Sharp Sobolev inequality involving a critical nonlinearity on a boundary

We consider the solvability of the Neumann problem for the equation -Delta u + lambda u = 0, partial derivative u/partial derivative v = Q(x)vertical bar u vertical bar(q-2)u on partial derivative Omega, where Q is a positive and continuous coefficient on partial derivative Omega, lambda is a parameter and q = 2(N - 1)/(N - 2) is a critical Sobolev exponent for the trace embedding of H-1(Omega) into L-q(partial derivative Omega). We investigate the joint effect of the mean curvature of partial derivative Omega and the shape of the graph of Q on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter lambda interferes with the spectrum of -Delta with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.

T-Q relation and exact solution for the XYZ chain with general non-diagonal boundary terms

We propose that the Baxter's Q-operator for the quantum XYZ spin chain with open boundary conditions is given by the j -> infinity limit of the corresponding transfer matrix with spin-j (i.e., (2j + I)-dimensional) auxiliary space. The associated T-Q relation is derived from the fusion hierarchy of the model. We use this relation to determine the Bethe Ansatz solution of the eigenvalues of the fundamental transfer matrix. The solution yields the complete spectrum of the Hamiltonian. (c) 2006 Elsevier B.V. All rights reserved.

The boundary lubricated friction and wear of low alloy steel

Pin on disc wear machines were used to study the boundary lubricated friction and wear of AISI 52100 steel sliding partners. Boundary conditions were obtained by using speed and load combinations which resulted in friction coefficients in excess of 0.1. Lubrication was achieved using zero, 15 and 1000 ppm concentrations of an organic dimeric acid additive in a hydrocarbon base stock. Experiments were performed for sliding speeds of 0.2, 0.35 and 0.5 m/s for a range of loads up to 220 N. Wear rate, frictional force and pin temperature were continually monitored throughout tests and where possible complementary methods of measurement were used to improve accuracy. A number of analytical techniques were used to examine wear surfaces, debris and lubricants, namely: Scanning Electron Microscopy (SEM), Auger Electron Spectroscopy (AES), Powder X-ray Diffraction (XRD), X-ray Photoelectron Spectroscopy (XPS), optical microscopy, Back scattered Electron Detection (BSED) and several metallographic techniques. Friction forces and wear rates were found to vary linearly with load for any given combination of speed and additive concentration. The additive itself was found to act as a surface oxidation inhibitor and as a lubricity enhancer, particularly in the case of the higher (1000 ppm) concentration. Wear was found to be due to a mild oxidational mechanism at low additive concentrations and a more severe metallic mechanism at higher concentrations with evidence of metallic delamination in the latter case. Scuffing loads were found to increase with increasing additive concentration and decrease with increasing speed as would be predicted by classical models of additive behaviour as an organo-metallic soap film. Heat flow considerations tended to suggest that surface temperature was not the overriding controlling factor in oxidational wear and a model is proposed which suggests oxygen concentration in the lubricant is the controlling factor in oxide growth and wear.

Determining the temperature from incomplete boundary data

An iterative procedure for determining temperature fields from Cauchy data given on a part of the boundary is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. A convergence proof of this method in a weighted L2-space is included, as well as a stopping criteria for the case of noisy data. Moreover, a solvability result in a weighted Sobolev space for a parabolic initial boundary value problem of second order with mixed boundary conditions is presented. Regularity of the solution is proved. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Stabilization of the Wave Equation by the Boundary

* Partially supported by CNPq (Brazil)

Asymptotic Property of Eigenvalues and Eigenfunctions of the Laplace Operator in Domain with a Perturbed Boundary

2000 Mathematics Subject Classification: 35J05, 35C15, 44P05

Anti-Periodic Boundary Value Problem for Impulsive Fractional Integro Differential Equations

MSC 2010: 34A37, 34B15, 26A33, 34C25, 34K37

Crack Theory with Singularities at the Boundary

2002 Mathematics Subject Classification: 35S15, 35J70, 35J40, 38J40

Two-phase flow analogy as an effective boundary condition for modelling liquids at atomistic resolution

A hybrid Molecular Dynamics/Fluctuating Hydrodynamics framework based on the analogy with two-phase hydrodynamics has been extended to dynamically tracking the feature of interest at all-atom resolution. In the model, the hydrodynamics description is used as an effective boundary condition to close the molecular dynamics solution without resorting to standard periodic boundary conditions. The approach is implemented in a popular Molecular Dynamics package GROMACS and results for two biomolecular systems are reported. A small peptide dialanine and a complete capsid of a virus porcine circovirus 2 in water are considered and shown to reproduce the structural and dynamic properties compared to those obtained in theory, purely atomistic simulations, and experiment.

Dynamic analysis of submerged microscale plates:the effects of acoustic radiation and viscous dissipation

The aim of this paper is to study the dynamic characteristics of micromechanical rectangular plates used as sensing elements in a viscous compressible fluid. A novel modelling procedure for the plate- fluid interaction problem is developed on the basis of linearized Navier-Stokes equations and noslip conditions. Analytical expression for the fluidloading impedance is obtained using a double Fourier transform approach. This modelling work provides us an analytical means to study the effects of inertial loading, acoustic radiation and viscous dissipation of the fluid acting on the vibration of microplates. The numerical simulation is conducted on microplates with different boundary conditions and fluids with different viscosities. The simulation results reveal that the acoustic radiation dominates the damping mechanism of the submerged microplates. It is also proved that microplates offer better sensitivities (Q-factors) than the conventional beam type microcantilevers beingmass sensing platforms in a viscous fluid environment. The frequency response features of microplates under highly viscous fluid loading are studied using the present model. The dynamics of the microplates with all edges clamped are less influenced by the highly viscous dissipation of the fluid than the microplates with other types of boundary conditions.

Graphene-based waveguide resonators for submillimeter-wave applications

Utilization of graphene covered waveguide inserts to form tunable waveguide resonators is theoretically explained and rigorously investigated by means of full-wave numerical electromagnetic simulations. Instead of using graphene-based switching elements, the concept we propose incorporates graphene sheets as parts of a resonator. Electrostatic tuning of the graphene surface conductivity leads to changes in the electromagnetic field boundary conditions at the resonator edges and surfaces, thus producing an effect similar to varying the electrical length of a resonator. The presented outline of the theoretical background serves to give phenomenological insight into the resonator behavior, but it can also be used to develop customized software tools for design and optimization of graphene-based resonators and filters. Due to the linear dependence of the imaginary part of the graphene surface impedance on frequency, the proposed concept was expected to become effective for frequencies above 100 GHz, which is confirmed by the numerical simulations. A frequency range from 100 GHz up to 1100 GHz, where the rectangular waveguides are used, is considered. Simple, all-graphene-based resonators are analyzed first, to assess the achievable tunability and to check the performance throughout the considered frequency range. Graphene–metal combined waveguide resonators are proposed in order to preserve the excellent quality factors typical for the type of waveguide discontinuities used. Dependence of resonator properties on key design parameters is studied in detail. Dependence of resonator properties throughout the frequency range of interest is studied using eight different waveguide sections appropriate for different frequency intervals. Proposed resonators are aimed at applications in the submillimeter-wave spectral region, serving as the compact tunable components for the design of bandpass filters and other devices.

Estudo teórico da tomografia por impedância elétrica

O presente trabalho descreve um estudo sobre a metodologia matemática para a solução do problema direto e inverso na Tomografia por Impedância Elétrica. Este estudo foi motivado pela necessidade de compreender o problema inverso e sua utilidade na formação de imagens por Tomografia por Impedância Elétrica. O entendimento deste estudo possibilitou constatar, através de equações e programas, a identificação das estruturas internas que constituem um corpo. Para isto, primeiramente, é preciso conhecer os potencias elétricos adquiridos nas fronteiras do corpo. Estes potenciais são adquiridos pela aplicação de uma corrente elétrica e resolvidos matematicamente pelo problema direto através da equação de Laplace. O Método dos Elementos Finitos em conjunção com as equações oriundas do eletromagnetismo é utilizado para resolver o problema direto. O software EIDORS, contudo, através dos conceitos de problema direto e inverso, reconstrói imagens de Tomografia por Impedância Elétrica que possibilitam visualizar e comparar diferentes métodos de resolução do problema inverso para reconstrução de estruturas internas. Os métodos de Tikhonov, Noser, Laplace, Hiperparamétrico e Variação Total foram utilizados para obter uma solução aproximada (regularizada) para o problema de identificação. Na Tomografia por Impedância Elétrica, com as condições de contorno preestabelecidas de corrente elétricas e regiões definidas, o método hiperparamétrico apresentou uma solução aproximada mais adequada para reconstrução da imagem.

Unfolding Operator Method for Thin Domains with a Locally Periodic Highly Oscillatory Boundary

We analyze the behavior of solutions of the Poisson equation with homogeneous Neumann boundary conditions in a two-dimensional thin domain which presents locally periodic oscillations at the boundary. The oscillations are such that both the amplitude and period of the oscillations may vary in space. We obtain the homogenized limit problem and a corrector result by extending the unfolding operator method to the case of locally periodic media.

Homogenization of the p-Laplacian with nonlinear boundary condition on critical size particles: Identifying the strange term for the some non smooth and multivalued operators

We extend previous papers in the literature concerning the homogenization of Robin type boundary conditions for quasilinear equations, in the case of microscopic obstacles of critical size: here we consider nonlinear boundary conditions involving some maximal monotone graphs which may correspond to discontinuous or non-Lipschitz functions arising in some catalysis problems.

Second-order elliptic PDE with discontinuous boundary data

We shall consider the weak formulation of a linear elliptic model problem with discontinuous Dirichlet boundary conditions. Since such problems are typically not well-defined in the standard H^1-H^1 setting, we will introduce a suitable saddle point formulation in terms of weighted Sobolev spaces. Furthermore, we will discuss the numerical solution of such problems. Specifically, we employ an hp-discontinuous Galerkin method and derive an L^2-norm a posteriori error estimate. Numerical experiments demonstrate the effectiveness of the proposed error indicator in both the h- and hp-version setting. Indeed, in the latter case exponential convergence of the error is attained as the mesh is adaptively refined.