zbMATH — the first resource for mathematics

Nonlocal first-order Hamilton-Jacobi equations modelling dislocations dynamics. (English) Zbl 1109.35026
The starting point of this work and its main motivation is the study of the following type of nonlocal equations arising in dislocations theory \[ u_t= c[\mathbf{1}_{[u(\cdot, t)\geq 0]}]|Du|\qquad\text{in }\mathbb{R}^N\times (0,T), \] where \(T> 0\), the solution \(u\) is a real-valued function, \(u_t\) and \(Du\) stand respectively for its time and space derivatives, and \(\mathbf{1}\) is the indicator function of \(A\) for any \(A\subset\mathbb{R}^N\). O. Alvarez, P. Cardaliaguet and R. Monneau [Interfaces Free Bound. 7, No. 4, 415–434 (2005; Zbl 1099.35148)] remarked that, in the situation where \(c[\rho]\) is positive for any indicator functions, the existence and uniqueness can be proven for any time interval. The aim of this paper is to simplify the arguments of Alvarez by using a different approach, closer to the spirit of the level-set approach. The first step is to obtain fine properties of the solution of the standard level-sets equation \[ u_t= c(x, t)|Du|\qquad\text{in }\mathbb{R}^N\times (0,T), \] where \(c\) is a continuous function, satisfying suitable assumptions and in particular \(c(x, t)\geq 0\) in \(\mathbb{R}^N\times(0, T)\). The key result is an \(L^1\)-estimates on the measure of sets like \(\{a\leq u(\cdot, t)\leq b\}\) where \(-\overline\delta\leq a< b\leq\overline\delta\) for some small enough \(\overline\delta\). The key difference with the paper of Alvarez et al. is that here the authors used here the classical level-sets approach with continuous (and even Lipschitz continuous) solutions \(u\) while, there just indicator functions are used.

35F20 Nonlinear first-order PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
Full Text: DOI arXiv
[1] Alvarez O., C. R. Math. Acad. Sci. Paris 338 pp 679– (2004)
[2] DOI: 10.4171/IFB/131 · Zbl 1099.35148 · doi:10.4171/IFB/131
[3] DOI: 10.1007/978-0-8176-4755-1 · doi:10.1007/978-0-8176-4755-1
[4] Barles G., Solutions de Viscosité des Équations de Hamilton–Jacobi (1994)
[5] DOI: 10.1137/0331021 · Zbl 0785.35049 · doi:10.1137/0331021
[6] Cannarsa C., Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control (2004) · Zbl 1095.49003
[7] DOI: 10.1051/cocv:2006002 · Zbl 1105.93007 · doi:10.1051/cocv:2006002
[8] Chen Y. G., J. Differential Geom. 33 pp 749– (1991)
[9] Clarke F. H., Nonsmooth Analysis and Control Theory (1998) · Zbl 1047.49500
[10] Evans L. C., Measure Theory and Fine Properties of Functions (1992) · Zbl 0804.28001
[11] Evans L. C., J. Differential Geom. 33 pp 635– (1991)
[12] Ley O., Adv. Differential Equations 6 pp 547– (2001)
[13] Ley , O. ( 2001b ). Thèse de doctorat . Équations Quasilinéaires Paraboliques Dégénérées et Équations de Hamilton–Jacobi: Équations Géométriques et Mouvements de Fronts. Université de Tours , Tours , France .
[14] DOI: 10.1016/0021-9991(88)90002-2 · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[15] DOI: 10.1016/S1359-6454(01)00379-2 · doi:10.1016/S1359-6454(01)00379-2
[16] DOI: 10.3934/cpaa.2004.3.757 · Zbl 1064.49024 · doi:10.3934/cpaa.2004.3.757
[17] DOI: 10.1016/S0362-546X(02)00098-6 · Zbl 1028.35068 · doi:10.1016/S0362-546X(02)00098-6
[18] Souganidis , P. E. ( 1995 ). Interface dynamics in phase transitions . In Proceedings of the International Congress of Mathematicians . Vol. 1, 2 . ( Zürich, 1994 ). Basel : Birkhäuser , pp. 1133 – 1144 . · Zbl 0845.35045
[19] DOI: 10.1007/BFb0094298 · doi:10.1007/BFb0094298
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.