A priori-Schranken für diskrete Lösungen monotoner Operatorgleichungen und Anwendungen

Summary Operator equationsTu=f are approximated by Galerkin's method, whereT is a monotone operator in the sense of Browder and Minty, so that existence results are available in a reflexive Banach spaceX. In a normed spaceY error estimates are established, which require a priori bounds for the discrete solutionsu _h in the norm of a suitable space . Sufficient conditions for the uniform boundedness u _{h Z} =O(1) ash0 are proved. Well-known error estimates in [3] for the special caseX=Y=Z are generalized by this. The theory is applied to quasilinear elliptic boundary value problems of order 2m in a bounded domain . The approximating subspaces are finite element spaces. Especially the caseX=W ₀ ^{m, p} (), 1<p<,Y=W ₀ ^{m. 2} (),Z=W ₀ ^{m. max (2,p)} ()W^m, () is treated. Some examples for 1<p<2 are considered. Forp2 a refined technique is introduced in the author's paper [7].

     

Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant

Summary This work is concerned with the existence and uniqueness of a class of semimartingale reflecting Brownian motions which live in the non-negative orthant of ^d . Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the orthant the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the (d-1)-dimensional faces that form the boundary of the orthant, the bounded variation part of the process increases in a given direction (constant for any particular face) so as to confine the process to the orthant. For historical reasons, this pushing at the boundary is called instantaneous reflection. In 1988, Reiman and Williams proved that a necessary condition for the existence of such a semimartingale reflecting Brownian motion (SRBM) is that the reflection matrix formed by the directions of reflection be completely-L. In this work we prove that condition is sufficient for the existence of an SRBM and that the SRBM is unique in law. It follows from the uniqueness that an SRBM defines a strong Markov process. Our results have potential application to the study of diffusions arising as approximations tomulti-class queueing networks.Research supported in part by NSF Grants DMS 8657483, 8722351 and 9023335, and a grant from AT&T Bell Labs. In addition, R.J. Williams was supported in part during the period of this research by an Alfred P. Sloan Research Fellowship     

On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations

Summary This paper is a continuation of our previous work [10] on projection methods. We study first existing higher order projection schemes in the semidiscretized form for the Navier-Stokes equations. One error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent Neumann boundary condition satisfied by the pressure approximations. We then propose a penalty-projection scheme for which we obtain improved error estimates.This work is partially supported by NSF grant MS-8802596.     

Chern forms,chern numbers and curvature conditions on a Kähler-Einstein surface

In this paper we considered curvature conditions on a Kähler-Einstein surface of general type. In particular we showed that it has negative holomorphic sectional curvature if theL ²-norm of (3C ₂ ?C ₁ ² )/C ₁ ² is sufficiently small, whereC ₁ andC ₂ are the first and second Chern classes of the surfaces. This generalizes a result of Yau on the uniformization of Kähler-Einstein surfaces of general type and with 3C ₂ ?C ₁ ² = 0. Also in the process, we obtain a necessary condition in terms of an inequality between Chern numbers for a Kähler-Einstein metric to have negative holomorphic sectional curvature.     

Improved biological performance of low modulus Ti-24Nb-4Zr-7.9Sn implants due to surface modification by anodic oxidation

Dental implants are usually made from commercially pure titanium or titanium alloys. The purpose of this study was to evaluate the influence of surface treatment to low modulus Ti-24Nb-4Zr-7.9Sn (TNZS) on cell and bone responses. The TNZS alloy samples were modified using anodic oxidation (AD). Surface oxide properties were characterized by using various surface analytic techniques, involving scanning electron microscopy (SEM) equipped with energy dispersive spectrometer (EDS), X-ray diffractometry (XRD) and surface profilometer. During the AD treatment, porous titanium oxide layer was formed and Ca ions were incorporated into the oxide layer. The viability and morphology of osteoblasts on Ca-incorporated TNZS were studied. The bone responses of Ca-incorporated TNZS were evaluated by pull-out tests and morphological analysis after implantation in rabbit tibiae. The non-treated Ti and TNZS samples were used as the control. Significant increases in cell viability and pull-out forces (p < 0.05) were observed for Ca-incorporated TNZS implants compared with those for the control groups. Porous structures supplied positive guidance cues for osteoblasts to attach. The enhanced cell and bone responses to Ca-incorporated TNZS implants could be explained by the surface chemistry and microtopography.     

Limitations and guidelines for measuring the spectral width of ultrashort light pulses with a scanning Fabry–Pérot interferometer

The measurement of the spectral width of ultrashort light pulses using a Fabry–Pérot interferometer (FPI) is investigated. It is shown, numerically and experimentally, that the measured width critically depends on the pulse properties (such as pulse shape, pulse duration, frequency chirp and wavelength) and on the properties of the FPI (such as the mirror spacing and the mirror reflectivities). The obtained results are of particular importance if the spatial length of the short light pulses is comparable or even shorter than the distance between the FPI mirrors. The derived guideline indicate that the actual spectral width of the ultrashort light pulses is measured with good accuracy only if the finesse F≥40 and the round trip time of the light pulses inside the Fabry–Pérot interferometer is approximately one to three times the pulse duration. Received: 17 December 1999 / Revised version: 14 April 2000 / Published online: 5 July 2000     

Gd-doping of HfO2

An increase in the density of states between the oxygen 2p bands and the Fermi level is seen with increasing Gd concentrations. In addition, for the Gd-doped HfO₂ films, the Gd 4f photoexcitation peak at 5.5 eV below the valence band maximum was identified using resonant photoemission. Electrical measurements show pronounced rectification properties for lightly-doped Gd:HfO₂ films on p-Si and for heavily-doped Gd:HfO₂ films on n-Si, suggesting a crossover from n-type to p-type behavior with increasing doping level. In addition, there is an increase in the reverse bias current with neutron irradiation.     

Dirichlet norms,capacities and generalized isoperimetric inequalities for Markov operators

We define the notion of p-capacity for a reversible Markov operator on a general measure space and prove that uniform estimates for the ratio of capacity and measure are equivalent to certain imbedding theorems for the Orlicz and Dirichlet norms. As a corollary we get results on connections between embedding theorems and isoperimetric properties for general Markov operators and, particularly, a generalization of the Kesten theorem on the spectral radius of random walks on amenable groups for the case of arbitrary graphs with non-finitely supported transition probabilities.     

Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in R^m,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0. R. H.’s research partially supported by the National Science Foundation.     

Convergence to equilibrium of critical branching particle systems and superprocesses,and related nonlinear partial differential equations

We consider a class of multitype particle systems in ^d undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).