**86d:90185****13.**S. N. Kruzkov, ``Generalized solutions of the Hamilton-Jacobi equations of eikonal type'',*Math. USSR Sbornik***27**(1975), 406-446. MR**53:8670****14.**M. Gromov,*Metric structures for Riemannian and non-Riemannian spaces*, Birkhäuser, Boston, 1998. MR**2000d:53065****15.**P. L. Lions,*Generalized solutions of Hamilton-Jacobi equations*, Pitman, London, 1982. MR**84a:49038**

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Additional Information

**Antonio Siconolfi**

Affiliation:
Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazzale Aldo Moro, 2, 00185 Roma, Italy

Email:
siconolfi@mat.uniroma1.it

DOI:
https://doi.org/10.1090/S0002-9947-03-03237-9

Keywords:
Hamilton--Jacobi equations,
viscosity solutions,
distance functions

Received by editor(s):
May 9, 2000

Received by editor(s) in revised form:
May 18, 2001

Published electronically:
January 8, 2003

Additional Notes:
Research partially supported by the TMR Network “Viscosity Solutions and Applications”

Article copyright:
© Copyright 2003
American Mathematical Society