Browsing Linear and NonLinear Waves by Title
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Reconstruction from boundary measurements for less regular conductivities
(20161001)In this paper, following Nachman's idea [14] and Haberman and Tataru's idea [9], we reconstruct $C^1$ conductivity $\gamma$ or Lipchitz conductivity $\gamma$ with small enough value of $\nabla log\gamma$ in a Lipschitz ... 
Regularity of fractional maximal functions through Fourier multipliers
(2018)We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function ... 
Relativistic Hardy Inequalities in Magnetic Fields
(20141231)We deal with Dirac operators with external homogeneous magnetic fields. Hardytype inequalities related to these operators are investigated: for a suitable class of transversal magnetic fields, we prove a Hardy inequality ... 
The relativistic spherical $\delta$shell interaction in $\mathbb{R}^3$: spectrum and approximation
(20170803)This note revolves on the free Dirac operator in $\mathbb{R}^3$ and its $\delta$shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by ... 
Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory
(2019)Abstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − Lσ,μ[φ(u)] = f(x,t) in RN × (0,T), where Lσ,μ is a general ... 
Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments
(2018)\noindent We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u\mathfrak{L}[\varphi(u)]=f(x,t) \qquad\text{in}\qquad ... 
The Schrödinger equation and Uncertainty Principles
(202009)The main task of this thesis is the analysis of the initial data u0 of Schrödinger’s initial value problem in order to determine certain properties of its dynamical evolution. First we consider the elliptic Schrödinger ... 
SelfAdjoint Extensions for the Dirac Operator with CoulombType Spherically Symmetric Potentials
(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 $\mathbb V(x)= x^{1} ( \nu \mathbb{I}_4 +\mu \beta i \lambda \alpha\cdot{x}/{x}\,\beta)$, ... 
Selfsimilar dynamics for the modified Kortewegde Vries equation
(20190409)We prove a local well posedness result for the modified Kortewegde Vries equa tion in a critical space designed so that is contains selfsimilar solutions. As a consequence, we can study the flow of this equation around ... 
Sensitivity of twin boundary movement to sample orientation and magnetic field direction in NiMnGa
(2019)When applying a magnetic field parallel or perpendicular to the long edge of a parallelepiped Ni MnGa stick, twin boundaries move instantaneously or gradullay through the sample. We evaluate the sample shape dependence ... 
Sharp bounds for the ratio of modified Bessel functions
(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 ... 
A sharp lorentzinvariant strichartz norm expansion for the cubic wave equation in \mathbb{R}^{1+3}
(2020)We provide an asymptotic formula for the maximal Stri chartz norm of small solutions to the cubic wave equation in Minkowski space. The leading coefficient is given by Foschi’s sharp constant for the linear Strichartz ... 
Shell interactions for Dirac operators: On the point spectrum and the confinement
(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
(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 lower bounds for solutions of Schrodinger evolutions
(20190821)We present some lower bounds for regular solutions of Schr odinger equations with bounded and time dependent complex potentials. Assuming that the solution has some positive mass at time zero within a ball of certain radius, ... 
Some remarks on the $L^p$ regularity of second derivatives of solutions to nondivergence elliptic equations and the Dini condition
(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
(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 ...