4,509 research outputs found
Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian
We establish sharp energy estimates for some solutions, such as global
minimizers, monotone solutions and saddle-shaped solutions, of the fractional
nonlinear equation in \re^n. Our energy estimates
hold for every nonlinearity and are sharp since they are optimal for
one-dimensional solutions, that is, for solutions depending only on one
Euclidian variable. As a consequence, in dimension , we deduce the
one-dimensional symmetry of every global minimizer and of every monotone
solution. This result is the analog of a conjecture of De Giorgi on
one-dimensional symmetry for the classical equation in
\re^n
Classical and quantum filaments in the ground state of trapped dipolar Bose gases
We study by quantum Monte Carlo simulations the ground state of a
harmonically confined dipolar Bose gas with aligned dipole moments, and with
the inclusion of a repulsive two-body potential of varying range. Two different
limits can be clearly identified, namely a classical one in which the
attractive part of the dipolar interaction dominates and the system forms an
ordered array of parallel filaments, and a quantum-mechanical one, wherein
filaments are destabilized by zero-point motion, and eventually the ground
state becomes a uniform cloud. The physical character of the system smoothly
evolves from classical to quantum mechanical as the range of the repulsive
two-body potential increases. An intermediate regime is observed, in which
ordered filaments are still present, albeit forming different structures from
the ones predicted classically; quantum-mechanical exchanges of
indistinguishable particles across different filaments allow phase coherence to
be established, underlying a global superfluid response.Comment: Replaced with published versio
Interpolation inequalities in pattern formation
We prove some interpolation inequalities which arise in the analysis of
pattern formation in physics. They are the strong version of some already known
estimates in weak form that are used to give a lower bound of the energy in
many contexts (coarsening and branching in micromagnetics and superconductors).
The main ingredient in the proof of our inequalities is a geometric
construction which was first used by Choksi, Conti, Kohn, and one of the
authors in the study of branching in superconductors
Sharp energy estimates for nonlinear fractional diffusion equations
We study the nonlinear fractional equation in
, for all fractions and all nonlinearities . For every
fractional power , we obtain sharp energy estimates for bounded
global minimizers and for bounded monotone solutions. They are sharp since they
are optimal for solutions depending only on one Euclidian variable.
As a consequence, we deduce the one-dimensional symmetry of bounded global
minimizers and of bounded monotone solutions in dimension whenever . This result is the analogue of a conjecture of De Giorgi on
one-dimensional symmetry for the classical equation in
. It remains open for and , and also for
and all .Comment: arXiv admin note: text overlap with arXiv:1004.286
A nonlocal supercritical Neumann problem
We establish existence of positive non-decreasing radial solutions for a
nonlocal nonlinear Neumann problem both in the ball and in the annulus. The
nonlinearity that we consider is rather general, allowing for supercritical
growth (in the sense of Sobolev embedding). The consequent lack of compactness
can be overcome, by working in the cone of non-negative and non-decreasing
radial functions. Within this cone, we establish some a priori estimates which
allow, via a truncation argument, to use variational methods for proving
existence of solutions. As a side result, we prove a strong maximum principle
for nonlocal Neumann problems, which is of independent interest.Comment: 32 pages, 0 figure
Bounds on short cylinders and uniqueness results for degenerate Kolmogorov equation
We consider the Cauchy problem for hypoelliptic Kolmogorov equations in both divergence and non divergence form. We prove that, if |u(x,t)| < M exp(a(t^{-\beta}+|x|^2)) for some positive constants a, M, \beta in ]0,1[ and u(x,0) = 0, then u(x,t) = 0 for positive t. The proof of the main result is based on some previous uniqueness result and on the application of some estimates in short cylinders, first introduced by Safonov in the study of uniformly parabolic operators
A nonlinear Liouville theorem for fractional equations in the Heisenberg group
We establish a Liouville-type theorem for a subcritical nonlinear problem,
involving a fractional power of the sub-Laplacian in the Heisenberg group. To
prove our result we will use the local realization of fractional CR covariant
operators, which can be constructed as the Dirichlet-to-Neumann operator of a
degenerate elliptic equation in the spirit of Caffarelli and Silvestre, as
established in \cite{FGMT}. The main tools in our proof are the CR inversion
and the moving plane method, applied to the solution of the lifted problem in
the half-space \mathbb H^n\times \mathbbR^+
Exchange-induced crystallization of soft core bosons
We study the phase diagram of a two-dimensional assembly of bosons
interacting via a soft core repulsive pair potential of varying strength, and
compare it to that of the equivalent system in which particles are regarded as
distinguishable. We show that quantum-mechanical exchanges stabilize a "cluster
crystal" phase in a wider region of parameter space than predicted by
calculations in which exchanges are neglected. This physical effect is
diametrically opposite to that which takes place in hard core Bose systems such
as He, wherein exchanges strengthen the fluid phase. It is underlain in the
cluster crystal phase of soft core bosons by the free energy gain associated to
the formation of local Bose-Einstein condensates.Comment: 10 pages, 4 figures, published versio
Quantitative flatness results and -estimates for stable nonlocal minimal surfaces
We establish quantitative properties of minimizers and stable sets for
nonlocal interaction functionals, including the -fractional perimeter as a
particular case.
On the one hand, we establish universal -estimates in every dimension
for stable sets. Namely, we prove that any stable set in has
finite classical perimeter in , with a universal bound. This nonlocal
result is new even in the case of -perimeters and its local counterpart (for
classical stable minimal surfaces) was known only for simply connected
two-dimensional surfaces immersed in .
On the other hand, we prove quantitative flatness estimates for minimizers
and stable sets in low dimensions . More precisely, we show that a
stable set in , with large, is very close in measure to being a half
space in ---with a quantitative estimate on the measure of the symmetric
difference. As a byproduct, we obtain new classification results for stable
sets in the whole plane
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