Luca Fanelli

Strichartz and Smoothing Estimates for Dispersive Equations with Magnetic Potentials

We prove global smoothing and Strichartz estimates for the Schrödinger, wave, Klein–Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical.

On the existence of maximizers for a family of restriction theorems

We prove the existence of maximizers for a general family of re- strictions operators, up to the end-point. We also provide some counterxam- ples in the end-point case. In the sequel we shall denote by dany positive measure on R d . For every fixed

On the blow-up threshold for weakly coupled nonlinear Schrödinger equations

We study the Cauchy problem for a system of two coupled nonlinear focusing Schrodinger equations arising in nonlinear optics. We discuss when the solutions are global in time or blow-up in finite time. Some results, in dependence of the data of the problem, are proved; in particular we prove, for suitable values of the parameters, that the blow-up threshold (if the nonlinearity has the critical growth) is a universal constant.

Counterexamples to strichartz estimates for the magnetic Schrödinger equation

In space dimension n ≥ 3, we consider the magnetic Schrodinger Hamiltonian H = -(∇ - iA(x))2 and the corresponding Schrodinger equation \[ i\partial_tu + Hu=0. \] We show some explicit examples of potentials A, with less than Coulomb decay, for which any solution of this equation cannot satisfy Strichartz estimates, in the whole range of Schrodinger admissibility.

Time Decay of Scaling Invariant Electromagnetic Schrödinger Equations on the Plane

We prove the sharp

Semilinear Hamiltonian Schrödinger systems

{L^1-L^\infty}

Relativistic Hardy Inequalities in Magnetic Fields

L1-L∞ time-decay estimate for the 2D -Schrödinger equation with a general family of scaling critical electromagnetic potentials.

Endpoint Strichartz estimates for the magnetic Schrödinger equation

We study the Helmholtz equation with electromagnetic-type perturbations, in the exterior of a domain, in dimension