On the extensions of the De Giorgi approach to nonlinear hyperbolic equations

In 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 Graph

The 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

A Note on the Dirac Operator with Kirchoff-Type Vertex Conditions on Metric Graphs

In 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

Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity

We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on S2 . Precisely, local well-posedness is proved for any C 2 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

On the extensions of the De Giorgi approach to nonlinear hyperbolic equations

In 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

Symmetry breaking in two–dimensional square grids: Persistence and failure of the dimensional crossover

We 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

An introduction to the two-dimensional Schrödinger equation with nonlinear point interactions

We 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

Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

We 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