Ricerc@Sapienza

We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator and a concave-convex term, together with mixed Dirichlet-Neumann boundary conditions.

We analyze the behavior of the eigenvalues of the following non local mixed
problem $\left\ \beginarrayrcll (-\Delta)^s u &=& \lambda_1(D) \ u
&\inn\Omega,\\ u&=&0&\inn D,\\ \mathcalN_su&=&0&\inn N. \endarray\right $
Our goal is to construct different sequences of problems by modifying the
configuration of the sets $D$ and $N$, and to provide sufficient and necessary
conditions on the size and the location of these sets in order to obtain
sequences of eigenvalues that in the limit recover the eigenvalues of the

In this work we study regularity properties of solutions to fractional
elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with
the Spectral Fractional Laplacian.

In this paper we study several aspects related with solutions of nonlocal
problems whose prototype is $$ u_t =displaystyle int_mathbbR^N J(x-y)
ig( u(y,t) -u(x,t) ig) mathcal Gig( u(y,t) -u(x,t) ig) dy qquad
mbox in , Omega imes (0,T),, $$ being $ u (x,t)=0 mbox in
(mathbbR^Nsetminus Omega ) imes (0,T),$ and $ u(x,0)=u_0 (x) mbox in
Omega$. We take, as the most important instance, $mathcal G (s) sim 1+
racmu2 racs1+mu^2 s^2 $ with $muin mathbbR$ as well as $u_0
in L^1 (Omega)$, $J$ is a smooth symmetric function with compact support and

In this paper we deal with uniqueness of solutions to the following problem
\[ \begincases \beginsplit & u_t-\Delta_p u=H(t,x,\nabla u) &\quad
\textin\quad Q_T,\\ & u (t,x) =0 &\quad \texton\quad(0,T)\times \partial
\Omega,\\ & u(0,x)=u_0(x) &\quad \displaystyle\textin \quad \Omega
\endsplit \endcases \] where $Q_T=(0,T)\times \Omega$ is the parabolic
cylinder, $\Omega$ is an open subset of $\mathbbR^N$, $N\ge2$, $1