Semicontinuous viscosity solutions for quasiconvex Hamiltonians

Emmanuel N. Barron

doi:10.1016/j.crma.2013.09.021

Partial differential equations/Optimal control

Semicontinuous viscosity solutions for quasiconvex Hamiltonians

Presented by: the Editorial Board

Emmanuel N. Barron ¹

¹ Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, USA

Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 737-741.

Abstract
Résumé

The main theorem connecting convex Hamiltonians and semicontinuous viscosity solutions due to Barron and Jensen is extended to quasiconvex Hamiltonians. Some applications are indicated.

Le théorème principal reliant les hamiltoniens convexes et les solutions de viscosité semicontinues, due à Barron et Jensen, est étendu aux hamiltoniens quasi-convexes. Quelques applications sont indiquées.

Received: 2013-06-15
Accepted: 2013-09-20
Published online: 2013-10-01

DOI: 10.1016/j.crma.2013.09.021

Author's affiliations:

Emmanuel N. Barron ¹

¹ Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, USA

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     title = {Semicontinuous viscosity solutions for quasiconvex {Hamiltonians}},
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Emmanuel N. Barron. Semicontinuous viscosity solutions for quasiconvex Hamiltonians. Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 737-741. doi : 10.1016/j.crma.2013.09.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.09.021/

References
Cited by

[1] O. Alvarez; E. Barron; H. Ishii Hopf–Lax formulas for semicontinuous data, Indiana Univ. Math. J., Volume 48 (1999), pp. 993-1035

[2] M. Bardi; I. Capuzzo-Dolcetta Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997

[3] G. Barles Solutions de viscosité des équations de Hamilton–Jacobi, Mathématiques et Applications, vol. 17, Springer, Paris, 1994

[4] E.N. Barron; R. Jensen Semicontinuous viscosity solutions of Hamilton–Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, Volume 15 (1990) no. 12, pp. 1713-1740

[5] E.N. Barron; W. Liu Calculus of variations in $L^{\infty}$ , Appl. Math. Optim., Volume 35 (1997), pp. 237-263

[6] E.N. Barron; W. Liu Semicontinuous solutions for Hamilton–Jacobi equations and the $L^{\infty}$ control problem, Appl. Math. Optim., Volume 34 (1996), pp. 325-360

[7] E.N. Barron; R. Jensen; W. Liu Hopf–Lax formula for $u_{t} + H (u, D u) = 0$ : II, Comm. Partial Differential Equations, Volume 22 (1997), pp. 1141-1160

[8] H. Frankowska Lower semicontinuous solutions of Hamilton–Jacobi–Bellman equations, SIAM J. Control Optim., Volume 31 (1993) no. 1, pp. 257-272

[9] H. Ishii A generalization of a theorem of Barron and Jensen and a comparison theorem for lower semicontinuous viscosity solutions, Proc. R. Soc. Edinb. A, Volume 131 (2001) no. 1, pp. 137-154

[10] P. Soravia Discontinuous viscosity solutions to Dirichlet problems for Hamilton–Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, Volume 18 (1993), pp. 1493-1514

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