Uniqueness results for nonlocal Hamilton–Jacobi equations

Barles, Guy; Cardaliaguet, Pierre; Ley, Olivier; Monteillet, Aurélien

doi:10.1016/j.jfa.2009.04.014

Cited by 18 publications

(29 citation statements)

References 18 publications

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“…We point out that, in the case of nonnegative velocities, the global existence and uniqueness were first obtained by Alvarez et al [1] and then by Barles et al [9] using different arguments. These uniqueness results were recently extended by Barles et al in [8], using a new approach allowing to relax the assumptions of [1,9]. Moreover, the proof proposed in [8] is simpler than that of [1,9] and requires a mild regularity on the velocity.…”

Section: Setting Of the Problemmentioning

confidence: 94%

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Global existence results for eikonal equation with BV initial data

Boudjerada

Hajj

2015

Nonlinear Differ. Equ. Appl.

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In this paper, we study a local and a non-local eikonal equations in one dimensional space describing the evolution of interfaces moving with non-signed velocity. For these equations, the global existence and uniqueness are available only of Lipschitz continuous viscosity solutions in some particular cases. In the present paper, we are interested in the study of the global in time existence of these equations, considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed propagation property of these equations, we show, in a particular setting, global existence results of discontinuous viscosity solutions of this problem. An interesting application of these results is shown in the case of dislocation dynamics.Mathematics Subject Classification. 35A01, 74G25, 35F20, 35F21, 70H20, 35Q74.

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Section: Setting Of the Problemmentioning

confidence: 94%

“…These uniqueness results were recently extended by Barles et al in [8], using a new approach allowing to relax the assumptions of [1,9]. Moreover, the proof proposed in [8] is simpler than that of [1,9] and requires a mild regularity on the velocity.…”

Section: Setting Of the Problemmentioning

confidence: 94%

Global existence results for eikonal equation with BV initial data

Boudjerada

Hajj

2015

Nonlinear Differ. Equ. Appl.

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“…We are also interested in the following system, 5) which is obtained as the asymptotics as ε → 0 of the following FitzhughNagumo system arising in neural wave propagation or chemical kinetics (see [18]):…”

Section: 22mentioning

confidence: 99%

Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations

Barles

Cardaliaguet

Ley

et al. 2009

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems.

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“…Our result is strictly related to Theorem 5.8 in [4], where the authors estimate the perimeter of sets enjoying an internal cone property. Indeed, the same arguments of Corollary 4.3 (for θ = 0, 0 < C < 1) easily gives the same conclusion of [4]. See also [26, Proposition 2.4].…”

Section: Standing Hypothesis and First Consequencesmentioning

confidence: 63%

Some regularity results for a class of upper semicontinuous functions

Marigonda¹,

Nguyen²,

Vittone³

2013

Indiana Univ. Math. J.

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Abstract. We study regularity properties enjoyed by a class of real-valued upper semicontinuous functions f : R d → R whose hypograph satisfies a geometric property implying, for each point P on the boundary of hypo f , the existence of a sort of (uniform) subquadratic tangent hypersurface whose intersection with hypo f in a neighbourhood of P reduces to P . This geometric property generalizes both the concepts of semiconcave functions and functions whose hypograph has positive reach in the sense of Federer; the associated class of functions arises in the study of regularity properties for the minimum time function of certain classes of nonlinear control systems and differential inclusions.We will prove that these functions share several regularity properties with semiconcave functions. In particular, they are locally BV and differentiable a.e. Our approach consists in providing upper bounds for the dimension of the set of nondifferentiability points. Moreover, a finer classification of the singularities can be performed according to the dimension of the normal cone to the hypograph, thus generalizing a similar result proved by Federer for sets with positive reach. Techniques of nonsmooth analysis and geometric measure theory are used.

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Uniqueness results for nonlocal Hamilton–Jacobi equations

Cited by 18 publications

References 18 publications

Global existence results for eikonal equation with BV initial data

Global existence results for eikonal equation with BV initial data

Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations

Some regularity results for a class of upper semicontinuous functions

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