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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.

MSC:
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
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