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#### Singular Perturbation of the Dirac Hamiltonian

(2017-12-15)

This thesis is devoted to the study of the Dirac Hamiltonian perturbed by delta-type potentials and Coulomb-type potentials.
We analysed the delta-shell interaction on bounded and smooth domains and its approximation by ...

#### Sphere-valued harmonic maps with surface energy and the K13 problem

(2017-11)

We consider an energy functional motivated by the celebrated K13 problem in the Oseen-Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an ...

#### On a hyperbolic system arising in liquid crystal modelling

(2017-11)

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 ...

#### Klein's Paradox and the Relativistic $\delta$-shell Interaction in $\mathbb{R}^3$

(2017-11)

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 re-scaling of $\mathbf{V}$, converges in the strong resolvent sense ...

#### Absence of eigenvalues of two-dimensional magnetic Schr ̈odinger operators

(2017-10-17)

By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding two-dimensional Schr ̈odinger operator possesses no point ...

#### Global well-posedness and twist-wave solutions for the inertial Qian-Sheng model of liquid crystals

(2017-10-02)

We consider the inertial Qian-Sheng model of liquid crystals which couples a hyperbolic-type equation involving a second-order material derivative with a forced incompressible Navier-Stokes system. We study the energy law ...

#### Asymptotic behaviour for fractional diffusion-convection equations

(2017-10)

We consider a convection-diffusion model with linear fractional diffusion in the sub-critical 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

(2017-10)

We study the general nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$ that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters ...

#### Three-dimensional coarsening dynamics of a conserved, nematic liquid crystal-isotropic fluid mixture

(2017-09)

We present a numerical investigation of the three-dimensional coarsening dynamics of a nematic liquid crystal-isotropic fluid mixture using a conserved phase field model. The model is a coupled system for a generalized ...

#### The relativistic spherical $\delta$-shell interaction in $\mathbb{R}^3$: spectrum and approximation

(2017-08-03)

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 ...