Regularity in a one-phase free boundary problem for the fractional Laplacian

For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,\alpha}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.Comment: Corrected some typo

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

Regularity properties of nonlocal minimal surfaces via limiting arguments

We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and $s$ is close to 1

Traveling Wave Solutions of Advection-Diffusion Equations with Nonlinear Diffusion

We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c {\in}]c*, +{\infty}, where c* > 0 is explicitly computed but may not be optimal. We also prove that a free boundary hy- persurface separates a region where u = 0 and a region where u > 0, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region u > 0, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed.Comment: 40 pages, 1 figur

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