1,569 research outputs found
Multiple positive solutions of parabolic systems with nonlinear, nonlocal initial conditions
In this paper we study the existence, localization and multiplicity of
positive solutions for parabolic systems with nonlocal initial conditions. In
order to do this, we extend an abstract theory that was recently developed by
the authors jointly with Radu Precup, related to the existence of fixed points
of nonlinear operators satisfying some upper and lower bounds. Our main tool is
the Granas fixed point index theory. We also provide a non-existence result and
an example to illustrate our theory.Comment: 28 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1401.135
Porous medium equation with nonlocal pressure
We provide a rather complete description of the results obtained so far on
the nonlinear diffusion equation , which describes a flow through a porous medium driven by a
nonlocal pressure. We consider constant parameters and , we assume
that the solutions are non-negative, and the problem is posed in the whole
space. We present a theory of existence of solutions, results on uniqueness,
and relation to other models. As new results of this paper, we prove the
existence of self-similar solutions in the range when and , and the
asymptotic behavior of solutions when . The cases and were
rather well known.Comment: 24 pages, 2 figure
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
- …