arXiv subjects

Maria J. Esteban

Publications and source records attributed to Maria J. Esteban.

At least 19 recordsLinked to original sources

arXiv2025

The CKN inequality for spinors: symmetry and symmetry breaking

This paper is devoted to Sobolev interpolation inequalities for spinors, with weights of Caffarelli-Kohn-Nirenberg (CKN) type. In view of the corresponding results for scalar functions, a natural question is to determine whether optimal spinors have symmetry properties, or whether spinors with symmetry properties are linearly unstable, in which case we shall say that symmetry breaking occurs. What symmetry means has to be carefully defined and the overall picture turns out to be richer than in the scalar case. So far, no symmetrization technique is available in the spinorial case. We can however determine a range of the parameters for which symmetry holds using a detailed analysis based mostly on spectral methods.

math.AP↗

arXiv2025

Mathematics in art, for art and as art

The fundamental role of mathematics as an inspiration for artists, but also as a tool for art creation, is presented in this paper following different art fields, like architecture, sculpture, painting, photography, literature and poetry, movie making and music. The historical viewpoint is completed with recent applications of mathematics to create art in the digital era. Finally, the article contains a discussion about the possibility of the mathematical creation being considered artistic.

math.HO↗

arXiv2025

The importance of Luis Caffarelli's work in the study of fluids

In this paper we describe the work of Luis Caffarelli in the area of fluid mechanics and related topics. Not only has his work on fluid mechanics been very influential, but many of his contributions that do not directly relate to fluid mechanics, such as his important results on the fractional Laplacian or the regularity of solutions to linear parabolic equations with oscillating coefficients, have been used in the study of fluids in many important ways. Thus, any review of his work has to include his contributions to the general (partial) regularity theory of solutions of Navier-Stokes equations and other studies related to fluid motion.

math.AP↗

arXiv2025

Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence

We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proofs rely on several ingredients such as competing symmetries, a flow based on continuous Steiner symmetrization that interpolates continuously between a function and its symmetric decreasing rearrangement, and refined estimates on the Sobolev functional in the neighborhood of the optimal Aubin--Talenti functions.

math.AP↗

arXiv2024

A short review on Improvements and stability for some interpolation inequalities

In this paper, we present recent stability results with explicit and dimensionally sharp constants and optimal norms for the Sobolev inequality and for the Gaussian logarithmic Sobolev inequality obtained by the authors in [24]. The stability for the Gaussian logarithmic Sobolev inequality was obtained as a byproduct of the stability for the Sobolev inequality. Here we give a new, direct, alternative proof. We also discuss improved versions of interpolation inequalities based on the carré du champ method.

math.AP↗

arXiv2023

Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac-Coulomb operators

We consider a linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property associated with the block decomposition. A typical example is the Dirac-Coulomb operator defined on $C^\infty_c(\mathbb R^3\setminus\{0\}, \mathbb C^4)$. In this paper we define a distinguished self-adjoint extension with a spectral gap and characterize its eigenvalues in that gap by a min-max principle. This has been done in the past under technical conditions. Here we use a different, geometric strategy, to achieve that goal by making only minimal assumptions. Our result applied to the Dirac-Coulomb-like Hamitonians covers sign-changing potentials as well as molecules with an arbitrary number of nuclei having atomic numbers less than or equal to 137.

math.SP↗

arXiv2022

Which nuclear shape generates the strongest attraction on a relativistic electron? An open problem in relativistic quantum mechanics

In this article we formulate several conjectures concerning the lowest eigenvalue of a Dirac operator with an external electrostatic potential. The latter describes a relativistic quantum electron moving in the field of some (pointwise or extended) nuclei. The main question we ask is whether the eigenvalue is minimal when the nuclear charge is concentrated at one single point. This well-known property in nonrelativistic quantum mechanics has escaped all attempts of proof in the relativistic case.

math.AP↗

arXiv2022

On the eigenvalues of operators with gaps. Application to Dirac operators

This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. The result is optimal for the Coulomb potential. An erratum is appended at the end of the tex.

math.SP↗

arXiv2022

Hardy-Littlewood-Sobolev and related inequalities: stability

The purpose of this text is twofold. We present a review of the existing stability results for Sobolev, Hardy-Littlewood-Sobolev (HLS) and related inequalities. We also contribute to the topic with some observations on constructive stability estimates for (HLS).

math.AP↗

arXiv2021

Gagliardo-Nirenberg-Sobolev inequalities on planar graphs

In this paper we study a family of interpolation Gagliardo-Nirenberg-Sololev inequalities on planar graphs. We are interested in knowing when the best constants in the inequalities are achieved. We also analyse the set of solutions of the corresponding Euler-Lagrange equations.

math.DS↗

arXiv2021

Dirac-Coulomb operators with general charge distribution. I. Distinguished extension and min-max formulas

This paper is the first of a series where we study the spectral properties of Dirac operators with the Coulomb potential generated by any finite signed charge distribution $μ$. We show here that the operator has a unique distinguished self-adjoint extension under the sole condition that $μ$ has no atom of weight larger than or equal to one. Then we discuss the case of a positive measure and characterize the domain using a quadratic form associated with the upper spinor, following earlier works by Esteban and Loss. This allows us to provide min-max formulas for the eigenvalues in the gap. In the event that some eigenvalues have dived into the negative continuum, the min-max formulas remain valid for the remaining ones. At the end of the paper we also discuss the case of multi-center Dirac-Coulomb operators corresponding to $μ$ being a finite sum of deltas.

math.SP↗

arXiv2020

Dirac-Coulomb operators with general charge distribution. II. The lowest eigenvalue

Consider the Coulomb potential $-μ\ast|x|^{-1}$ generated by a non-negative finite measure $μ$. It is well known that the lowest eigenvalue of the corresponding Schrödinger operator $-Δ/2-μ\ast|x|^{-1}$ is minimized, at fixed mass $μ(\mathbb{R}^3)=ν$, when $μ$ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator $-iα\cdot\nabla+β-μ\ast|x|^{-1}$. In a previous work on the subject we proved that this operator is self-adjoint when $μ$ has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min-max formulas. Here we consider the critical mass $ν_1$, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all $μ\geq0$ with $μ(\mathbb{R}^3)<ν_1$. We first show that $ν_1$ is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all $0\leqν<ν_1$, there exists an optimal measure $μ\geq0$ giving the lowest possible eigenvalue at fixed mass $μ(\mathbb{R}^3)=ν$, which concentrates on a compact set of Lebesgue measure zero. The last property is shown using a new unique continuation principle for Dirac operators. The existence proof is based on the concentration-compactness principle.

math.SP↗

arXiv2020

Critical magnetic field for 2d magnetic Dirac-Coulomb operators and Hardy inequalities

This paper is devoted to the study of the two-dimensional Dirac-Coulomb operator in presence of an Aharonov-Bohm external magnetic potential. We characterize the highest intensity of the magnetic field for which a two-dimensional magnetic Hardy inequality holds. Up to this critical magnetic field, the operator admits a distinguished self-adjoint extension and there is a notion of ground state energy, defined as the lowest eigenvalue in the gap of the continuous spectrum.

math.AP↗

arXiv2020

Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schr{ö}dinger operators involving Aharonov-Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions two and three. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3.

math.AP↗

arXiv2019

Improved interpolation inequalities and stability

For exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carré du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.

math.AP↗

arXiv2019

Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields

This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schr{ö}dinger operator involving an Aharonov-Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller-Lieb-Thir-ring inequality. We prove that this potential is radially symmetric if the intensity of the magnetic field is below an explicit threshold, while symmetry is broken above a second threshold corresponding to a higher magnetic field. The method relies on the study of the magnetic kinetic energy of the wave function and amounts to study the symmetry properties of the optimal functions in a magnetic Hardy-Sobolev interpolation inequality. We give a quantified range of symmetry by a non-perturbative method. To establish the symmetry breaking range, we exploit the coupling of the phase and of the modulus and also obtain a quantitative result.

math.AP↗

arXiv2019

Domains for Dirac-Coulomb min-max levels

We consider a Dirac operator in three space dimensions, with an electrostatic (i.e. real-valued) potential $V(x)$, having a strong Coulomb-type singularity at the origin. This operator is not always essentially self-adjoint but admits a distinguished self-adjoint extension $D\_V$. In a first part we obtain new results on the domain of this extension, complementing previous works of Esteban and Loss. Then we prove the validity of min-max formulas for the eigenvalues in the spectral gap of $D\_V$, in a range of simple function spaces independent of $V$. Our results include the critical case $\liminf\_{x \to 0} |x| V(x)= -1$, with units such that $\hbar=mc^2=1$, and they are the first ones in this situation. We also give the corresponding results in two dimensions.

math-ph↗

arXiv2018

Interpolation inequalities and spectral estimates for magnetic operators

We prove magnetic interpolation inequalities and Keller-Lieb-Thir-ring estimates for the principal eigenvalue of magnetic Schr{ö}dinger operators. We establish explicit upper and lower bounds for the best constants and show by numerical methods that our theoretical estimates are accurate.

math.AP↗