Analysis of Partial Differential Equations (APDE)
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Entire vortex solutions of negative degree for the anisotropic Ginzburg-Landau system
Kowalczyk, M.; Lamy, X.; Smyrnelis, P.A.
(2022)
The anisotropic Ginzburg-Landau system $\Delta u+\delta \nabla (div u) +\delta curl^*(curl u)=(|u|^2-1) u$, for $u\colon\mathbb R^2\to\mathbb R^2$ and $\delta\in (-1,1)$, models the formation of vortices in liquid crystals. ... -
Cubic microlattices embedded in nematic liquid crystals: A Landau-de Gennes study
Ceuca, R.D. (2021-01-01)
We consider a Landau-de Gennes model for a connected cubic lattice scaffold in a nematic host, in a dilute regime. We analyse the homogenised limit for both cases in which the lattice of embedded particles presents or not ... -
On the Hausdorff dimension of Riemann's non-differentiable function
(2021-01-01)Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we ... -
A pseudospectral method for the one-dimensional fractional Laplacian on R
(2021-01-15)In this paper, we propose a novel pseudospectral method to approximate accurately and efficiently the fractional Laplacian without using truncation. More precisely, given a bounded regular function defined over R, we map ... -
Discrepancy of Minimal Riesz Energy Points
(2021-12-01)We find upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz s-energy on the sphere Sd. Our results are based on bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished ... -
Self-adjointness of two-dimensional Dirac operators on corner domains
(2021-01-01)We investigate the self-adjointness of the two-dimensional Dirac operator D, with quantum-dot and Lorentz-scalar i-shell boundary conditions, on piecewise C2 domains (with finitely many corners). For both models, we prove ... -
Dirac Operators and Shell Interactions: A Survey
(2021-01-01)In this survey we gather recent results on Dirac operators coupled with δ-shell interactions. We start by discussing recent advances regarding the question of self-adjointness for these operators. Afterwards we switch to ... -
On the regularity of solutions to the k-generalized korteweg-de vries equation
Kenig, C. E.; Linares, F.; Ponce, G.; Vega, L. (2018-01-01)
This work is concerned with special regularity properties of solutions to the k-generalized Korteweg-de Vries equation. In [Comm. Partial Differential Equations 40 (2015), 1336–1364] it was established that if the initial ... -
Topics in the mathematical design of materials
Chen, X.; Fonseca, I.; Ravnik, M.; Slastikov, V.; Zannoni, C.; Zarnescu, A. (2021-01-01)
We present a perspective on several current research directions relevant to the mathematical design of new materials. We discuss: (i) design problems for phase-transforming and shape-morphing materials, (ii) epitaxy as an ... -
The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
Kumar, S.; Ponce Vanegas, F.; Roncal, L.; Vega, L. (2022)
We consider the solution of the Schrödinger equation $u$ in $\mathbb{R}$ when the initial datum tends to the Dirac comb. Let $h_{\text{p}, \delta}(t)$ be the fluctuations in time of $\int\lvert x \rvert^{2\delta}\lvert ... -
On the Schrödinger map for regular helical polygons in the hyperbolic space
(2022-01-01)The main purpose of this article is to understand the evolution of X t = X s ∧− X ss , with X(s, 0) a regular polygonal curve with a nonzero torsion in the three-dimensional Minkowski space. Unlike in the case of the ... -
The Two Dimensional Liquid Crystal Droplet Problem with Tangential Boundary Condition
Geng, Z.; Lin, F. (2022-01-18)
This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition ... -
Rotational smoothing
(2022-01-05)Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators ... -
Pointwise Convergence of the Schr\"odinger Flow
Compaan, E.; Lucà, R.; Staffilani, G. (2021-01)
In this paper we address the question of the pointwise almost everywhere limit of nonlinear Schr\"odinger flows to the initial data, in both the continuous and the periodic settings. Then we show how, in some cases, certain ... -
Geometric Harmonic Analysis
Canto, J. (2021)
This thesis is the compilation of the results obtained during my PhD, which started in January 2018 and is being completed in the end of 2021. The main matter is divided into ve chapters, Chapters 2 6. Each of these ... -
Counterexamples for the fractal Schrödinger convergence problem with an Intermediate Space Trick
Eceizabarrena, D.; Ponce Vanegas, F. (2021-12-09)
We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of Du--Kim--Wang--Zhang. We confirm that the same ... -
On the uniqueness of minimisers of Ginzburg-Landau functionals
Ignat, R.; Nguyen, L.; Slastikov, V.; Zarnescu, A. (2020)
We provide necessary and sufficient conditions for the uniqueness of minimisers of the Ginzburg-Landau functional for Rn-valued maps under a suitable convexity assumption on the potential and for H1=2 \ L1 boundary data ... -
Mathematical problems of nematic liquid crystals: between dynamical and stationary problems
Zarnescu, A. (2021-05-24)
Mathematical studies of nematic liquid crystals address in general two rather different perspectives: That of fluid mechanics and that of calculus of variations. The former focuses on dynamical problems while the latter ... -
Weak sequential stability for a nonlinear model of nematic electrolytes
Fereisl, E.; Rocca, E.; Schimperna, G.; Zarnescu, A. (2021-01-01)
In this article we study a system of nonlinear PDEs modelling the electrokinetics of a nematic electrolyte material consisting of various ions species contained in a nematic liquid crystal. The evolution is described by a ... -
Entire Minimizers of Allen–Cahn Systems with Sub-Quadratic Potentials
Alikakos, N.; Gazoulis, D.; Zarnescu, A. (2021-01-01)
We study entire minimizers of the Allen–Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding ...
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