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.

}, keywords = {Finite-dimensional reductions, Local inversion, Singularly perturbed elliptic problems}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2012.08.017}, url = {http://www.sciencedirect.com/science/article/pii/S0022039612003312}, author = {Serena Dipierro} } @article {2013, title = {Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting}, number = {SISSA preprint;43/2013/MATE}, year = {2013}, institution = {SISSA}, abstract = {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.}, keywords = {nonlocal Allen-Cahn equation}, url = {http://hdl.handle.net/1963/7124}, author = {Serena Dipierro and Giampiero Palatucci and Enrico Valdinoci} } @article {2013, title = {Existence and symmetry results for a Schrodinger type problem involving the fractional Laplacian}, journal = {Le Matematiche (Catania), Vol. 68 (2013), no. 1: 201-216}, number = {arXiv:1202.0576v1;}, year = {2013}, publisher = {University of Catania}, abstract = {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 $s=1$.

}, author = {Serena Dipierro and Giampiero Palatucci and Enrico Valdinoci} } @article {2012, title = {Asymptotics of the s-perimeter as s {\textrightarrow}0 }, journal = {Discrete Contin. Dyn. Syst. 33, nr.7 (2012): 2777-2790}, number = {arXiv:1204.0750v2;}, year = {2012}, publisher = {American Institute of Mathematical Sciences}, abstract = {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.

}, doi = {10.3934/dcds.2013.33.2777}, author = {Serena Dipierro and Alessio Figalli and Giampiero Palatucci and Enrico Valdinoci} } @article {AIHPC_2011__28_1_107_0, title = {Concentration of solutions for a singularly perturbed Neumann problem in non-smooth domains}, journal = {Annales de l{\textquoteright}I.H.P. Analyse non lin{\'e}aire}, volume = {28}, number = {1}, year = {2011}, pages = {107-126}, publisher = {Elsevier}, doi = {10.1016/j.anihpc.2010.11.003}, url = {http://www.numdam.org/item/AIHPC_2011__28_1_107_0}, author = {Serena Dipierro} }