Random homogenization of an obstacle problem

We study the homogenization of an obstacle problem in a perforated domain. The holes are periodically distributed but have random size and shape. The capacity of the holes is assumed to be stationary ergodic. As in the periodic case, we show that the asymptotic behavior of the solutions is described by an elliptic equation involving an additional term that takes into account the effects of the obstacle.Comment: 28 page

Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

On a price formation free boundary model by Lasry & Lions: The Neumann problem

We discuss local and global existence and uniqueness for the price formation free boundary model with homogeneous Neumann boundary conditions introduced by Lasry & Lions in 2007. The results are based on a transformation of the problem to the heat equation with nonstandard boundary conditions. The free boundary becomes the zero level set of the solution of the heat equation. The transformation allows us to construct an explicit solution and discuss the behavior of the free boundary. Global existence can be verified under certain conditions on the free boundary and examples of non-existence are given

Counter-example in 3D and homogenization of geometric motions in 2D

In this paper we give a counter-example to the homogenization of the forced mean curvature motion in a periodic setting in dimension $N\ge 3$ when the forcing is positive. We also prove a general homogenization result for geometric motions in dimension $N=2$ under the assumption that there exists a constant $\delta>0$ such that every straight line moving with a normal velocity equal to $\delta$ is a subsolution for the motion. We also present a generalization in dimension $2$, where we allow sign changing normal velocity and still construct bounded correctors, when there exists a subsolution with compact support expanding in all directions