We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^n$ whose boundary has an $(n−2)$-dimensional singularity. Assuming $1<p<\frac{n+2}{n−2}$, we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.

VL - 254 UR - http://www.sciencedirect.com/science/article/pii/S0022039612003312 ER - TY - RPRT T1 - Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting Y1 - 2013 A1 - Serena Dipierro A1 - Giampiero Palatucci A1 - Enrico Valdinoci KW - nonlocal Allen-Cahn equation AB - We consider an evolution equation arising in the Peierls-Nabarro model for crystal dislocation. We study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior repulsive potential. PB - SISSA UR - http://hdl.handle.net/1963/7124 U1 - 7124 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Existence and symmetry results for a Schrodinger type problem involving the fractional Laplacian JF - Le Matematiche (Catania), Vol. 68 (2013), no. 1: 201-216 Y1 - 2013 A1 - Serena Dipierro A1 - Giampiero Palatucci A1 - Enrico Valdinoci AB -This paper deals with the following class of nonlocal Schr\"odinger equations $$ \displaystyle (-\Delta)^s u + u = |u|^{p-1}u \ \ \text{in} \ \mathbb{R}^N, \quad \text{for} \ s\in (0,1). $$ We prove existence and symmetry results for the solutions $u$ in the fractional Sobolev space $H^s(\mathbb{R}^N)$. Our results are in clear accordance with those for the classical local counterpart, that is when $s=1$.

PB - University of Catania U1 - 7318 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - Asymptotics of the s-perimeter as s →0 JF - Discrete Contin. Dyn. Syst. 33, nr.7 (2012): 2777-2790 Y1 - 2012 A1 - Serena Dipierro A1 - Alessio Figalli A1 - Giampiero Palatucci A1 - Enrico Valdinoci AB -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.

PB - American Institute of Mathematical Sciences U1 - 7317 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - Concentration of solutions for a singularly perturbed Neumann problem in non-smooth domains JF - Annales de l'I.H.P. Analyse non linéaire Y1 - 2011 A1 - Serena Dipierro PB - Elsevier VL - 28 UR - http://www.numdam.org/item/AIHPC_2011__28_1_107_0 ER -