Partial differential equations/Optimal control
Semicontinuous viscosity solutions for quasiconvex Hamiltonians
Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 737-741.

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.

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DOI: 10.1016/j.crma.2013.09.021
Barron, Emmanuel N. 1

1 Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, USA
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Barron, Emmanuel N. 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. http://www.numdam.org/articles/10.1016/j.crma.2013.09.021/

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

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

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

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

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

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

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

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

[9] Ishii, H. 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] Soravia, P. 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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