Monge solutions for discontinuous Hamiltonians

Briani, Ariela; Davini, Andrea

doi:10.1051/cocv:2005004

Cited by 21 publications

(36 citation statements)

References 8 publications

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“…They introduced a suitable notion called Monge solutions, which, in the case of continuous Hamiltonians, are consistent with the usual viscosity solution. Briani and Davini [4] generalized the approach of Monge solutions for the equation H(x, ∇u) = 0, where H(x, p) is only assumed to be Borel measurable and quasi-convex in p. Although we did not check, we expect that our envelope solution should agree with the Monge solution when the latter is available. The work by Soravia [24] is related to our results concerning the optimal control theory.…”

Section: H(x P) = −|P| − Ci(x)mentioning

confidence: 91%

Hamilton-Jacobi Equations with Discontinuous Source Terms

Giga

Hamamuki

2012

Communications in Partial Differential Equations

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We study the initial-value problem for a Hamilton-Jacobi equation whose Hamiltonian is discontinuous with respect to state variables. Our motivation comes from a model describing the two dimensional nucleation in crystal growth phenomena. A typical equation has a semicontinuous source term. We introduce a new notion of viscosity solutions and prove among other results that the initial-value problem admits a unique global-in-time uniformly continuous solution for any bounded uniformly continuous initial data. We also give a representation formula of the solution as a value function by the optimal control theory with a semicontinuous running cost function.

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Section: H(x P) = −|P| − Ci(x)mentioning

confidence: 91%

Hamilton-Jacobi Equations with Discontinuous Source Terms

Giga

Hamamuki

2012

Communications in Partial Differential Equations

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“…Several papers in the literature deal with HJB equations with discontinuous coefficients; see for instance [6,34,28,37,39,9,38,13,12]. Note that in these works the optimal trajectories do cross the regions of discontinuities (i.e.…”

Section: Discontinuitymentioning

confidence: 99%

A Hamilton-Jacobi approach to junction problems and application to traffic flows

2012

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This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a "junction", that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.

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“…Viscosity solutions of Hamilton-Jacobi equations with discontinuous Hamiltonians have been studied extensively by many authors, in different settings; we refer to the books by Barles [2] and Bardi and CapuzzoDolcetta [1] for a general treatment. They have been used in the analysis of geodesic distances and in the study of some discontinuous control problems, combustion phenomena in nonhomogeneous media, and geometric optic propagation in the presence of layers; see [6,20,22,23]. Measurable Hamiltonians have been considered in [7][8][9]11].…”

Section: L(x(t)ẋ(t)) Dtmentioning

confidence: 99%

The Mather problem for lower semicontinuous Lagrangians

Gomes

Terrone²

2013

Nonlinear Differ. Equ. Appl.

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Abstract. In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler-Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity. Mathematics Subject Classification (2010). List of symbolsThe canonical basis of R . , ∂ xN ψ) x(t),ẍ(t)The first and second derivative of a functionThe tangent space of a smooth manifold M at the pointThe Dirac mass concentrated at x 0 μ X (x)The projection on X of a measure μ(x, y) onThe jump of the function r : I ⊂ R → R at t 0 , that is lim t→t

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Monge solutions for discontinuous Hamiltonians

Abstract: Abstract.We consider an Hamilton-Jacobi equation of the formwhere H(x, p) is assumed Borel measurable and quasi-convex in p.

Cited by 21 publications

References 8 publications

Hamilton-Jacobi Equations with Discontinuous Source Terms

Hamilton-Jacobi Equations with Discontinuous Source Terms

A Hamilton-Jacobi approach to junction problems and application to traffic flows

The Mather problem for lower semicontinuous Lagrangians

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