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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 investigate a two-player zero-sum stochastic differential game in which the players have an asymmetric information on the random payoff. We prove that the game has a value and characterize this value in terms of dual solutions of some second order Hamilton-Jacobi equation.
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)
We consider the so-called G-equation, a level set Hamilton-Jacobi equation, used as a sharp interface model for flame propagation, perturbed by an oscillatory advection in a spatio-temporal periodic environment. Assuming that the advection has suitably small spatial divergence, we prove that, as the size of the oscillations diminishes, the solutions… (More)
Existence and uniqueness of solutions for Hele-shaw moving boundary problem for power-law fluid is established in the framework of viscosity solutions.
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)
A Hamilton-Jacobi equation involving a double obstacle problem is investigated. The link between this equation and the notion of dual solutions—introduced in [1, 2, 3] in the framework of differential games with lack of information—is established. As an application we characterize the convex hull of a function in the simplex as the unique solution of some… (More)