Measures on double or resonant eigenvalues for linear Schrödinger operator

In this paper we consider linear Schrödinger operator with double or resonant eigenvalues. The main result is the bound of the measure (in a suitable space of functions) of the potentials leading to such double or resonant eigenvalues. Namely we present measure type estimates evaluating neighborhoods of the so-called double or resonant set.     

A posteriori error estimates by recovered gradients in parabolic finite element equations

This paper considers a posteriori error estimates by averaged gradients in second order parabolic problems. Fully discrete schemes are treated. The theory from the elliptic case as to when such estimates are asymptotically exact, on an element, is carried over to the error on an element at a given time. The basic principle is that the elliptic theory can be extended to the parabolic problems provided the time-step error is smaller than the space-discretization error. Numerical illustrations confirming the theoretical results are given. Our results are not practical in the sense that various constants can not be estimated realistically. They are conceptual in nature. AMS subject classification (2000)  65M60, 65M20, 65M15     

Spectral radius, index estimates for Schrödinger operators and geometric applications

In this paper we study the existence of a first zero and the oscillatory behavior of solutions of the ordinary differential equation ^′(vz^′)+Avz=0, where A, v are functions arising from geometry. In particular, we introduce a new technique to estimate the distance between two consecutive zeros. These results are applied in the setting of complete Riemannian manifolds: in particular, we prove index bounds for certain Schrödinger operators, and an estimate of the growth of the spectral radius of the Laplacian outside compact sets when the volume growth is faster than exponential. Applications to the geometry of complete minimal hypersurfaces of Euclidean space, to minimal surfaces and to the Yamabe problem are discussed.     

一般非线性渗流方程正性的扩展

本文研究一般非线性渗流方程解的一种性态.利用De Giorgi方法与BCP估计相结合,得到了一般非线性渗流方程正性的扩展性质,推广了非线性渗流方程的结果.     

On the spatial blow-up and decay for some nonlinear parabolic equations with nonlinear boundary conditions

In this note we investigate the spatial behavior of several nonlinear parabolic equations with nonlinear boundary conditions. Under suitable conditions on the nonlinear terms we prove that the solutions either cease to exist for a finite value of the spatial variable or else they decay algebraically. The main tool used is the weighted energy method. Our results can be applied to several situations concerning heat conduction. Received: April 4, 2004; revised: September 20, 2004     

A Phragmèn-Lindelöf principle for the thermoelastic cylinder of variable cross-section

We consider a non-prismatic cylinder consisting of an anisotropic, compressible, inhomogeneous thermoelastic material, subject to either null tractions or null displacements on the lateral boundary and loaded by a self-equilibrated force system at one end. We show that the mean square cross-sectional measures of the temperature and the coupled displacement—temperature either grow faster than an exponentially increasing function of axial distance or decrease faster than a decaying exponential function of axial distance. We thus establish a Phragmèn-Lindelöf principle for the thermoelastic cylinder of variable cross-section.

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Sommario Viene considerato un cilindro non-prismatico costituito da un materiale termoelastico, anisotropo, comprimibile ed omogeneo. Il cilindro é soggetto a zero trazioni o zero dislocamenti sul limite laterale ed é caricato su uno dei termini con un sistema autoequilibrilata delle forze. Mostriamo che il valore medio quadratico della temperatura e dell'accoppiata della temperatura ed il dislocamento cresce più veloce che una funzione esponenziale crescente della distanza assiale o diminuisce più veloce che una funzione esponenziale declinante della distanza assiale. Viene stabilito un principio di Phragmèn-Lindelöf per il cilindro termoelastico di sezione trasversale variabile.

     

Quelques modèles diffusifs capillaires de type KortewegSome diffusive capillary models of Korteweg type.

The purpose of this Note is to propose new diffusive capillary models of Korteweg type and discuss their mathematical properties. More precisely, we introduce viscous models which provide some additional information on the behavior of the density close to vacuum. We actually prove that if some compatibility conditions between diffusion and capillarity are satisfied, some extra regularity information on a quantity involving the density is available. We obtain a non-trivial equality deduced from the special structure of the momentum equation. This Note generalizes to some extent the authors' previous works on the Korteweg model (with constant capillary coefficient) and on the shallow water equation. To cite this article: D. Bresch, B. Desjardins, C. R. Mecanique 332 (2004).     

Detection of bio-constituents in complex biological tissue using Raman microscopy. Application to human nail clippings

Raman spectra of human nail clippings from various sources were collected and then deconvoluted to obtain the pure component spectra of the underlying constituents present. This blind-deconvolution was performed using a self-modeling curve resolution technique, namely band-target entropy minimization (BTEM). The aim was to simplify the complexity of the Raman spectra and hence to identify the underlying biological molecules in more detail. BTEM analysis could recover 13 pure component Raman spectral estimates from the collected 438 spectra measured from 113 nail samples. Six recovered pure component spectral estimates correspond to proteins or polypeptides that contain various amino acids such as phenylalanine, tyrosine, tryptophan, and cysteine. Two are associated with the secondary structures of proteins, and five are associated with two carotenoid species, lipid, ferulic acid, and calcium phosphate. Subsequently, the relative concentrations of these bio-constituents were calculated from the measured mixture spectra and the pure component BTEM estimates. These profiles indicated that the concentrations of some bio-constituents are correlated while others are not. A further analysis using target transformation factor analysis (TTFA) revealed the possible presence of curcumin in the human nails. Since the present approach and analysis is rather general, it might be extended to many other biological tissues in a rather straightforward and similar manner, thus revealing more detailed underlying biochemical information such as biomarkers that may be useful for diagnostic purposes.     

Efficient and stable perfectly matched layer for CEM

An efficient unsplit perfectly matched layer for numerical simulation of electromagnetic waves in unbounded domains is derived via a complex change of variables. In order to surround a Cartesian grid with the PML, the time-dependent PML requires only one (scalar) auxiliary variable in two space dimensions and six (scalar) auxiliary variables in three space dimensions. It is therefore cheap and straightforward to implement. We use Fourier and energy methods to prove the stability of the PML. We extend the stability result to a semi-discrete PML approximated by central finite differences of arbitrary order of accuracy and to a fully discrete problem for the ‘Leap-Frog’ schemes. This makes precise the usefulness of the derived PML model for longtime simulations. Numerical experiments are presented, illustrating the accuracy and stability of the PML.     

Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.