Degenerate Eikonal equations with discontinuous refraction index

Soravia, Pierpaolo

doi:10.1051/cocv:2005033

Cited by 19 publications

(23 citation statements)

References 28 publications

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“…For the stationary problem, the equation of eikonal type was studied by Newcomb and Su [20], Ostrov [21], Deckelnick and Elliott [10] and Soravia [25]. In [20] the authors considered the equation…”

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

confidence: 99%

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: 99%

Hamilton-Jacobi Equations with Discontinuous Source Terms

Giga

Hamamuki

2012

Communications in Partial Differential Equations

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“…17) We see that functiond is similar to the one used in Sect. 19) there exists k j > 0 and K j > 0 such that, ifd ε (x, y) is defined by 20) then, for all sequences (x ε , y ε ) such that x ε ∈ J j and y ε ∈ J i with i = j, and lim ε→0d 2 ε (xε,yε) ε = 0, it is possible to extract a subsequence such that…”

Section: The Comparison Principlementioning

confidence: 99%

Hamilton–Jacobi equations constrained on networks

Achdou¹,

Camilli²,

Cutrì³

et al. 2012

Nonlinear Differ. Equ. Appl.

133

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Abstract. We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. In our setting, the value function is continuous. We define a notion of constrained viscosity solution of Hamilton-Jacobi equations on the network and we study related comparison principles. Under suitable assumptions, we prove in particular that the value function is the unique constrained viscosity solution of the Hamilton-Jacobi equation on the network. Mathematics Subject Classification (2010). Primary 35R02, 35F21, 35Q93; Secondary 34H05, 49J15.

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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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Degenerate Eikonal equations with discontinuous refraction index

Cited by 19 publications

References 28 publications

Hamilton-Jacobi Equations with Discontinuous Source Terms

Hamilton-Jacobi Equations with Discontinuous Source Terms

Hamilton–Jacobi equations constrained on networks

The Mather problem for lower semicontinuous Lagrangians

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