Analysis of Partial Differential Equations (APDE)
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Quantitative weighted estimates for singular integrals and commutators
(20180227)In this dissertation several quantitative weighted estimates for singular integral op erators, commutators and some vector valued extensions are obtained. In particular strong and weak type $(p, p)$ estimates, CoifmanFe ... 
Variable Lorentz estimate for nonlinear elliptic equations with partially regular nonlinearities
(Nonlinear Analysis, 20180215)We prove global Calder\'onZygmund type estimate in Lorentz spaces for variable power of the gradients to weak solution of nonlinear elliptic equations in a nonsmooth domain. We mainly assume that the nonlinearities are ... 
Zero limit of entropic relaxation time for the Shliomis model of ferrofluids
(20180211)We construct solutions for the Shilomis model of ferrofluids in a critical space, uniformly in the entropic relaxation time $ \tau \in (0, \tau_0) $. This allows us to study the convergence when $ \tau\to 0 $ for such solutions. 
Lorentz estimates for the gradient of weak solutions to elliptic obstacle problems with partially BMO coefficients
(Boundary Value Problems, 2017)We prove global Lorentz estimates for variable power of the gradient of weak solution to linear elliptic obstacle problems with small partially BMO coefficients over a bounded nonsmooth domain. Here, we assume that the ... 
Weghted Lorentz and LorentzMorrey estimates to viscosity solutions of fully nonlinear elliptic equations
(Complex Variables and Elliptic Equations, 2018)We prove a global weighted Lorentz and LorentzMorrey estimates of the viscosity solutions to the Dirichlet problem for fully nonlinear elliptic equation $F(D^{2}u,x)=f(x)$ defined in a bounded $C^{1,1}$ domain. The ... 
Bilinear representation theorem
(Transactions of the American Mathematical Society, 20180101)We represent a general bilinear CalderónZygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so ... 
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 ... 
Improved A1 − A∞ and related estimates for commutators of rough singular integrals
(Proceedings of the Edinburgh Mathematical Society, 2017)An $A_1A_\infty$ estimate improving a previous result in [22] for $[b, T_\Omega]$ with $\Omega\in L^\infty(S^{n1})$ and $b\in BMO$ is obtained. Also a new result in terms of the $A_\infty$ constant and the one ... 
On a hyperbolic system arising in liquid crystal modelling
(Journal of Hyperbolic Differential Equations, 201711)We consider a model of liquid crystals, based on a nonlinear hyperbolic system of differential equations, that represents an inviscid version of the model proposed by Qian and Sheng. A new concept of dissipative solution ... 
Spherevalued harmonic maps with surface energy and the K13 problem
(Advances in the Calculus of Variations, 201711)We consider an energy functional motivated by the celebrated K13 problem in the OseenFrank theory of nematic liquid crystals. It is defined for spherevalued functions and appears as the usual Dirichlet energy with an ... 
Global wellposedness and twistwave solutions for the inertial QianSheng model of liquid crystals
(Journal of Differential Equations, 20171002)We consider the inertial QianSheng model of liquid crystals which couples a hyperbolictype equation involving a secondorder material derivative with a forced incompressible NavierStokes system. We study the energy law ... 
Klein's Paradox and the Relativistic $\delta$shell Interaction in $\mathbb{R}^3$
(Analysis & PDE, 201711)Under certain hypothesis of smallness of the regular potential $\mathbf{V}$, we prove that the Dirac operator in $\mathbb{R}^3$ coupled with a suitable rescaling of $\mathbf{V}$, converges in the strong resolvent sense ... 
Asymptotic behaviour for fractional diffusionconvection equations
(201710)We consider a convectiondiffusion model with linear fractional diffusion in the subcritical range. We prove that the large time asymptotic behavior of the solution is given by the unique entropy solution of the convective ... 
Existence of weak solutions for a general porous medium equation with nonlocal pressure
(submitted, 201710)We study the general nonlinear diffusion equation $u_t=\nabla\cdot (u^{m1}\nabla (\Delta)^{s}u)$ that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters ... 
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 ... 
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 ... 
Weighted mixed weaktype inequalities for multilinear operators
(Studia Mathematica, 2017)In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,...,w_m)$ and $\nu = w_1^\frac{1}{m}...w_m^\frac{1}{m}$, the main ... 
Mixed norm estimates for the Cesàro means associated with DunklHermite expansions
(Transactions of the American Mathematical Society, 2017)Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with DunklHermite expansions on $\mathbb{R}^d$. These expansions arise when one considers the DunklHermite operator ... 
Dispersive effects of weakly compressible and fast rotating inviscid fluids
(Discrete and Continuous Dynamical Systems  Series A, 201708)We consider a system describing the motion of an isentropic, inviscid, weakly compressible, fast rotating fluid in the whole space $\mathbb{R}^3$, with initial data belonging to $ H^s \left( \mathbb{R}^3 \right), s>5/2 $. ... 
On pointwise and weighted estimates for commutators of CalderónZygmund operators
(Advances in Mathematics, 2017)In recent years, it has been well understood that a CalderónZygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar ...