168 research outputs found
Some monotonicity results for general systems of nonlinear elliptic PDEs
In this paper we show that minima and stable solutions of a general energy
functional of the form enjoy
some monotonicity properties, under an assumption on the growth at infinity of
the energy.
Our results are quite general, and comprise some rigidity results which are
known in the literature
A density property for fractional weighted Sobolev spaces
In this paper we show a density property for fractional weighted Sobolev
spaces. That is, we prove that any function in a fractional weighted Sobolev
space can be approximated by a smooth function with compact support.
The additional difficulty in this nonlocal setting is caused by the fact that
the weights are not necessarily translation invariant
(Non)local and (non)linear free boundary problems
We discuss some recent developments in the theory of free boundary problems,
as obtained in a series of papers in collaboration with L. Caffarelli, A.
Karakhanyan and O. Savin.
The main feature of these new free boundary problems is that they deeply take
into account nonlinear energy superpositions and possibly nonlocal functionals.
The nonlocal parameter interpolates between volume and perimeter functionals,
and so it can be seen as a fractional counterpart of classical free boundary
problems, in which the bulk energy presents nonlocal aspects.
The nonlinear term in the energy superposition takes into account the
possibility of modeling different regimes in terms of different energy levels
and provides a lack of scale invariance, which in turn may cause a structural
instability of minimizers that may vary from one scale to another
Geometric inequalities and symmetry results for elliptic systems
We obtain some Poincar\'{e} type formulas, that we use, together with the
level set analysis, to detect the one-dimensional symmetry of monotone and
stable solutions of possibly degenerate elliptic systems of the form
{eqnarray*} {{array}{ll} div(a(|\nabla u|) \nabla u) = F_1(u, v), div(b(|\nabla
v|) \nabla v) = F_2(u, v), {array}. {eqnarray*} where . Our setting is very general, and it comprises, as a
particular case, a conjecture of De Giorgi for phase separations in .Comment: Minor change
On stable solutions of boundary reaction-diffusion equations and applications to nonlocal problems with Neumann data
We study reaction-diffusion equations in cylinders with possibly nonlinear
diffusion and possibly nonlinear Neumann boundary conditions. We provide a
geometric Poincar\'e-type inequality and classification results for stable
solutions, and we apply them to the study of an associated nonlocal problem. We
also establish a counterexample in the corresponding framework for the
fractional Laplacian
All functions are locally -harmonic up to a small error
We show that we can approximate every function with a
-harmonic function in that vanishes outside a compact set.
That is, -harmonic functions are dense in . This result
is clearly in contrast with the rigidity of harmonic functions in the classical
case and can be viewed as a purely nonlocal feature.Comment: To appear in J. Eur. Math. Soc. (JEMS
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