Variational models for nematic liquid crystals with subquadratic growth

Nematic liquid crystals are matter in an intermediate phase between the solid and the liquid ones. Materials in the nematic phase may have topological defects, that is, regions of sharp optical contrast that carry topological invariants. In this talk, we will briefly discuss the mathematical modelling of nematic liquid crystals and their defects. In particular, we will focus on some variational continuum models with subquadratic growth in the gradient. The talk is based on joint works with Apala Majumdar (University of Bath, UK), Giandomenico Orlandi (University of Verona, Italy), and Bianca Stroffolini (University of Naples Federico II, Italy).

Some recent results on quantum Boltzman equation with singular kernel.

I will present recent results and open questions on some kinetic equations describing large systems of quantum particles

The Calderón problem with corrupted data.

In a joint work with Pedro Caro (BCAM), we consider the inverse Calderón problem consisting of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, one usually assumes the data to be given by such map. This situation corresponds to having access to infinite-precision measurements, which is totally unrealistic. In this work, we study the Calderón problem assuming the data to contain measurement errors and provide formulas to reconstruct the conductivity and its normal derivative on the surface. Additionally, we state the rate convergence of the method. Our approach is theoretical and has a stochastic flavour.

High frequencies for some spectral problems in thin structures

We analyze the behavior of the spectrum of the Laplacian in a planar domain. The thickness of the domain or of one of the components depends on a small parameter that we shall make to go to zero. The boundary conditions can be Dirichlet or Neumann depending on the problem. As it is well known, in some of these models, the low frequencies can give rise to vibrations affecting only a part of the structure or ignore, e.g., transverse vibrations. We characterize the asymptotic behavior of the high frequencies, in a certain range which depends on the structure.

Spectral convergence in high contrast boundary homogenization problems.

We consider the homogenization of boundary value problems with periodic alternating boundary conditions of Steklov type. The periodicity depends on a small parameter, and we look at the asymptotic behavior of the spectra when the parameter converges towards zero. In particular, we provide an overview of some results for the Laplace operator and for the elasticity system obtained within the framework of research of the group “Matematicas de las Vibraciones” of the UC from 2006 till 2018

On nonlocal discrete equations

Porous medium equations with nonlocal pressure

Discretizations of fractional powers of the Laplacian in bounded domains.

The fractional Laplacian is a differential operator of non-integer order that has been extensively studied in the last few decades and is naturally defined on the whole $\mathbb{R}^N$. As many other fractional order derivatives and integrals, this operator has been often used to model transport processes which generalize classical Brownian motion. However, many physical problems of interest are defined in bounded domains and the use of the fractional Laplacian as modelling tool in this context poses the challenge of providing a meaningful interpretation of the operator in these settings. Following the heat semi-group formalism, we consider a family of operators which are boundary conditions dependent and discuss a suitable approach for their numerical discretizations by combining quadratures rules with finite element methods. This approach will provide a flexible strategy for numerical computations of fractional powers of operators in bounded settings with different homogeneous boundary conditions in multi-dimensional (possibly irregular) domains. We also discuss the corresponding fractional Poisson problem and provide a natural way of defining different types of non-homogeneous boundary conditions.

Dynamics and defect formation in the nematohydrodynamics

We consider the Beris-Edwards system modelling incompressible liquid crystal flows of nematic type. This couples a Navier-Stokes system for the fluid velocity with a parabolic reaction-convection-diffusion equation for the Q-tensors describing the direction of liquid crystal molecules. In this paper, we study the effect that the flow has on the dynamics of the Q-tensors, by considering two fundamental aspects: the preservation of eigenvalue-range and the dynamical emergence of defects in the limit of high Ericksen number. This is joint work with Hao Wu and Xiang Xu.