A fractional framework for perimeters and phase transitions

We review some recent results on minimisers of a non-local perimeter functional, in connection with some phase coexistence models whose diffusion term is given by the fractional Laplacian

From the long jump random walk to the fractional Laplacian

This note illustrates how a simple random walk with possibly long jumps is related to fractional powers of the Laplace operator. The exposition is elementary and self-contained.Comment: Submitted to the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Regularity of nonlocal minimal cones in dimension 2

We show that the only nonlocal $s$-minimal cones in $\R^2$ are the trivial ones for all $s \in (0,1)$. As a consequence we obtain that the singular set of a nonlocal minimal surface has at most $n-3$ Hausdorff dimension

Some elliptic PDEs on Riemannian manifolds with boundary

The goal of this paper is to investigate some rigidity properties of stable solutions of elliptic equations set on manifolds with boundary. We provide several types of results, according to the dimension of the manifold and the sign of its Ricci curvature

The Ginzburg-Landau equation in the Heisenberg group

We consider a functional related with phase transition models in the Heisenberg group framework. We prove that level sets of local minimizers satisfy some density estimates, that is, they behave as "codimension one" sets. We thus deduce a uniform convergence property of these level sets to interfaces with minimal area. These results are then applied in the construction of (quasi)periodic, plane-like minimizers, i.e., minimizers of our functional whose level sets are contained in a spacial slab of universal size in a prescribed direction. As a limiting case, we obtain the existence of hypersurfaces contained in such a slab which minimize the surface area with respect to a given periodic metric.Comment: 49 page

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

Regularity and Bernstein-type results for nonlocal minimal surfaces

We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi stating that the validity of Bernstein's theorem in dimension $n+1$ is a consequence of the nonexistence of $n$-dimensional singular minimal cones in $\R^n$

Some monotonicity results for minimizers in the calculus of variations

We obtain monotonicity properties for minima and stable solutions of general energy functionals of the type $$\int F(\nabla u, u, x) dx$$ under the assumption that a certain integral grows at most quadratically at infinity. As a consequence we obtain several rigidity results of global solutions in low dimensions

Uniqueness in weighted Lebesgue spaces for a class of fractional parabolic and elliptic equations

We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of fractional parabolic and elliptic equations