Discontinuous viscosity solutions to dirichlet problems for hamilton-jacob1 equations with

Soravia, Pierpaolo

doi:10.1080/03605309308820983

Cited by 46 publications

(39 citation statements)

References 2 publications

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“…Bardi [1], BardiSoravia [3], Soravia [21], or the books by Bardi and Capuzzo-Dolcetta [2], Barles [4] and the references therein. Here we will instead concentrate on the case of f discontinuous.…”

Section: Introductionmentioning

confidence: 99%

Degenerate Eikonal equations with discontinuous refraction index

Soravia

2006

ESAIM: COCV

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Abstract.We study the Dirichlet boundary value problem for eikonal type equations of ray light propagation in an inhomogeneous medium with discontinuous refraction index. We prove a comparison principle that allows us to obtain existence and uniqueness of a continuous viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value function of the generalized minimum time problem with discontinuous running cost.Mathematics Subject Classification. 35A05, 35F30, 49L20, 49L25.

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Section: Introductionmentioning

confidence: 99%

Degenerate Eikonal equations with discontinuous refraction index

Soravia

2006

ESAIM: COCV

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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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“…A solution of (HJ S ) means a lower semicontinuous function ϕ : R S + −→ IR ∪ {+∞} such that ϕ(S) = 0 and for every x ∈ R S + , for every ζ ∈ ∂ P ϕ(x) (if any), we have h(x, ζ) + 1 = 0 (we say that ϕ(x) is a proximal solution, see [8]). This is equivalent to the statement that ϕ is a lower semicontinuous viscosity solution of the following Hamilton-Jacobi equation: It is well-known that the minimal time function T (·, S) is a solution of (HJ S ) if we replace R S + by R S + \ S (see for example [1,3,7,18,22]), but it is never a solution on R S + since for all α ∈ S we have 0 ∈ ∂ P T (·, S)(α) and h(α, 0) = 0. In [11], Clarke and Nour study the Hamilton-Jacobi equation (HJ S ) in the case S = {α 0 } (we denote this equation by (HJ α0 )) 2 .…”

Section: (T) ∈ F (X(t)) Ae T ∈ [0 T ] X(0) = α and X(t ) = βmentioning

confidence: 99%

Semigeodesics and the minimal time function

Nour

2005

ESAIM: COCV

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Abstract.We study the Hamilton-Jacobi equation of the minimal time function in a domain which contains the target set. We generalize the results of Clarke and Nour [J. Convex Anal., 2004], where the target set is taken to be a single point. As an application, we give necessary and sufficient conditions for the existence of solutions to eikonal equations.Mathematics Subject Classification. 49J52, 49L20, 49L25.

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Discontinuous viscosity solutions to dirichlet problems for hamilton-jacob1 equations with

Cited by 46 publications

References 2 publications

Degenerate Eikonal equations with discontinuous refraction index

Degenerate Eikonal equations with discontinuous refraction index

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

Semigeodesics and the minimal time function

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