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Nicolas Ginoux

Publications and source records attributed to Nicolas Ginoux.

At least 19 recordsLinked to original sources
arXiv2025

On the spectrum of the magnetic Dirac operator

The magnetic Dirac operator describes the relativistic motion of a charged particle in a magnetic field. Although this operator got a lot of attention in physics many of its fundamental mathematical properties remain unexplored and this article is a first step towards filling this gap. To this end we provide a number of eigenvalue estimates for the magnetic Dirac operator on closed Riemannian manifolds and explicitly compute its spectrum for specific choices of the magnetic field on the flat torus and on the three-dimensional round sphere.

math.DG↗
arXiv2024

The Buckling and Clamped Plate Problems on Differential Forms

We extend the buckling and clamped-plate problems to the context of differential forms on compact Riemannian manifolds with smooth boundary. We characterize their smallest eigenvalues and prove that, in the case of bounded Euclidean domains, their spectra without multiplicities on forms coincide with the spectra of the corresponding problems on functions. We obtain various estimates involving the first eigenvalues of the mentioned problems and the ones of the Hodge Laplacian with respect to Dirichlet and absolute boundary conditions on forms. These estimates generalize previous ones in the case of functions.

math.DG↗
arXiv2024

On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary

In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are shown. Furthermore, if the Friedrichs system is hyperbolic, the Cauchy problem is proved to be well-posed in the sense of Hadamard. Finally, examples of Friedrichs systems with admissible boundary conditions are provided. Keywords: symmetric hyperbolic systems, symmetric positive systems, admissible boundary conditions, Dirac operator, normally hyperbolic operator, Klein-Gordon operator, heat operator, reaction-diffusion operator, globally hyperbolic manifolds with timelike boundary.

math.AP↗
arXiv2023

A Poincaré formula for differential forms and applications

We prove a new general Poincaré-type inequality for differential forms on compact Riemannian manifolds with nonempty boundary. When the boundary is isometrically immersed in Euclidean space, we derive a new inequality involving mean and scalar curvatures of the boundary only and characterize its limiting case in codimension one. A new Ros-type inequality for differential forms is also derived assuming the existence of a nonzero parallel form on the manifold.

math.DG↗
arXiv2022

Biharmonic Steklov operator on differential forms

We introduce the biharmonic Steklov problem on differential forms by considering suitable boundary conditions. We characterize its smallest eigenvalue and prove elementary properties of the spectrum. We obtain various estimates for the first eigenvalue, some of which involve eigenvalues of other problems such as the Dirichlet, Neumann, Robin and Steklov ones. Independently, new inequalities relating the eigenvalues of the latter problems are proved.

math.DG↗
arXiv2021

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

The aim of this paper is to prove the existence of Hadamard states for Dirac fields coupled with MIT boundary conditions on any globally hyperbolic manifold with timelike boundary. This is achieved by introducing a geometric Møller operator which implements a unitary isomorphism between the spaces of $L^2$ -initial data of particular symmetric systems we call weakly-hyperbolic and which are coupled with admissible boundary conditions. In particular, we show that for Dirac fields with MIT boundary conditions, this isomorphism can be lifted to a $*$-isomorphism between the algebras of Dirac fields and that any Hadamard state can be pulled back along this $*$-isomorphism preserving the singular structure of its two-point distribution.

math-ph↗
arXiv2020

Skew Killing spinors in four dimensions

This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor $ψ$ is a spinor that satisfies the equation $\nabla$X$ψ$ = AX $\times$ $ψ$ with a skew-symmetric endomorphism A. We consider the degenerate case, where the rank of A is at most two everywhere and the non-degenerate case, where the rank of A is four everywhere. We prove that in the degenerate case the manifold is locally isometric to the Riemannian product R x N with N having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to S 2 x R 2 occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe.

math.DG↗
arXiv2020

An Obata-type characterization of doubly-warped product K\''ahler manifolds

We give a characterization {\sl à la Obata} for certain families of K\''ahler manifolds. These results are in the same line as other extensions of the well-known Obata's rigidity theorem from \cite{Obata62}, like for instance the generalizations in \cite{RanjSant97} and \cite{Santhanam07}. Moreover, we give a complete description of the so-called K\''ahler doubly-warped product structures whose underlying metric is Einstein.

math.DG↗
arXiv2019

New eigenvalue estimates involving Bessel functions

Given a compact Riemannian manifold (M n , g) with boundary $\partial$M , we give an estimate for the quotient $\partial$M f d$μ$ g M f d$μ$ g , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [37], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a direct proof is given of the Faber-Krahn inequalities for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Independently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.

math.DG↗
arXiv2018

Some examples of Dirac-harmonic maps

We discuss a method to construct Dirac-harmonic maps developed by J.~Jost, X.~Mo and M.~Zhu in J.~Jost, X.~Mo, M.~Zhu, \emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. \textbf{59} (2009), no. 11, 1512--1527.The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmonic maps of codimension $1$ with target spaces of constant sectional curvature.Before the present article, it remained unclear when the conditions of the theorems in J.~Jost, X.~Mo, M.~Zhu, \emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. \textbf{59} (2009), no. 11, 1512--1527, were fulfilled. We show that for isometric immersions into spaceforms, these conditions are fulfilled only under special assumptions.In several cases we show the existence of solutions.

math.AP↗
arXiv2018

Tire Noise Optimization Problem: a Mixed Integer Linear Program Approach

We present a Mixed Integer Linear Program (MILP) approach in order to model the nonlinear problem of minimizing the tire noise. We first take more industrial constraints into account than in a former work of the authors. Then, we associate a Branch-and-Cut algorithm to the MILP to obtain exact solutions. We compare our experimental results with those obtained by other methods.

cs.DS↗
arXiv2016

Rigidity results for Riemannian spin^c manifolds with foliated boundary

Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O'Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kähler-Einstein manifold or a Riemannian product of a Kähler-Einstein manifold with R (or with the circle S^1).

math.DG↗
arXiv2016

Global solvability of massless Dirac-Maxwell systems

We consider the Cauchy problem of massless Dirac-Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be extended via analogous methods to Dirac-Higgs-Yang-Mills theories.

math.AP↗
arXiv2015

Dirac-harmonic maps from index theory

We prove existence results for Dirac-harmonic maps using index theoretical tools. They are mainly interesting if the source manifold has dimension 1 or 2 modulo 8. Our solutions are uncoupled in the sense that the underlying map between the source and target manifolds is a harmonic map.

math.DG↗