H−H-convergence result for nonlocal elliptic-type problems via Tartar's method

In this work we obtain a compactness result for the $H-$convergence of a family of nonlocal and nonlinear monotone elliptic-type problems by means of Tartar's method of oscillating test functions.Comment: In this revision we added a new section that shows the Gamma-convergence of the associated energy functional

Well-posedness via Monotonicity. An Overview

The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a comprehensive class of such problems. We elaborate the applicability of our scheme with a number examples. A brief discussion of stability and homogenization issues is also provided.Comment: Thoroughly revised version. Examples correcte

Large Time Behavior of Periodic Viscosity Solutions for Uniformly Parabolic Integro-Differential Equations

International audienceIn this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form $\lambda t + v(x)$. We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of ''mixed operators'' for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis

Homogenization of the Peierls-Nabarro model for dislocation dynamics

This paper is concerned with a result of homogenization of an integro-differential equation describing dislocation dynamics. Our model involves both an anisotropic L\'{e}vy operator of order 1 and a potential depending periodically on u/\ep. The limit equation is a non-local Hamilton-Jacobi equation, which is an effective plastic law for densities of dislocations moving in a single slip plane.Comment: 39 page