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Federico Cacciafesta

Publications and source records attributed to Federico Cacciafesta.

29 records · Page 2Linked to original sources
arXiv2015

Local smoothing estimates for the massless Dirac-coulomb equation in 2 and 3 dimensions

We prove local smoothing estimates for the massless Dirac equation with a Coulomb potential in 2 and 3 space dimensions. Our strategy of proof is inspired by a paper of Burq et al. (2003) about Schroedinger and wave equations with inverse-square potentials, and relies on partial wave subspaces decomposition and spectral analysis of the Dirac-Coulomb operator.

math.AP↗
arXiv2014

Invariant measure for the Schrödinger equation on the real line

In this paper, we build a Gibbs measure for the cubic defocusing Schrödinger equation on the real line with a decreasing interaction potential, in the sense that the non linearity $|u|^2u$ is multiplied by a function $χ$ which we assume integrable and smooth enough. We prove that this equation is globally well-posed in the support of this measure and that the measure is invariant under the flow of the equation. What is more, the support of the measure (the set of initial data) is disjoint from $L^2$.

math.AP↗
arXiv2014

Helmholtz and dispersive equations with variable coefficients on exterior domains

We prove smoothing estimates in Morrey-Campanato spaces for a Helmholtz equation $$ -Lu+zu=f, \qquad -Lu:=\nabla^{b}(a(x)\nabla^{b}u)-c(x)u, \qquad \nabla^{b}:=\nabla+ib(x) $$ with fully variable coefficients, of limited regularity, defined on the exterior of a starshaped compact obstacle in $\mathbb{R}^{n}$, $n\ge3$, with Dirichlet boundary conditions. The principal part of the operator is a long range perturbation of a constant coefficient operator, while the lower order terms have an almost critical decay. We give explicit conditions on the size of the perturbation which prevent trapping. As an application, we prove smoothing estimates for the Schrödinger flow $e^{itL}$ and the wave flow $e^{it \sqrt{L}}$ with variable coefficients on exterior domains and Dirichlet boundary conditions.

math.AP↗
arXiv2013

The cubic nonlinear Dirac equation

We present some results obtained in collaboration with prof. Piero D'Ancona concerning global existence for the 3D cubic non linear massless Dirac equation with a potential for small initial data in $H^1$ with slight additional assumptions. The new crucial tool is given by the proof of some refined endpoint Strichartz estimates.

math.AP↗
arXiv2011

Weighted L^p estimates for powers of selfadjoint operators

We prove $L^{p}$ and weighted $L^{p}$ estimates for bounded functions of a selfadjoint operator satisfying both a pointwise gaussian estimate for its heat kernel and a finite speed of propagation property. As an application, we obtain weighted estimates for fractiional powers of an electromagnetic Schrödinger operator with singular coefficienta. The proofs are based on a modification of techniques due to Hebisch and Auscher and Martell. EDIT: second version, November 2010, a few inaccuracies have been corrected.

math.AP↗
arXiv2011

Endpoint estimates and global existence for the nonlinear Dirac equation with potential

We prove endpoint estimates with angular regularity for the wave and Dirac equations perturbed with a small potential. The estimates are applied to prove global existence for the cubic Dirac equation perturbed with a small potential, for small initial $H^{1}$ data with additional angular regularity. This implies in particular global existence in the critical energy space $H^{1}$ for small radial data.

math.AP↗