%0 Journal Article %J Journal of Differential Equations %D 2013 %T Concentration of solutions for a singularly perturbed mixed problem in non-smooth domains %A Serena Dipierro %K Finite-dimensional reductions %K Local inversion %K Singularly perturbed elliptic problems %X

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.

%B Journal of Differential Equations %V 254 %P 30 - 66 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039612003312 %R https://doi.org/10.1016/j.jde.2012.08.017 %0 Report %D 2013 %T Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting %A Serena Dipierro %A Giampiero Palatucci %A Enrico Valdinoci %K nonlocal Allen-Cahn equation %X 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. %I SISSA %G en %U http://hdl.handle.net/1963/7124 %1 7124 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Giampiero Palatucci (palatucc@sissa.it) on 2013-09-23T12:18:14Z No. of bitstreams: 1 Dipierro-Palatucci-Valdinoci.pdf: 651692 bytes, checksum: 839508f3ff7cdc4417c33991ebf3a9f3 (MD5) %0 Journal Article %J Le Matematiche (Catania), Vol. 68 (2013), no. 1: 201-216 %D 2013 %T Existence and symmetry results for a Schrodinger type problem involving the fractional Laplacian %A Serena Dipierro %A Giampiero Palatucci %A Enrico Valdinoci %X

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 $s=1$.

%B Le Matematiche (Catania), Vol. 68 (2013), no. 1: 201-216 %I University of Catania %G en %1 7318 %2 Mathematics %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-03-11T16:07:36Z No. of bitstreams: 1 1202.0576v1.pdf: 219939 bytes, checksum: 822c73753d6d4194cf48cb0ff9ad0e48 (MD5) %0 Journal Article %J Discrete Contin. Dyn. Syst. 33, nr.7 (2012): 2777-2790 %D 2012 %T Asymptotics of the s-perimeter as s →0 %A Serena Dipierro %A Alessio Figalli %A Giampiero Palatucci %A Enrico Valdinoci %X

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.

%B Discrete Contin. Dyn. Syst. 33, nr.7 (2012): 2777-2790 %I American Institute of Mathematical Sciences %G en %1 7317 %2 Mathematics %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-03-11T15:55:21Z No. of bitstreams: 1 1204.0750v2.pdf: 216883 bytes, checksum: 3ee8d497a2c0f9a211ec5327e8aa6b9a (MD5) %R 10.3934/dcds.2013.33.2777 %0 Journal Article %J Annales de l'I.H.P. Analyse non linéaire %D 2011 %T Concentration of solutions for a singularly perturbed Neumann problem in non-smooth domains %A Serena Dipierro %B Annales de l'I.H.P. Analyse non linéaire %I Elsevier %V 28 %P 107-126 %G eng %U http://www.numdam.org/item/AIHPC_2011__28_1_107_0 %R 10.1016/j.anihpc.2010.11.003