Analysis of Partial Differential Equations (APDE)
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Rotational smoothing
(20220105)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
(202101)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
(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
(20211209)We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of DuKimWangZhang. We confirm that the same ... 
On the uniqueness of minimisers of GinzburgLandau functionals
(2020)We provide necessary and sufficient conditions for the uniqueness of minimisers of the GinzburgLandau functional for Rnvalued 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
(20210524)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
(20210101)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 SubQuadratic Potentials
(20210101)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 subquadratic behaviour locally near their minima. The corresponding ... 
Quasiinvariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLS
(20220101)The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasiinvariance of the Gaussian measure on L2(T) with covariance [1+(−Δ)s]−1 ... 
Cavity Volume and Free Energy in ManyBody Systems
(20211001)Within this work, we derive and analyse an expression for the free energy of a singlespecies system in the thermodynamic limit in terms of a generalised cavity volume, that is exact in general, and in principle applicable ... 
Hölder regularity and convergence for a nonlocal model of nematic liquid crystals in the largedomain limit
(20220201)We consider a nonlocal free energy functional, modelling a competition between entropy and pairwise interactions reminiscent of the second order virial expansion, with applications to nematic liquid crystals as a particular ... 
Discrete Carleman estimates and three balls inequalities
(20211016)We prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schrödinger operators. These quantitatively connect the discrete setting in which the unique continuation property fails and the ... 
Numerical approximation of the fractional Laplacian on R using orthogonal families
(20201201)In this paper, using wellknown complex variable techniques, we compute explicitly, in terms of the F12 Gaussian hypergeometric function, the onedimensional fractional Laplacian of the complex Higgins functions, the complex ... 
Vortex Filament Equation for a regular polygon in the hyperbolic plane
(20200709)The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and ... 
Riemann's nondifferentiable function and the binormal curvature flow
(20200714)We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object ... 
A Hardytype inequality and some spectral characterizations for the Dirac–Coulomb operator
(20200101)We prove a sharp Hardytype inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrixvalued potentials V of Coulomb type: we characterise ... 
Uniqueness properties of solutions to the BenjaminOno equation and related models
(20200315)We prove that if u1,u2 are real solutions of the BenjaminOno equation defined in (x,t)∈R×[0,T] which agree in an open set Ω⊂R×[0,T], then u1≡u2. We extend this uniqueness result to a general class of equations of BenjaminOno ... 
Asymptotics in Fourier space of selfsimilar solutions to the modified Kortewegde Vries equation
(20200501)We give the asymptotics of the Fourier transform of selfsimilar solutions for the modified Kortewegde Vries equation. In the defocussing case, the selfsimilar profiles are solutions to the Painlevé II equation; although ... 
Evolution of Polygonal Lines by the Binormal Flow
(20200601)The aim of this paper is threefold. First we display solutions of the cubic nonlinear Schrödinger equation on R in link with initial data a sum of Dirac masses. Secondly we show a Talbot effect for the same equation. Finally ... 
On the energy of critical solutions of the binormal flow
(20200702)The binormal flow is a model for the dynamics of a vortex filament in a 3D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1D cubic ...