DCDS

Given $s\in(0,1)$, we consider the problem of minimizing the fractional
Gagliardo
seminorm in $H^s$ with prescribed condition outside the ball
and under the further constraint of attaining zero value in a given set $K$.

We investigate how the energy changes in dependence of such set.
In particular, under mild regularity conditions, we show
that adding a set $A$ to $K$ increases the energy of at most the measure of $A$
(this may be seen as a perturbation result for small sets $A$).

Also, we point out a monotonicity feature of the energy with respect
to the prescribed sets and the boundary conditions.

DCDS

We consider a mesoscopic model of phase transitions and investigate
the geometric properties of the interfaces of the associated minimal
solutions. We provide density estimates for level sets and, in the
periodic setting, we construct minimal interfaces at a universal
distance from any given hyperplane.

CPAA

The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities
\begin{eqnarray}
(-\Delta)^s u-\lambda u=|u|^{2^*-2}u, in \Omega \\
u=0, in R^n\setminus \Omega,
\end{eqnarray}
where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive parameter, $2^*$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $R^n$, $n>2s$, with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when $\Omega$ is an open bounded subset of $R^n$ with $n\geq 4s$ and, in this framework, we prove some existence results.
Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when $2s < n < 4s$. In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension.
In particular when $s=1$ (and consequently $n=3$) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4].
In this sense the present work may be considered as the extension
of some classical results for the Laplacian to the case of non-local fractional operators.

DCDS

We consider
solutions of the
Allen-Cahn
equation in the whole Grushin plane
and we show that if they are monotone
in the vertical direction, then they are stable
and they satisfy a good energy estimate.

However, they are not necessarily
one-dimensional, as a counter-example shows.

DCDS-S

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.

NHM

We prove that
for any $H: R^2 \to R$ which
is $Z^2$-periodic,
there exists $H_\varepsilon$, which is smooth, $\varepsilon$-close
to $H$ in $L^1$, with $L^\infty$-norm
controlled by the one of $H$,
and with the same average
of $H$,
for which there exists a smooth closed curve $\gamma_\varepsilon$
whose curvature is $H_\varepsilon$.
A pinning phenomenon
for curvature driven flow with a periodic forcing term
then follows. Namely,
curves in fine periodic media may be moved
only by small amounts, of the order of
the period.

DCDS

We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$
as $s\searrow0$.
We prove necessary and sufficient conditions for the existence of such limit,
by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\Omega$.
Moreover, we construct examples of sets for which the limit does not exist.

DCDS

In this paper we study the existence of non-trivial solutions for
equations driven by a non-local integrodifferential
operator $\mathcal L_K$ with homogeneous Dirichlet boundary
conditions. More precisely, we consider the problem
$$ \left\{
\begin{array}{ll}
\mathcal L_K u+\lambda u+f(x,u)=0 in Ω \\
u=0 in \mathbb{R}^n \backslash Ω ,
\end{array} \right.
$$
where $\lambda$ is a real parameter and the nonlinear term $f$
satisfies superlinear and subcritical growth conditions at zero and
at infinity. This equation has a variational nature, and so its
solutions can be found as critical points of the energy functional
$\mathcal J_\lambda$ associated to the problem. Here we get such
critical points using both the Mountain Pass Theorem and the Linking
Theorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq
\lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of the
operator $-\mathcal L_K$.
As a particular case, we derive an existence theorem for the
following equation driven by the fractional Laplacian
$$ \left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=f(x,u) in Ω \\
u=0 in \mathbb{R}^n \backslash Ω.
\end{array} \right.
$$
Thus, the results presented here may be seen as the extension
of some classical nonlinear analysis theorems to the case of fractional
operators.