117,303 research outputs found
L-2-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features
No full textCarrying on the discussion initiated in Dovetta and Tentarelli (Ground states of the L2-critical NLS equation with localized nonlinearity on a tadpole graph, 2018. arXiv:1804.11107 [math.AP]), we investigate the existence of ground states of prescribed mass for the L2-critical NonLinear Schrodinger Equation on noncompact metric graphs with localized nonlinearity. Precisely, we show that the existence (or nonexistence) of ground states mainly depends on a parameter called reduced critical mass, and then we discuss how the topological and metric features of the graphs affect such a parameter, establishing some relevant differences with respect to the case of the extended nonlinearity studied by Adami et al. (Commun Math Phys 352(1):387-406, 2017). Our results rely on a thorough analysis of the optimal constant of a suitable variant of the L2-critical Gagliardo-Nirenberg inequality
On the extensions of the De Giorgi approach to nonlinear hyperbolic equations
No full textIn this talk we present an overview on the extensions of the De Giorgi approach to general second order nonlinear hyperbolic equations. We start with an introduction to the original conjecture by E. De Giorgi ([1, 2]) and to its solution by E. Serra and P. Tilli ([4]). Then, we discuss a first extension of this idea (Serra&Tilli, [5]) aimed at investigating a wide class of homogeneous equations. Finally, we announce a further extension to nonhomogeneous equations, obtained by the author in [9] in collaboration with P. Tilli
Ground states of the l 2-Critical NLS equation with localized nonlinearity on a Tadpole traph
No full textThe paper aims at giving a first insight on the existence/nonexistence of ground states for the L2-critical NLS equation on metric graphs with localized nonlinearity. As a consequence, we focus on the tadpole graph, which, albeit being a toy model, allows to point out some specific features of the problem, whose understanding will be useful for future investigations. More precisely, we prove that there exists an interval of masses for which ground states do exist, and that for large masses the functional is unbounded from below, whereas for small masses ground states cannot exist although the functional is bounded
Symmetry breaking in two–dimensional square grids: Persistence and failure of the dimensional crossover
No full textWe discuss the model robustness of the infinite two–dimensional square grid with respect to symmetry breakings due to the presence of defects, that is, lacks of finitely or infinitely many edges. Precisely, we study how these topological perturbations of the square grid affect the so–called dimensional crossover identified in [4]. Such a phenomenon has two evidences: the coexistence of the one and the two–dimensional Sobolev inequalities and the appearance of a continuum of L^2–critical exponents for the ground states at fixed mass of the nonlinear Schrödinger equation. From this twofold perspective, we investigate which classes of defects do preserve the dimensional crossover and which classes do not
Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit
No full textIn this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the L-2-subcritical case, they converge to the bound states of the nonlinear Schrodinger equation in the nonrelativistic limit
Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity
No full textWe consider a two-dimensional nonlinear Schr\"odinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise
An introduction to the two-dimensional Schrödinger equation with nonlinear point interactions
No full textWe present an introduction to the nonlinear Schroedinger equation (NLSE) with concentrated nonlinearities in R2. Precisely, taking a cue from the linear problem, we sketch the main challenges and the typical difficulties that arise in the twodimensional
case, and mention some recent results obtained by the authors on local and global wellposedness
Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity
No full textWe discuss strong local and global well-posedness for the three-dimensional
NLS equation with nonlinearity concentrated on . Precisely, local
well-posedness is proved for any power-nonlinearity, while global
well-posedness is obtained either for small data or in the defocusing case
under some growth assumptions. With respect to point-concentrated NLS models,
widely studied in the literature, here the dimension of the support of the
nonlinearity does not allow a direct extension of the known techniques and
calls for new ideas.Comment: 40 pages. Keywords: NLS equation, concentrated nonlinearity, unit
sphere, well-posedness, spherical harmonics. Minor changes have been made
with respect to the previous versio
Ground states for the planar NLSE with a point defect as minimizers of the constrained energy
No full textWe investigate the ground states for the focusing, subcritical nonlinear Schrodinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass. We prove that ground states exist for every positive mass and show a logarithmic singularity at the defect. Moreover, up to a multiplication by a constant phase, they are positive, radially symmetric, and decreasing along the radial direction. In order to overcome the obstacles arising from the uncommon structure of the energy space, that complicates the application of standard rearrangement theory, we move to the study of the minimizers of the action functional on the Nehari manifold and then establish a connection with the original problem. A refinement of a classical result on rearrangements is proved to obtain qualitative features of the ground states
A Note on the Dirac Operator with Kirchoff-Type Vertex Conditions on Metric Graphs
No full textIn this note we present some properties of the Dirac operator on noncompact metric graphs with Kirchoff-type vertex conditions. In particular, we discuss its spectral features and describe the associated quadratic form
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