685 research outputs found
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 -minimal cones in are the trivial
ones for all . As a consequence we obtain that the singular set of
a nonlocal minimal surface has at most 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 when .
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 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 is a
consequence of the nonexistence of -dimensional singular minimal cones in
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 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
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