Monge solutions for discontinuous hamiltonians
ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 2, pp. 229-251.

We consider an Hamilton-Jacobi equation of the form

 $H\left(x,Du\right)=0\phantom{\rule{1em}{0ex}}x\in \Omega \subset {ℝ}^{N},\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\left(1\right)$
where $H\left(x,p\right)$ is assumed Borel measurable and quasi-convex in $p$. 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 and supersolution, existence and uniqueness for equation (1) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed.

DOI: 10.1051/cocv:2005004
Classification: 49J25,  35C15,  35R05
Keywords: viscosity solution, lax formula, Finsler metric
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title = {Monge solutions for discontinuous hamiltonians},
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Briani, Ariela; Davini, Andrea. Monge solutions for discontinuous hamiltonians. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 2, pp. 229-251. doi : 10.1051/cocv:2005004. http://archive.numdam.org/articles/10.1051/cocv:2005004/

[1] L. Ambrosio and P. Tilli, Selected topics on “Analysis on Metric spaces”. Scuola Normale Superiore di Pisa (2000). | Zbl

[2] M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Syst. Control Found. Appl. (1997). | MR | Zbl

[3] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Math. Appl. 17 (1994). | MR | Zbl

[4] E.N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial Diff. Equ. 15 (1990) 1713-1742. | Zbl

[5] G. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Res. Notes Math. Ser. 207 (1989). | MR | Zbl

[6] G. Buttazzo, L. De Pascale and I. Fragalà, Topological equivalence of some variational problems involving distances. Discrete Contin. Dyn. Syst. 7 (2001) 247-258. | Zbl

[7] L. Caffarelli, M.G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math. 49 (1996) 365-397. | Zbl

[8] F. Camilli and A. Siconolfi, Hamilton-Jacobi equations with measurable dependence on the state variable. Adv. Differ. Equ. 8 (2003) 733-768. | Zbl

[9] F.H. Clarke, Optimization and Nonsmooth Analysis. John Wiley & Sons, New York (1983). | MR | Zbl

[10] A. Davini, On the relaxation of a class of functionals defined on Riemannian distances. J. Convex Anal., to appear. | MR | Zbl

[11] A. Davini, Smooth approximation of weak Finsler metrics. Adv. Differ. Equ., to appear. | MR

[12] G. De Cecco and G. Palmieri, Length of curves on LIP manifolds. Rend. Accad. Naz. Lincei, Ser. 9 1 (1990) 215-221. | Zbl

[13] G. De Cecco and G. Palmieri, Integral distance on a Lipschitz Riemannian Manifold. Math. Z. 207 (1991) 223-243. | Zbl

[14] 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. | Zbl

[15] G. De Cecco and G. Palmieri, LIP manifolds: from metric to Finslerian structure. Math. Z. 218 (1995) 223-237. | EuDML | MR | Zbl

[16] H. Ishii, A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Sc. Norm. Sup. Pisa 16 (1989) 105-135. | EuDML | Numdam | MR | Zbl

[17] H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets. Bull. Facul. Sci. & Eng., Chuo Univ., Ser I 28 (1985) 33-77. | MR | Zbl

[18] P.L. Lions, Generalized solutions of Hamilton Jacobi equations. Pitman (Advanced Publishing Program). Res. Notes Math. 69 (1982). | MR | Zbl

[19] R.T. Newcomb Ii and J. Su, Eikonal equations with discontinuities. Differ. Integral Equ. 8 (1995) 1947-1960. | MR | Zbl

[20] P. Soravia, Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J. 51 (2002) 451-477. | MR | Zbl

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