Pointwise regularity of the free boundary for the parabolic obstacle problem

We study the parabolic obstacle problem \lap u-u_t=f\chi_{\{u>0\}}, \quad u\geq 0,\quad f\in L^p \quad \mbox{with}\quad f(0)=1 and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular free boundary points. These are used to prove a second order Taylor expansion at singular points (under a pointwise Dini condition), with an estimate of the error (under a pointwise double Dini condition). Moreover, under the assumption that $f$ is Dini continuous, we prove that the set of regular points is locally a (parabolic) $C^1$-surface and that the set of singular points is locally contained in a union of (parabolic) $C^1$ manifolds

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

The method of differential contractions

In this Note we present a general and fairly simple method to design families of contractions for nonlinear partial differential equations, either of evolution type, or of stationary type. As a particular example, we apply this method to the porous medium equation, for which we get new contractions. This method opens new directions to explore

A kinetic formulation of moving fronts and application to dislocations dynamics

In this article, we consider hypersurfaces moving with normal velocity depending on the time-space coordinates and on the normal to the hypersurface. We naturally define a measure associated to this hypersurface. This measure is defined on a suitable space/unit normal/curvature configuration space. We show that, while the hypersurface stays smooth, then the measure is a solution to a linear transport equation, that we call a kinetic formulation. In the particular case of curves moving in the plane, we get a simple kinetic formulation. With this kinetic formulation in hands, it is then easy to complete the models of dislocations densities that were proposed in the 60's. As a consequence, we therefore propose a closed mean field model for the dynamics of dislocations densities

A junction condition by specified homogenization and application to traffic lights

Given a coercive Hamiltonian which is quasi-convex with respect to the gradient variable and periodic with respect to time and space at least "far away from the origin", we consider the solution of the Cauchy problem of the corresponding Hamilton-Jacobi equation posed on the real line. Compact perturbations of coercive periodic quasi-convex Hamiltonians enter into this framework for example. We prove that the rescaled solution converges towards the solution of the expected effective Hamilton-Jacobi equation, but whose "flux" at the origin is "limited" in a sense made precise by the authors in \cite{im}. In other words, the homogenization of such a Hamilton-Jacobi equation yields to supplement the expected homogenized Hamilton-Jacobi equation with a junction condition at the single discontinuous point of the effective Hamiltonian. We also illustrate possible applications of such a result by deriving, for a traffic flow problem, the effective flux limiter generated by the presence of a finite number of traffic lights on an ideal road. We also provide meaningful qualitative properties of the effective limiter.Comment: 41 page

Uniqueness and existence of spirals moving by forced mean curvature motion

In this paper, we study the motion of spirals by mean curvature type motion in the (two dimensional) plane. Our motivation comes from dislocation dynamics; in this context, spirals appear when a screw dislocation line reaches the surface of a crystal. The first main result of this paper is a comparison principle for the corresponding parabolic quasi-linear equation. As far as motion of spirals are concerned, the novelty and originality of our setting and results come from the fact that, first, the singularity generated by the attached end point of spirals is taken into account for the first time, and second, spirals are studied in the whole space. Our second main result states that the Cauchy problem is well-posed in the class of sub-linear weak (viscosity) solutions. We also explain how to get the existence of smooth solutions when initial data satisfy an additional compatibility condition.Comment: This new version contains new results: we prove that the weak (viscosity) solutions of the Cauchy problem are in fact smooth. This is a consequence of some gradient estimates in time and spac

Explicit phase diagram for a one-dimensional blister model

In this article, we consider a simple one-dimensional variational model, describing the delamination of thin films under cooling. We characterize the global minimizers, which correspond to films of three possible types: non delaminated, partially delaminated (called blisters), or fully delaminated. Two parameters play an important role: the length of the film and the cooling parameter. In the phase plane of those two parameters, we classify all the minimizers. As a consequence of our analysis, we identify explicitly the smallest possible blisters for this model

The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model

We study a generalization of the fully overdamped Frenkel-Kontorova model in dimension $n\geq 1.$ This model describes the evolution of the position of each atom in a crystal, and is mathematically given by an infinite system of coupled first order ODEs. We prove that for a suitable rescaling of this model, the solution converges to the solution of a Peierls-Nabarro model, which is a coupled system of two PDEs (typically an elliptic PDE in a domain with an evolution PDE on the boundary of the domain). This passage from the discrete model to a continuous model is done in the framework of viscosity solutions

A new contraction family for porous medium and fast diffusion equation

In this paper, we present a surprising two-dimensional contraction family for porous medium and fast diffusion equations. This approach provides new a priori estimates on the solutions, even for the standard heat equation

Derivation and study of dynamical models of dislocation densities

In this paper, starting from the microscopic dynamics of isolated dislocations, we explain how to derive formally mean field models for the dynamics of dislocation densities. Essentially these models are tranport equations, coupled with the equations of elasticity. Rigorous results of existence of solutions are presented for some of these models and the main ideas of the proofs are given