Contribution to the study of local and non-local front propagations equations.

Summary This work deals with the study of front propagations governed by local and non-local laws. In the level line method, the front is seen as a 0-level line of an auxiliary function. The geometrical law of the front evolution corresponds to a Hamilton-Jacobi equation on this function, which we consider in the context of viscosity solutions. In non-local models, the main difficulty in proving existence or uniqueness results is the absence of inclusion principle between the fronts. In the level line method, this corresponds to an absence of comparison principle between functions, which makes the use of usual techniques impossible. The alternative use of fixed point methods associates to any non-local equation a family of local equations. Understanding the regularity of the solutions of the local equations, and in particular the perimeter of their level lines, then appears crucial in fixed point arguments. In chapter 1, we prove integral formulations of the local eikonal equation, from which we derive estimates on the perimeter of the level lines of its solutions. In the rest of the work, we are interested in non-local equations, and in particular in a notion of weak solution for these equations. Two non-local models, the dynamics of dislocations and a Fitzhugh-Nagumo system, are also studied in detail. In particular, results of existence, uniqueness and numerical approximation of weak solutions are given.