Browsing Linear and NonLinear Waves by Title
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SelfAdjoint Extensions for the Dirac Operator with CoulombType Spherically Symmetric Potentials
(Letters in Mathematical Physics, 2018)We describe the selfadjoint realizations of the operator $H:=i\alpha\cdot \nabla + m\beta + \mathbb V(x)$, for $m\in\mathbb R $, and $\mathbb V(x)= x^{1} ( \nu \mathbb{I}_4 +\mu \beta i \lambda \alpha\cdot{x}/{x}\,\beta)$, ... 
Sharp bounds for the ratio of modified Bessel functions
(Mediterranean Journal of Mathematics, 20170621)Let $I_{\nu }\left( x\right) $ be the modified Bessel functions of the first kind of order $\nu $, and $S_{p,\nu }\left( x\right) =W_{\nu }\left( x\right) ^{2}2pW_{\nu }\left( x\right) x^{2}$ with $W_{\nu }\left( x\right) ... 
Sharp exponential localization for eigenfunctions of the Dirac Operator
(2018)We determine the fastest possible rate of exponential decay at infinity for eigenfunctions of the Dirac operator $\mathcal D_n + \mathbb V$, being $\mathcal D_n$ the massless Dirac operator in dimensions $n=2,3$ and ... 
Shell interactions for Dirac operators: On the point spectrum and the confinement
(SIAM Journal on Mathematical Analysis, 20151231)Spectral properties and the confinement phenomenon for the coupling $H + V$ are studied, where $H =i\alpha \cdot \nabla + m\beta$ is the free Dirac operator in $\mathbb{R}^3$ and $V$ is a measurevalued potential. The ... 
Singular Perturbation of the Dirac Hamiltonian
(20171215)This thesis is devoted to the study of the Dirac Hamiltonian perturbed by deltatype potentials and Coulombtype potentials. We analysed the deltashell interaction on bounded and smooth domains and its approximation by ... 
Singularity formation for the 1D cubic NLS and the Schrödinger map on $\mathbb{S}^2$
(20170202)In this note we consider the 1D cubic Schrödinger equation with data given as small perturbations of a Dirac$\delta$ function and some other related equations. We first recall that although the problem for this type of ... 
Some geometric properties of Riemann’s nondifferentiable function
(Comptes Rendus Mathematique, 20191106)Riemann’s nondifferentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory ... 
Some remarks on the $L^p$ regularity of second derivatives of solutions to nondivergence elliptic equations and the Dini condition
(Rendiconti Lincei  Matematica e Applicazioni, 20170530)In this note we prove an endpoint regularity result on the $L^P$ integrability of the second derivatives of solutions to nondivergence form uniformly elliptic equations whose second derivatives are a priori only known ... 
Sparse bounds for pseudodifferential operators
(Journal d'Analyse Mathématique, 2018)We prove sparse bounds for pseudodifferential operators associated to H\"ormander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of ... 
Spectral asymptotics for $\delta$interactions on sharp cones
(Journal of Mathematical Analysis and Applications, 2017)We investigate the spectrum of threedimensional Schr\"odinger operators with $\delta$interactions of constant strength supported on circular cones. As shown in earlier works, such operators have infinitely many eigenvalues ... 
Spectral asymptotics of the Dirichlet Laplacian in a conical layer
(Communications on Pure and Applied Analysis, 20150501)The spectrum of the Dirichlet Laplacian on conical layers is analysed through two aspects: the infiniteness of the discrete eigenvalues and their expansions in the small aperture limit. On the one hand, we prove that, for ... 
Spectral stability of Schrödinger operators with subordinated complex potentials
(Journal of Spectral Theory, 20180628)We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the nonnegative semiaxis for all potentials satisfying a formsubordinate smallness condition. By developing ... 
Spectral Transitions for AharonovBohm Laplacians on Conical Layers
(20160711)We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an AharonovBohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary ... 
A strategy for selfadjointness of Dirac operators: Applications to the MIT bag model and deltashell interactions
(20161221)We develop an approach to prove selfadjointness of Dirac operators with boundary or transmission conditions at a $C^2$compact surface without boundary. To do so we are lead to study the layer potential induced by the ... 
The dynamics of vortex filaments with corners
(Communications on Pure and Applied Analysis (CPAA), 20150701)This paper focuses on surveying some recent results obtained by the author together with V. Banica on the evolution of a vortex filament with one corner according to the socalled binormal flow. The case of a regular polygon ... 
The initial value problem for the binormal flow with rough data
(Annales Scientifiques de l'Ecole Normale Superieure, 20151231)In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the ... 
The Vortex Filament Equation as a Pseudorandom Generator
(Acta Applicandae Mathematicae, 20150801)In this paper, we consider the evolution of the socalled vortex filament equation (VFE), $$ X_t = X_s \wedge X_{ss},$$ taking a planar regular polygon of M sides as initial datum. We study VFE from a completely novel ... 
Threedimensional coarsening dynamics of a conserved, nematic liquid crystalisotropic fluid mixture
(Journal of NonNewtonian Fluid Mechanics, 201709)We present a numerical investigation of the threedimensional coarsening dynamics of a nematic liquid crystalisotropic fluid mixture using a conserved phase field model. The model is a coupled system for a generalized ... 
Uniqueness and Properties of Distributional Solutions of Nonlocal Equations of Porous Medium Type
(Advances in Mathematics, 20160901)We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation $\partial_tu\mathcal{L}^\mu [\varphi (u)]=0$. Here $\mathcal{L}^\mu$ ... 
Uniqueness properties for discrete equations and Carleman estimates
(Journal of Functional Analysis, 20170325)Using Carleman estimates, we give a lower bound for solutions to the discrete Schrödinger equation in both dynamic and stationary settings that allows us to prove uniqueness results, under some assumptions on the decay of ...