arXiv subjects

Albert Mas

Publications and source records attributed to Albert Mas.

At least 19 recordsLinked to original sources

arXiv2026

A Payne-Weinberger inequality for quantum dot Dirac operators

In this work we prove a Payne-Weinberger type inequality for quantum dot Dirac operators defined on bounded and simply connected planar domains. This is a sharp upper bound for their first positive eigenvalue depending only on the isoperimetric deficit of the domain. To this end, we use a recently studied connection with the so-called $\overline\partial$-Robin Laplacian and we establish an analogous inequality for its first eigenvalue, relying on the corresponding inequality for the Robin Laplacian.

math.AP↗

arXiv2025

A connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians in the context of shape optimization problems

This work addresses Faber-Krahn-type inequalities for quantum dot Dirac operators with nonnegative mass on bounded domains in $\mathbb{R}^2$. We show that this family of inequalities is equivalent to a family of Faber-Krahn-type inequalities for $\overline\partial$-Robin Laplacians. Thanks to this, we prove them in the case of simply connected domains for quantum dot boundary conditions asymptotically close to zigzag boundary conditions. Finally, we also study the case of negative mass.

math.AP↗

arXiv2024

Examples of optimal H\"older regularity in semilinear equations involving the fractional Laplacian

We discuss the H\"older regularity of solutions to the semilinear equation involving the fractional Laplacian $(-\Delta)^s u=f(u)$ in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the nonlocality and the semilinearity of the equation, since it does not happen neither for local semilinear equations, nor for nonlocal linear equations. Namely, for nonlinearities $f$ in $C^\beta$ and when $2s+\beta <1$, the solution is not always $C^{2s+\beta-\epsilon}$ for all $\epsilon >0$. Instead, in general the solution $u$ is at most $C^{2s/(1-\beta)}.$

math.AP↗

arXiv2024

Strict rearrangement inequalities: nonexpansivity and periodic Gagliardo seminorms

This paper deals with the behavior of the periodic Gagliardo seminorm under two types of rearrangements, namely under a periodic, and respectively a cylindrical, symmetric decreasing rearrangement. Our two main results are P\'olya-Szeg\H{o} type inequalities for these rearrangements. We also deal with the cases of equality. Our method uses, among others, some classical nonexpansivity results for rearrangements for which we provide some slight improvements. Our proof is based on the ideas of [Frank and Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 2008], where a new proof to deal with the cases of equality in the nonexpansivity theorem was given, albeit in a special case involving the rearrangement of only one function.

math.AP↗

arXiv2024

Periodic solutions to integro-differential equations: variational formulation, symmetry, and regularity

We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in $\mathbb{R}$. We prove that, after an appropriate translation, each of them is necessarily an even function which is decreasing in half its period. In particular, it has only two critical points in half its period, the absolute maximum and minimum. If these statements hold for all nonconstant periodic solutions, and not only for constrained minimizers, remains as an open problem. Our results apply to operators with kernels in two different classes: kernels $K$ which are convex and kernels for which $K(\tau^{1/2})$ is a completely monotonic function of $\tau$. This last new class arose in our previous work on nonlocal Delaunay surfaces in $\mathbb{R}^n$. Due to their symmetry of revolution, it gave rise to a 1d problem involving an operator with a nonconvex kernel. Our proofs are based on a not so well-known Riesz rearrangement inequality on the circle $\mathbb{S}^1$ established in 1976. We also put in evidence a new regularity fact which is a truly nonlocal-semilinear effect and also occurs in the nonperiodic setting. Namely, for nonlinearities in $C^\beta$ and when $2s+\beta <1$ ($2s$ being the order of the operator), the solution is not always $C^{2s+\beta-\epsilon}$ for all $\epsilon>0$.

math.AP↗

arXiv2024

Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions

This work addresses the resolvent convergence of generalized MIT bag operators to Dirac operators with zigzag type boundary conditions. We prove that the convergence holds in strong but not in norm resolvent sense. Moreover, we show that the only obstruction for having norm resolvent convergence is the existence of an eigenvalue of infinite multiplicity for the limiting operator. More precisely, we prove the convergence of the resolvents in operator norm once projected into the orthogonal of the corresponding eigenspace.

math.AP↗

arXiv2022

Existence and symmetry of periodic nonlocal-CMC surfaces via variational methods

This paper provides the first variational proof of the existence of periodic nonlocal-CMC surfaces. These are nonlocal analogues of the classical Delaunay cylinders. More precisely, we show the existence of a set in $\mathbb{R}^n$ which is periodic in one direction, has a prescribed (but arbitrary) volume within a slab orthogonal to that direction, has constant nonlocal mean curvature, and minimizes an appropriate periodic version of the fractional perimeter functional under the volume constraint. We show, in addition, that the set is cylindrically symmetric and, more significantly, that it is even as well as nonincreasing on half its period. This monotonicity property solves an open problem and an obstruction which arose in an earlier attempt, by other authors, to show the existence of minimizers.

math.AP↗

arXiv2021

Eigenvalue curves for generalized MIT bag models

We study spectral properties of Dirac operators on bounded domains $\Omega \subset \mathbb{R}^3$ with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter $\tau\in\mathbb{R}$; the case $\tau = 0$ corresponds to the MIT bag model. We show that the eigenvalues are parametrized as increasing functions of $\tau$, and we exploit this monotonicity to study the limits as $\tau \to \pm \infty$. We prove that if $\Omega$ is not a ball then the first positive eigenvalue is greater than the one of a ball with the same volume for all $\tau$ large enough. Moreover, we show that the first positive eigenvalue converges to the mass of the particle as $\tau \downarrow -\infty$, and we also analyze its first order asymptotics.

math.AP↗

arXiv2021

General $\delta$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation

In this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials.

math.AP↗

arXiv2019

Self-adjoint Dirac operators on domains in $\mathbb{R}^3$

In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(\Omega; \mathbb{C}^4)$, where $\Omega \subset \mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with Robin boundary conditions. Among the Dirac operators treated here is also the so-called MIT bag operator, which has been used by physicists and more recently was discussed in the mathematical literature. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman-Schwinger principle, a qualitative understanding of the scattering properties in the case that $\Omega$ is unbounded, and corresponding trace formulas.

math.SP↗

arXiv2019

Discrepancy of minimal Riesz energy points

We find upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz $s$-energy on the sphere $\mathbb S^d.$ Our results are based in bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished manuscript where estimates for the spherical cap discrepancy of the logarithmic energy minimizers in $\mathbb S^2$ were obtained. Our result improves previously known bounds for $0\le s<2$ and $s\neq 1$ in $\mathbb S^2,$ where $s=0$ is Wolff's result, and for $d-t_0<s<d$ with $t_0\approx 2.5$ when $d\ge 3$ and $s\neq d-1.$

math.CA↗

arXiv2019

Failure of rational approximation on some Cantor type sets

Let $A(K)$ be the algebra of continuous functions on a compact set $K\subset\mathbb C$ which are analytic on the interior of $K$, and $R(K)$ the closure (with the uniform convergence on $K$) of the functions that are analytic on a neighborhood of $K$. A counterexample of a question made by A. O'Farrell about the equality of the algebras $R(K)$ and $A(K)$ when $K=(K_{1}\times[0,1])\cup([0,1]\times K_{2})\subseteq\mathbb C$, with $K_{1}$ and $K_{2}$ compact subsets of $[0,1]$, is given. Also, the equality is proved with the assumption that $K_{1}$ has no interior.

math.CA↗

arXiv2018

The MIT Bag Model as an infinite mass limit

The Dirac operator, acting in three dimensions, is considered. Assuming that a large mass $m>0$ lies outside a smooth and bounded open set $\Omega\subset\R^3$, it is proved that its spectrum is approximated by the one of the Dirac operator on $\Omega$ with the MIT bag boundary condition. The approximation, which is developed up to and error of order $o(1/\sqrt m)$, is carried out by introducing tubular coordinates in a neighborhood of $\partial\Omega$ and analyzing the corresponding one dimensional optimization problems in the normal direction.

math.AP↗

arXiv2018

Pointwise monotonicity of heat kernels

In this paper the authors present a proof of a pointwise radial monotonicity property of heat kernels that is shared by the euclidean spaces, spheres and hyperbolic spaces. The main result deals with monotonicity from special points on revolution hypersurfaces from which the aforementioned are deduced. The proof relies on a non straightforward but elementary application of the parabolic maximum principle.

math.CA↗

arXiv2018

Characterization of balls as minimizers of an endpoint Gagliardo seminorm on the boundary

Given a bounded $C^2$ domain $\Omega\subset{\mathbb R}^d$ with $d\geq3$, we prove a sharp inequality which relates the perimeter of ${\partial\Omega}$ to the endpoint Gagliardo seminorm in $W^{r,2}({\partial\Omega})$, corresponding to $r=0$, of the normal vector field on ${\partial\Omega}$. The proof of the inequality relies on the use of Bessel potentials and a monotonicity formula; we also show that balls are the unique minimizers. For $1/2<r<1$, the Gagliardo seminorm of the normal vector field on ${\partial\Omega}$ is related to a fractional second fundamental form which arises in the study of nonlocal perimeters and nonlocal minimal surfaces.

math.AP↗

arXiv2017

The relativistic spherical $\delta$-shell interaction in $\mathbb{R}^3$: spectrum and approximation

This note revolves on the free Dirac operator in $\mathbb{R}^3$ and its $\delta$-shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by finding sharp constants and minimizers of some precise inequalities related to an uncertainty principle. On the other hand, we prove that the domains given by Dittrich, Exner and \v{S}eba [Dirac operators with a spherically symmetric $\delta$-shell interaction, J. Math. Phys. 30.12 (1989), 2875-2882] and by Arrizabalaga, Mas and Vega [Shell interactions for Dirac operators, J. Math. Pures et Appl. 102.4 (2014), 617-639] for the realization of an electrostatic spherical shell interaction coincide. Finally, we explore the spectral relation between the shell interaction and its approximation by short range potentials with shrinking support, improving previous results in the spherical case.

math.AP↗

arXiv2016

Klein's Paradox and the Relativistic $\delta$-shell Interaction in $\mathbb{R}^3$

Under certain hypothesis of smallness of the regular potential $\mathbf{V}$, we prove that the Dirac operator in $\mathbb{R}^3$ coupled with a suitable re-scaling of $\mathbf{V}$ converges in the strong resolvent sense to the Hamiltonian coupled with a $\delta$-shell potential supported on $\Sigma$, a bounded $C^2$ surface. Nevertheless, the coupling constant depends non-linearly on the potential $\mathbf{V}$: the Klein's Paradox comes into play.

math.AP↗

arXiv2015

Dirac operators, shell interactions and discontinuous gauge functions across the boundary

Given a bounded smooth domain $\Omega\subset\mathbb{R}^3$, we explore the relation between couplings of the free Dirac operator $-i\alpha\cdot\nabla+m\beta$ with pure electrostatic shell potentials $\lambda\delta_{\partial\Omega}$ ($\lambda\in\mathbb{R}$) and some perturbations of those potentials given by the normal vector field $N$ on the shell $\partial\Omega$, namely $\{\lambda_e+\lambda_n(\alpha\cdot N)\}\delta_{\partial\Omega}$ ($\lambda_e$, $\lambda_n\in\mathbb{R}$). Under the appropiate change of parameters, the couplings with perturbed and unperturbed electrostatic shell potentials yield unitary equivalent self-adjoint operators. The proof relies on the construction of an explicit family of unitary operators that is well adapted to the study of shell interactions, and fits within the framework of gauge theory. A generalization of such unitary operators also allow us to deal with the self-adjointness of couplings of $-i\alpha\cdot\nabla+m\beta$ with some shell potentials of magnetic type, namely $\lambda(\alpha\cdot N)\delta_{\partial\Omega}$ with $\lambda\in C^1(\partial\Omega)$.

math-ph↗