Analysis of Partial Differential Equations (APDE)
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Asymptotic behaviour of some nonlocal equations in mathematical biology and kinetic theory
(201909)We study the longtime behaviour of solutions to some partial differential equations arising in modeling of several biological and physical phenomena. In this work, the type of the equations we consider is mainly nonlocal, ... 
Minimizers of a Landaude Gennes energy with a subquadratic elastic energy
(Archive for Rational Mechanics and Analysis, 2019)We study a modified Landaude Gennes model for nematic liq uid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two and threedimensional ... 
Reconstruction of the Derivative of the Conductivity at the Boundary
(201908)We describe a method to reconstruct the conductivity and its normal derivative at the boundary from the knowledge of the potential and current measured at the boundary. This boundary determination implies the uniqueness ... 
A Bilinear Strategy for Calderón's Problem
(201908)Electrical Impedance Imaging would suffer a serious obstruction if for two different conductivities the potential and current measured at the boundary were the same. The Calder\'on's problem is to decide whether the ... 
$A_1$ theory of weights for rough homogeneous singular integrals and commutators
(Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, 2019)Quantitative $A_1A_\infty$ estimates for rough homogeneous singular integrals $T_{\Omega}$ and commutators of $\BMO$ symbols and $T_{\Omega}$ are obtained. In particular the following estimates are proved: \[ \T_\Omega ... 
On the absolute divergence of Fourier series in the infinite dimensional torus
(Colloquium Mathematicum, 20190322)In this note we present some simple counterexamples, based on quadratic forms in infinitely many variables, showing that the implication $f\in C^{(\infty}(\mathbb{T}^\omega)\Longrightarrow\sum_{\bar{p}\in\mathbb{Z}^\inf ... 
Hypocoercivity of linear kinetic equations via Harris's Theorem
(Kinetic & Related Models, 20190227)We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ or on the whole ... 
Improved fractional Poincaré type inequalities in John domains
(Arkiv för Matematik, 2019)We obtain improved fractional Poincaré inequalities in John domains of a metric space $(X, d)$ endowed with a doubling measure $\mu$ under some mild regularity conditions on the measure $\mu$. We also give sufficient ... 
Bilinear identities involving the $k$plane transform and Fourier extension operators
(2019)We prove certain $L^2(\mathbb{R}^n)$ bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the $k$plane transform. As the estimates are $L^2$based, they follow from ... 
Endpoint Sobolev continuity of the fractional maximal function in higher dimensions
(2019)We establish continuity mapping properties of the noncentered fractional maximal operator $M_{\beta}$ in the endpoint input space $W^{1,1}(\mathbb{R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. ... 
A Hardytype inequality and some spectral characterizations for the Dirac–Coulomb operator
(Revista Matemática Complutense, 201906)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 $\mathbf V$ of Coulomb type: ... 
On extension problem, trace hardy and Hardy’s inequalities for some fractional Laplacians
(Communications on pure and applied analysis, 201909)We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation ... 
Bloom type upper bounds in the product BMO setting
(Journal of Geometric Analysis, 20190408)We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral $T_n$ in $\mathbb R^n$ and a bounded singular integral $T_m$ in $\mathbb R^m$ we prove that $$ \ [T_n^1, ... 
Convex Integration Arising in the Modelling of ShapeMemory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations
(Journal of Nonlinear Science, 20190330)We study convex integration solutions in the context of the modelling of shapememory alloys. The purpose of the article is twofold, treating both rigidity and flexibility prop erties: Firstly, we relate the maximal ... 
Topological singular set of vectorvalued maps, I: application to manifoldconstrained Sobolev and BV spaces
(Calculus of Variations and Partial Differential Equations, 20190330)We introduce an operator $\mathbf{S}$ on vectorvalued maps $u$ which has the ability to capture the relevant topological information carried by $u$. In particular, this operator is defined on maps that take values in a ... 
Order Reconstruction for neatics on squares with isotropic inclusions: A Landaude Gennes study
(SIAM Journal on Applied Mathematics, 20190330)e study a modified Landaude Gennes model for nematic liq uid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two and threedimensional ... 
Bloom type inequality for biparameter singular integrals: efficient proof and iterated commutators
(International Mathematics Research Notices, 20190314)Utilising some recent ideas from our bilinear biparameter theory, we give an efficient proof of a twoweight Bloom type inequality for iterated commutators of linear biparameter singular integrals. We prove that if $T$ ... 
Boundary Triples for the Dirac Operator with CoulombType Spherically Symmetric Perturbations
(Journal of Mathematical Physics, 201902)We determine explicitly a boundary triple for the Dirac operator $H:=i\alpha\cdot \nabla + m\beta + \mathbb V(x)$ in $\mathbb R^3$, for $m\in\mathbb R$ and $\mathbb V(x)= x^{1} ( \nu \mathbb{I}_4 +\mu \beta i \lambda ... 
Sparse bounds for maximal rough singular integrals via the Fourier transform
(Annales de l'institut Fourier, 20190312)We prove a quantified sparse bound for the maximal truncations of convolutiontype singular integrals with suitable Fourier decay of the kernel. Our result extends the sparse domination principle by CondeAlonso, Culiuc, ... 
The excluded volume of twodimensional convex bodies: shape reconstruction and nonuniqueness
(Journal of Physics A: Mathematical and Theoretical, 20190205)In the Onsager model of onecomponent hardparticle systems, the entire phase behaviour is dictated by a function of relative orientation, which represents the amount of space excluded to one particle by another at this ...