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We are interested in nonlocal Eikonal Equations arising in the study of the dynamics of dislocations lines in crystals. For these nonlocal but also non monotone equations, only the existence and uniqueness of Lipschitz and local-in-time solutions were available in some particular cases. In this paper, we propose a definition of weak solutions for which we(More)
We study the Hamilton-Jacobi equation F (Du) = 0 a.e. in Ω u = ϕ on ∂Ω (0.1) where F : I R N −→ I R is not necessarily convex. When Ω is a convex set, under technical assumptions our first main result gives a necessary and sufficient condition on the geometry of Ω and on Dϕ for (0.1) to admit a Lipschitz viscosity solution. When we drop the convexity(More)
The paper investigates the long time average of the solutions of Hamilton-Jacobi equations with a non coercive, non convex Hamiltonian in the torus R 2 /Z 2. We give nonreson-nance conditions under which the long-time average converges to a constant. In the resonnant case, we show that the limit still exists, although it is non constant in general. We(More)