00951nas a2200145 4500008004100000245009100041210006900132260001000201520045700211653003300668100002100701700002500722700002200747856003600769 2013 en d00aDislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting0 aDislocation dynamics in crystals a macroscopic theory in a fract bSISSA3 aWe 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.10anonlocal Allen-Cahn equation1 aDipierro, Serena1 aPalatucci, Giampiero1 aValdinoci, Enrico uhttp://hdl.handle.net/1963/712400989nas a2200133 4500008004100000245010200041210006900143260002600212520042000238100002100658700002500679700002200704856012900726 2013 en d00aExistence and symmetry results for a Schrodinger type problem involving the fractional Laplacian0 aExistence and symmetry results for a Schrodinger type problem in bUniversity of Catania3 a
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 $s=1$.
1 aDipierro, Serena1 aPalatucci, Giampiero1 aValdinoci, Enrico uhttps://www.math.sissa.it/publication/existence-and-symmetry-results-schrodinger-type-problem-involving-fractional-laplacian00868nas a2200145 4500008004100000245004700041210004400088260004800132520037400180100002100554700002100575700002500596700002200621856007900643 2012 en d00aAsymptotics of the s-perimeter as s →0 0 aAsymptotics of the sperimeter as s →0 bAmerican Institute of Mathematical Sciences3 aWe 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.
1 aDipierro, Serena1 aFigalli, Alessio1 aPalatucci, Giampiero1 aValdinoci, Enrico uhttps://www.math.sissa.it/publication/asymptotics-s-perimeter-s-%E2%86%92000666nas a2200157 4500008004100000022001300041245006000054210005700114300001200171490000600183520016900189100002400358700001400382700002200396856009000418 2008 eng d a1534039200aOn periodic elliptic equations with gradient dependence0 aperiodic elliptic equations with gradient dependence a601-6150 v73 aWe construct entire solutions of Δu = f(x, u, ∇u) which are superpositions of odd, periodic functions and linear ones, with prescribed integer or rational slope.1 aBerti, Massimiliano1 aMatzeu, M1 aValdinoci, Enrico uhttps://www.math.sissa.it/publication/periodic-elliptic-equations-gradient-dependence01171nas a2200133 4500008004300000245008600043210006900129260003700198520070300235100002400938700001700962700002200979856003601001 2004 en_Ud 00aPeriodic orbits close to elliptic tori and applications to the three-body problem0 aPeriodic orbits close to elliptic tori and applications to the t bScuola Normale Superiore di Pisa3 aWe prove, under suitable non-resonance and non-degeneracy ``twist\\\'\\\' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the ``planets\\\'\\\'. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.)1 aBerti, Massimiliano1 aBiasco, Luca1 aValdinoci, Enrico uhttp://hdl.handle.net/1963/2985