arXiv subjects

Monique Dauge

Publications and source records attributed to Monique Dauge.

At least 19 recordsLinked to original sources

arXiv2026

End-point non-regularity of the Dirichlet problem on Lipschitz domains -- An elementary proof

We construct a plane Lipschitz domain for which the range of the Laplace operator from $H^{3/2}\cap H^1_0$ to $H^{-1/2}$ is not closed. We show that rapid oscillations of the boundary create a boundary layer that leads to unbounded $H^{3/2}$ norm of the solution of the Dirichlet problem. Our proof is elementary in the sense that it is directly based on homogeneity arguments and does not use deep results of harmonic analysis such as estimates of area integrals or harmonic measures of Dahlberg or Jerison and Kenig.

math.AP↗

arXiv2025

Estimates for harmonic functions near pseudo-corners

It is well known that derivatives of solutions to elliptic boundary value problems may become unbounded near the corner of a domain with a conical singularity, even if the data are smooth. When the corner domain is approximated by more regular domains, then higher order Sobolev norms of the solutions on these domains can blow up in the limit. We study this blow-up in the simple example of the Laplace operator with Dirichlet conditions in two situations: The rounding of a corner in any dimension, and the two-dimensional situation where a polygonal corner is replaced by two or more corners with smaller angles. We show how an inner expansion derived from a more general recent result on converging expansions into generalized power series can be employed to prove simple and explicit estimations for Sobolev norms and singularity coefficients by powers of the approximation scale.

math.AP↗

arXiv2025

Numerical simulation of Lugiato-Lefever equation for Kerr combs generation in Fabry-Perot resonators

Lugiato-Lefever equation (LLE) is a nonlinear Schrödinger equation with damping, detuning and driving terms, introduced as a model for Kerr combs generation in ring-shape resonators and more recently, in the form of a variant, in Fabry-Perot (FP) resonators. The aim of this paper is to present some numerical methods that complement each other to solve the LLE in its general form both in the dynamic and in the steady state regimes. We also provide some mathematical properties of the LLE likely to help the understanding and interpretation of the numerical simulation results.

physics.optics↗

arXiv2024

Dirichlet problem on perturbed conical domains via converging generalized power series

We consider the Poisson equation with homogeneous Dirichlet conditions in a family of domains in $R^{n}$ indexed by a small parameter $\epsilon$. The domains depend on $\epsilon$ only within a ball of radius proportional to $\epsilon$ and, as $\epsilon$ tends to zero, they converge in a self-similar way to a domain with a conical boundary singularity. We construct an expansion of the solution as a series of fractional powers of $\epsilon$, and prove that it is not just an asymptotic expansion as $\epsilon\to0$, but that, for small values of $\epsilon$, it converges normally in the Sobolev space $H^{1}$. The phenomenon that solutions to boundary value problems on singularly perturbed domains may have convergent expansions is the subject of the Functional Analytic Approach by Lanza de Cristoforis and his collaborators. This approach was originally adopted to study small holes shrinking to interior points of a smooth domain and heavily relies on integral representations obtained through layer potentials. To relax all regularity assumptions, we forgo boundary layer potentials and instead exploit expansions in terms of eigenfunctions of the Laplace-Beltrami operator on the intersection of the cone with the unit sphere. Our analysis is based on a two-scale cross-cutoff ansatz for the solution. Specifically, we write the solution as a sum of a function in the slow variable multiplied by a cutoff function depending on the fast variable, plus a function in the fast variable multiplied by a cutoff function depending on the slow variable. While the cutoffs are considered fixed, the two unknown functions are solutions to a $2\times2$ system of partial differential equations that depend on $\epsilon$ in a way that can be analyzed in the framework of generalized power series when the right-hand side of the Poisson equation vanishes in a neighborhood of the perturbation.

math.AP↗

arXiv2023

Stability Analysis of a Simple Discretization Method for a Class of Strongly Singular Integral Equations

Motivated by the discrete dipole approximation (DDA) for the scattering of electromagnetic waves by a dielectric obstacle that can be considered as a simple discretization of a Lippmann-Schwinger style volume integral equation for time-harmonic Maxwell equations, we analyze an analogous discretization of convolution operators with strongly singular kernels. For a class of kernel functions that includes the finite Hilbert transformation in 1D and the principal part of the Maxwell volume integral operator used for DDA in dimensions 2 and 3, we show that the method, which does not fit into known frameworks of projection methods, can nevertheless be considered as a finite section method for an infinite block Toeplitz matrix. The symbol of this matrix is given by a Fourier series that does not converge absolutely. We use Ewald's method to obtain an exponentially fast convergent series representation of this symbol and show that it is a bounded function, thereby allowing to describe the spectrum and the numerical range of the matrix. It turns out that this numerical range includes the numerical range of the integral operator, but that it is in some cases strictly larger. In these cases the discretization method does not provide a spectrally correct approximation, and while it is stable for a large range of the spectral parameter $\lambda$, there are values of $\lambda$ for which the singular integral equation is well posed, but the discretization method is unstable.

math.NA↗

arXiv2023

Characterization of Sobolev spaces by their Fourier coefficients in axisymmetric domains

Using Fourier series representations of functions on axisymmetric domains, we find weighted Sobolev norms of the Fourier coefficients of a function that yield norms equivalent to the standard Sobolev norms of the function. This characterization is universal in the sense that the equivalence constants are independent of the domain. In particular it is uniform whether the domain contains a part of its axis of rotation or is disjoint from, but maybe arbitrarily close to, the axis. Our characterization using step-weighted norms involving the distance to the axis is different from the one obtained earlier in the book [Bernardi, Dauge, Maday "Spectral methods for axisymmetric domains", Gauthier-Villars, 1999], which involves trace conditions and is domain dependent. We also provide a complement for non cylindrical domains of the proof given in loc. cit. .

math.FA↗

arXiv2022

Asymptotic Lattices, Good Labellings, and the Rotation Number for Quantum Integrable Systems

This article introduces the notion of good labellings for asymptotic lattices in order to study joint spectra of quantum integrable systems from the point of view of inverse spectral theory. As an application, we consider a new spectral quantity for a quantum integrable system, the quantum rotation number. In the case of two degrees of freedom, we obtain a constructive algorithm for the detection of appropriate labellings for joint eigenvalues, which we use to prove that, in the semiclassical limit, the quantum rotation number can be calculated on a joint spectrum in a robust way, and converges to the well-known classical rotation number. The general results are applied to the semitoric case where formulas become particularly natural.

math.SP↗

arXiv2022

Corner asymptotics of the magnetic potential in the eddy-current model

In this paper, we describe the magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi-dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific non-standard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials and further terms are genuine non-smooth functions generated by the piecewise constant zeroth order term of the operator.

math.NA↗

arXiv2022

On the influence of the geometry on skin effect in electromagnetism

We consider the equations of electromagnetism set on a domain made of a dielectric and a conductor subdomain in a regime where the conductivity is large. Assuming smoothness for the dielectric--conductor interface, relying on recent works we prove that the solution of the Maxwell equations admits a multiscale asymptotic expansion with profile terms rapidly decaying inside the conductor. This skin effect is measured by introducing a skin depth function that turns out to depend on the mean curvature of the boundary of the conductor. We then confirm these asymptotic results by numerical experiments in various axisymmetric configurations. We also investigate numerically the case of a nonsmooth interface, namely a cylindrical conductor.

math.AP↗

arXiv2021

Asymptotics for 2D whispering gallery modes in optical micro-disks with radially varying index

Whispering gallery modes [WGM] are resonant modes displaying special features: They concentrate along the boundary of the optical cavity at high polar frequencies and they are associated with complex scattering resonances very close to the real axis. As a classical simplification of the full Maxwell system, we consider two-dimensional Helmholtz equations governing transverse electric [TE] or magnetic [TM] modes. Even in this 2D framework, very few results provide asymptotic expansion of WGM resonances at high polar frequency $m\to\infty$ for cavities with radially varying optical index. In this work, using a direct Schrödinger analogy we highlight three typical behaviors in such optical micro-disks, depending on the sign of an effective curvature that takes into account the radius of the disk and the values of the optical index and its derivative. Accordingly, this corresponds to abruptly varying effective potentials (step linear or step harmonic) or more classical harmonic potentials, leading to three distinct asymptotic expansions for ground state energies. Using multiscale expansions, we design a unified procedure to construct families of quasi-resonances and associate quasi-modes that have the WGM structure and satisfy eigenequations modulo a super-algebraically small residual $O(m^{-\infty})$. We show using the black box scattering approach that quasi-resonances are $O(m^{-\infty})$ close to true resonances.

math.SP↗

arXiv2020

Trace Hardy inequality for the Euclidean space with a cut and its applications

We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $Σ\subset\mathbb R^d$, $d \ge 2$. In this novel geometric setting, the Hardy-type inequality non-typically holds also for $d = 2$. The respective Hardy weight is given in terms of the geodesic distance to the boundary of $Σ$. We provide its applications to the heat equation on $\mathbb R^d$ with an insulating cut at $Σ$ and to the Schrödinger operator with a $δ'$-interaction supported on $Σ$. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.

math.AP↗

arXiv2018

Maxwell eigenmodes in product domains

This paper is devoted to Maxwell modes in three-dimensional bounded electromagnetic cavities that have the form of a product of lower dimensional domains in some system of coordinates. The boundary conditions are those of the perfectly conducting or perfectly insulating body. The main case of interest is products in Cartesian variables. Cylindrical and spherical variables are also addressed. We exhibit common structures of polarization type for eigenmodes. In the Cartesian case, the cavity eigenvalues can be obtained as sums of Dirich-let or Neumann eigenvalues of positive Laplace operators and the corresponding eigenvectors have a tensor product form. We compare these descriptions with the spherical wave function Ansatz for a ball and show why the cavity eigenvalue of the ball are also Dirichlet or Neumann eigenvalues of some scalar operators. As application of our general formulas, we find explicit eigenpairs in a cuboid, in a circular cylinder, and in a cylinder with a coaxial circular hole. This latter example exhibit interesting '' TEM '' eigenmodes that have a one-dimensional vibrating string structure, and contribute to the least energy modes if the cylinder is long enough.

math-ph↗

arXiv2018

On the semiclassical Laplacian with magnetic field having self-intersecting zero set

This paper is devoted to the spectral analysis of the Neumann realization of the 2D magnetic Laplacian with semiclassical parameter h > 0 in the case when the magnetic field vanishes along a smooth curve which crosses itself inside a bounded domain. We investigate the behavior of its eigenpairs in the limit h $\rightarrow$ 0. We show that each crossing point acts as a potential well, generating a new decay scale of h 3/2 for the lowest eigenvalues, as well as exponential concentration for eigenvectors around the set of crossing points. These properties are consequences of the nature of associated model problems in R 2 for which the zero set of the magnetic field is the union of two straight lines. In this paper we also analyze the spectrum of model problems when the angle between the two straight lines tends to 0.

math.AP↗

arXiv2018

Dirichlet spectrum of the Fichera layer

We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a prototypal infinite polyhedral layer, that is formed by three perpendicular quarter-plane walls of constant width joining each other. Alternatively, this domain can be viewed as an octant from which another "parallel" octant is removed. It contains six edges (three convex and three non-convex) and two corners (one convex and one non-convex). It is a canonical example of non-smooth conical layer. We name it after Fichera because near its non-convex corner, it coincides with the famous Fichera cube that illustrates the interaction between edge and corner singularities. This domain could also be called an octant layer. We show that the essential spectrum of the Laplacian on such a domain is a half-line and we characterize its minimum as the first eigenvalue of the two-dimensional Laplacian on a broken guide. By a Born-Oppenheimer type strategy, we also prove that its discrete spectrum is finite and that a lower bound is given by the ground state of a special Sturm-Liouville operator. By finite element computations taking singularities into account, we exhibit exactly one eigenvalue under the essential spectrum threshold leaving a relative gap of 3%. We extend these results to a variant of the Fichera layer with rounded edges (for which we find a very small relative gap of 0.5%), and to a three-dimensional cross where the three walls are full thickened planes.

math.SP↗

arXiv2017

Converging expansions for Lipschitz self-similar perforations of a plane sector

In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the " functional analytic approach " of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size $ε$ for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as $ε$ tends to zero is described not only by asymptotic series in powers of $ε$, but by convergent power series. Here we use this method to investigate the Dirichlet problem for the Laplace operator where holes are collapsing at a polygonal corner of opening $ω$. Then in addition to the scale $ε$ there appears the scale $η= ε^{π/ω}$. We prove that when $π/ω$ is irrational, the solution of the Dirichlet problem is given by convergent series in powers of these two small parameters. Due to interference of the two scales, this convergence is obtained, in full generality, by grouping together integer powers of the two scales that are very close to each other. Nevertheless, there exists a dense subset of openings $ω$ (characterized by Diophantine approximation properties), for which real analyticity in the two variables $ε$ and $η$ holds and the power series converge unconditionally. When $π/ω$ is rational, the series are unconditionally convergent, but contain terms in log $ε$.

math.AP↗

arXiv2016

Magnetic Laplacian in sharp three dimensional cones

The core result of this paper is an upper bound for the ground state energyof the magnetic Laplacian with constant magnetic field on cones that are contained in ahalf-space. This bound involves a weighted norm of the magnetic field related to momentson a plane section of the cone. When the cone is sharp, i.e. when its section is small, thisupper bound tends to 0. A lower bound on the essential spectrum is proved for familiesof sharp cones, implying that if the section is small enough the ground state energy is aneigenvalue. This circumstance produces corner concentration in the semi-classical limit forthe magnetic Schrödinger operator when such sharp cones are involved.

math.SP↗

arXiv2016

High frequency oscillations of first eigenmodes in axisymmetric shells as the thickness tends to zero

The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2$\epsilon$) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue $\lambda$($\epsilon$) and the eigenvector angular frequency k($\epsilon$) for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on the other parts. First, the scalar Laplace operator for acoustics is addressed, for which k($\epsilon$) is always zero. In contrast to it, for the Lam{\'e} system of linear elasticity several different types of shells are defined, characterized by their geometry, for which k($\epsilon$) tends to infinity as $\epsilon$ tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide $\lambda$($\epsilon$) and k($\epsilon$) and demonstrate by numerical examples the different behavior as $\epsilon$ tends to zero.

math.NA↗

arXiv2016

Free Vibrations of Axisymmetric Shells: Parabolic and Elliptic cases

Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lam{\'e} system) are determined by an asymptotic analysis as the thickness ($2\varepsilon$) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar modelthat depends on two parameters: the angular frequency $k$ and the half-thickness $\varepsilon$. Optimizing $k$ for each chosen $\varepsilon$, we find power laws for $k$ in function of $\varepsilon$ that provide the smallest eigenvalues of the scalar reductions.Corresponding eigenpairs generate quasimodes for the 3D Lam{\'e} system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lam{\'e} system.Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as $\varepsilon$ tends to $0$. Its angular frequency exhibits a power law relationof the form $k=\gamma \varepsilon^{-\beta}$ with $\beta=\frac14$ in the parabolic case (cylinders and trimmed cones), and the various $\beta$s $\frac25$, $\frac37$, and $\frac13$ in the elliptic case.For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented.

math.NA↗