# Stability of infinite dimensional control problems with pointwise state constraints

*Authors*

- Hinze, Michael
- Meyer, Christian

*2010 Mathematics Subject Classification*

- 49K20 49N10 49M20

*Keywords*

- Optimal control of semi-linear elliptic equations, pointwise state constraints, finite element approximation

*DOI*

*Abstract*

A general class of nonlinear infinite dimensional optimization problems is considered that covers semi-linear elliptic control problems with distributed control as well as boundary control. Moreover, pointwise inequality constraints on the control and the state are incorporated. The general optimization problem is perturbed by a certain class of perturbations, and we establish convergence of local solutions of the perturbed problems to a local solution of the unperturbed optimal control problem. These class of perturbations include finite element discretization as well as data perturbation such that the theory implies convergence of finite element approximation and stability w.r.t. noisy data.

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# A mathematical model for case hardening of steel

*Authors*

- Fasano, Antonio
- Hömberg, Dietmar
- Panizzi, Lucia

*2010 Mathematics Subject Classification*

- 35K60 35R05 82B26

*Keywords*

- Heat treatment, phase transitions, coupled PDE, case hardening

*DOI*

*Abstract*

A mathematical model for the case hardening of steel is presented. Carbon is dissolved in the surface layer of a low-carbon steel part at a temperature sufficient to render the steel austenitic, followed by quenching to form a martensitic microstructure. The model consists of a nonlinear evolution equation for the temperature, coupled with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations to describe the evolution of phase fractions. We investigate questions of existence and uniqueness of a solution and finally present some numerical simulations.

*Appeared in*

- Math. Models Methods Appl. Sci., 19 (2009) pp. 2101--2126.

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# Complete damage in elastic and viscoelastic media and its energetics

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Roubíček, Tomáš
- Zeman, Jan

*2008 Physics and Astronomy Classification Scheme*

- 46.15.-x 62.20.-x

*Keywords*

- Inelastic damage, small strain, energetic formulation

*DOI*

*Abstract*

A model for the evolution of damage that allows for complete disintegration is addressed. Small strains and a linear response function are assumed. The ``flow rule'' for the damage parameter is rate-independent. The stored energy involves the gradient of the damage variable, which determines an internal length-scale. Quasi-static fully rate-independent evolution is considered as well as rate-dependent evolution including viscous/inertial effects. Illustrative 2-dimensional computer simulations are presented, too.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 199 (2010) pp. 1242--1253.

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# Asymptotic convergence results for a system of partial differential equations with hysteresis

*Authors*

- Eleuteri, Michela
- Krejčí, Pavel

*2010 Mathematics Subject Classification*

- 35K55 47J40 35B40

*Keywords*

- partial differential equations, hysteresis, asymptotic convergence, Preisach operator

*DOI*

*Abstract*

A partial differential equation motivated by electromagnetic field equations in ferromagnetic media is considered with a relaxed rate dependent constitutive relation. It is shown that the solutions converge to the unique solution of the limit parabolic problem with a rate independent Preisach hysteresis constitutive operator as the relaxation parameter tends to zero.

*Appeared in*

- Commun. Pure Appl. Anal., 6 (2007) pp. 1131-1143.

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# On an evolutionary model for complete damage based on energies and stresses

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K65 35K85 49S05 74C05 74R05

*Keywords*

- Weak energetic solution, rate independent energetic system, complete damage, Gamma convergence

*DOI*

*Abstract*

A recent model allows for complete damage, such that the deformation is not well-defined. The evolution can be described in terms of energy densities and stresses. We introduce the notion of *weak energetic solution* and show how the existence theory can be generalized to convex, but non-quadratic elastic energies.

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# Variation of constants formula for hyperbolic systems

*Authors*

- Lichtner, Mark

*2010 Mathematics Subject Classification*

- 35L90 37L05 35B30 58D25 37L50 34K20 34K19

*Keywords*

- Semilinear hyperbolic systems, variations of constants formula, sun star calculus, smooth dependence on data, linarized stability, semigroup theory

*DOI*

*Abstract*

A smooth variation of constants formula for semilinear hyperbolic systems is established using a suitable Banach space $X$ of continuous functions together with its sun dual space $X^odot ast$. It is shown that mild solutions of this variation of constants formula generate a smooth semiflow in $X$. This proves that the stability of stationary states for the nonlinear flow is determined by the stability of the linearized semigroup.

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# Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions

*Authors*

- Meyer, Christian
- Yousept, Irwin

*2010 Mathematics Subject Classification*

- 35J60 49K20 49M05 65K10

*Keywords*

- Nonlinear optimal control, nonlocal radiation interface conditions, state constraints, first-order necessary conditions, second-order sufficient conditions, Moreau-Yosida approximation

*DOI*

*Abstract*

A state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented.

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# A higher gradient theory of mixtures for multi-component materials with numerical examples for binary alloys

*Authors*

- Böhme, Thomas
- Dreyer, Wolfgang
- Duderstadt, Frank
- Müller, Wolfgang H.

*2010 Mathematics Subject Classification*

- 74A15 74A50 74N15 74N25 80A17 80A20 82D35

*Keywords*

- Thermodynamics, structured surfaces and interfaces, coexistent phases, analysis of microstructure, transformations involving diffusion

*DOI*

*Abstract*

A theory of mixture for multi-component materials is presented based on a novel, straightforward method for the exploitation of the Second Law of thermodynamics. In particular the constitutive equations for entropy, heat and diffusion flux as well as the stress tensor are formulated as a consequence of the non-negative entropy production. Furthermore we derive the established Gibbs equation as well as the Gibbs Duhem relation which also follow from the formalism. Moreover, it is illustrated, how local mechanical strains due to eigenstrains or external loadings, modify the free energy and, consequently, change the chemical potentials of the components. All consecutive steps are illustrated, first, for simple mixtures and, second, for a system containing two different phases. So-called higher gradients of the concentrations are considered, which take the nonuniform composition into account. It will also become apparent that more/other variables of modified/different physical pr oblems beyond the illustrated ones can be easily treated within the presented framework. This work ends with the specification to binary alloys and with the presentation of various numerical simulations.

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# Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in $L^p$ with $pge 1$

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35D05 35J60

*Keywords*

- non local boundary condition, right-hand side in $L^p (p geq 1)$

*DOI*

*Abstract*

Accurate modeling of heat transfer in high-temperatures situations requires to account for the effect of heat radiation. In complex applications such as Czochralski's method for crystal growth, in which the conduction radiation heat transfer problem couples to an induction heating problem and to the melt flow problem, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L-1. In such situations, the known results on the unique solvability of the heat conduction problem with surface radiation do not apply, since a right-hand side in L-p with p < 6/5 no longer belongs to the dual of the Banach space in which coercivity is obtained. In this paper, we focus on a stationary heat equation with non-local boundary conditions and right-hand side in L-p with p>=1 arbitrary. Essentially, we construct an approximation procedure and, thanks to new coercivity results, we are able to produce energy estimates that involve only the L-p-norm of the heat-sources, and to pass to the limit.

*Appeared in*

- Math. Methods Appl. Sci., 32 (2008) pp. 135 - 166.

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# Universality of REM-like ageing in mean field spin glasses

*Authors*

- Ben Arous, Gérard
- Bovier, Anton
- Černý, Jiři

*2008 Physics and Astronomy Classification Scheme*

- 75.10.Nr, 75.10.Jm, 75.10.Hk, 05.30.-d

*Keywords*

- aging, universality, spin glasses, SK model, random walk

*DOI*

*Abstract*

Aging has become the paradigm to describe dynamical behavior of glassy systems, and in particular spin glasses. Trap models have been introduced as simple caricatures of effective dynamics of such systems. In this Letter we show that in a wide class of mean field models and on a wide range of time scales, aging occurs precisely as predicted by the REM-like trap model of Bouchaud and Dean. This is the first rigorous result about aging in mean field models except for the REM and the spherical model

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# Monotonicity properties of the quantum mechanical particle density

*Authors*

- Kaiser, Hans-Christoph
- Neidhardt, Hagen
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 47H05 47B10

*Keywords*

- Trace functionals, trace class operators, monotonicity

*DOI*

*Abstract*

An elementary proof of the anti-monotonicity of the quantum mechanical particle density with respect to the potential in the Hamiltonian is given for a large class of admissible thermodynamic equilibrium distribution functions. In particular the zero temperature case is included.

*Appeared in*

- Monatsh. Math., 158 (2009) pp. 179--185.

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# An inverse problem for fluid-solid interaction

*Authors*

- Elschner, Johannes
- Hsiao, George C.
- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 35R30 76Q05 35J05 35J20 70G75

*Keywords*

- Acoustic and elastic waves, inverse scattering, gradients, Gauss-Newton method

*DOI*

*Abstract*

Any acoustic plane wave incident to an elastic obstacle results in a scattered field with a corresponding far field pattern. Mathematically, the scattered field is the solution of a transmission problem coupling the reduced elastodynamic equations over the domain occupied by the obstacle with the Helmholtz equation in the exterior. The far field pattern is obtained applying an integral operator to the scattered field function restricted to a simple smooth surface surrounding the obstacle. The subject of our paper is the inverse problem, where the shape of the elastic body represented by a parametrization of its boundary is to be reconstructed from a finite number of measured far field patterns. We define a family of objective functionals depending on a non-negative regularization parameter such that, for regularization parameter zero, the shape of the sought elastic obstacle is a minimizer of the functional. For any positive regularization parameter, there exists a regularized solution minimizing the functional. Moreover, for the regularization parameter tending to zero, these regularized solutions converge to the solution of the inverse problem provided the latter is uniquely determined by the given far field patterns. The whole approach is based on the variational form of the partial differential operators involved. Hence, numerical approximations can be found applying finite element discretization. Note that, though the transmission problem in its weak formulation may have non-unique solutions for domains with so-called Jones frequencies, the scattered field and its far field pattern is unique and depend continuously on the shape of the obstacle.

*Appeared in*

- Inverse Probl. Imaging, 2 (2008) pp. 83--120.

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# Effect of higher-order dispersion on modulation instability, soliton propagation and pulse splitting

*Authors*

- Demircan, Ayhan
- Pietrzyk, Monika
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 35Q55 35Q60 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.81.Dp, 42.65.Sf

*Keywords*

- nonlinear fibers, pulse splitting, third-order dispersion, Modulation instability

*DOI*

*Abstract*

By solving numerically the extended nonlinear Schrödinger equation we investigate the influence of higher-order dispersion effects on the propagation of optical pulses in highly nonlinear fibers. In the anomalous dispersion regime third-order dispersion can, in general, induce soliton fission and yields asymmetric spectra, whereas modulation instability can be slightly suppressed. In the normal dispersion regime we demonstrate pulse splitting by third-order dispersion, as well as its later suppression by fourth-order dispersion.

*Appeared in*

- Opt. Quantum Electron., 40 (2008) pp. 455-460.

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# The effect of time-dependent coupling on non-equilibrium steady states

*Authors*

- Cornean, Horia D.
- Neidhardt, Hagen
- Zagrebnov, Valentin A.

*2010 Mathematics Subject Classification*

- 46N55 47N55 47A40 35L90 47E05

*Keywords*

- non-equilibrium steady states, Landauer-Büttiker formula, Landau-Lifschitz formula, quantum Liouville equation, wave and scattering operator

*DOI*

*Abstract*

Consider (for simplicity) two one-dimensional semi-infinite leads coupled to a quantum well via time dependent point interactions. In the remote past the system is decoupled, and each of its components is at thermal equilibrium. In the remote future the system is fully coupled. We define and compute the non equilibrium steady state (NESS) generated by this evolution. We show that when restricted to the subspace of absolute continuity of the fully coupled system, the state does not depend at all on the switching. Moreover, we show that the stationary charge current has the same invariant property, and derive the Landau-Lifschitz and Landauer-Büttiker formulas.

*Appeared in*

- Ann. Henri Poincare, 10 (2009) pp. 61--93.

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# Diffusion Tensor Imaging: Structural adaptive smoothing

*Authors*

- Tabelow, Karsten

ORCID: 0000-0003-1274-9951 - Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427 - Voss, Henning U.

*2010 Mathematics Subject Classification*

- 62P10 92C55 62G05

*Keywords*

- diffusion tensor imaging, structural adaptive smoothing

*DOI*

*Abstract*

Diffusion Tensor Imaging (DTI) data is characterized by a high noise level. Thus, estimation errors of quantities like anisotropy indices or the main diffusion direction used for fiber tracking are relatively large and may significantly confound the accuracy of DTI in clinical or neuroscience applications. Besides pulse sequence optimization, noise reduction by smoothing the data can be pursued as a complementary approach to increase the accuracy of DTI. Here, we suggest an anisotropic structural adaptive smoothing procedure, which is based on the Propagation-Separation method and preserves the structures seen in DTI and their different sizes and shapes. It is applied to artificial phantom data and a brain scan. We show that this method significantly improves the quality of the estimate of the diffusion tensor and hence enables one either to reduce the number of scans or to enhance the input for subsequent analysis such as fiber tracking.

*Appeared in*

- NeuroImage, 39 (2008) pp. 1763--1773.

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# Reverse approximation of energetic solutions to rate-independent processes

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Rindler, Filip

*2010 Mathematics Subject Classification*

- 49J40 49S05 65J15 74C05 74H15

*Keywords*

- Rate-independent processes, energetic solutions, approximate incremental problems, Gamma convergence

*DOI*

*Abstract*

Energetic solutions to rate-independent processes are usually constructed via time-incremental minimization problems. In this work we show that all energetic solutions can be approximated by incremental problems if we allow approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Gamma limit.

*Appeared in*

- NoDEA Nonlinear Differential Equations Appl., 16 (2009) pp. 17--40.

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# On the construction of bifurcation curves related to limit cycles of multiplicity three for planar vector fields

*Authors*

- Cherkas, Leonid
- Grin, Alexander
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 34C05 34C23

*Keywords*

- Multiple limit cycle, degenerate Hopf bifurcation, continuation method

*DOI*

*Abstract*

For plane vector fields depending on three parameters we describe an algorithm to construct a curve in the parameter space such that to each point of this curve there belongs a vector field possessing a limit cycle of multiplicity three. One point of this curve is related to the bifurcation of a limit cycle of multiplicity three from an equilibrium point. The underlying procedure is a continuation method.

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# Trace formulae for dissipative and coupled scattering systems

*Authors*

- Behrndt, Jussi
- Malamud, Mark
- Neidhardt, Hagen

*2010 Mathematics Subject Classification*

- 47A40 47A55 47B44

*Keywords*

- Scattering system, scattering matrix, boundary triplet, Titchmarsh-Weyl function, spectral shift function, Krein-Birman formula

*DOI*

*Abstract*

For scattering systems consisting of a (family of) maximal dissipative extension(s) and a selfadjoint extension of a symmetric operator with finite deficiency indices, the spectral shift function is expressed in terms of an abstract Titchmarsh-Weyl function and a variant of the Birman-Krein formula is proved.

*Appeared in*

- vol. 188 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2008, pp. 57--93

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# Estimation of the infinitesimal generator by square-root approximation

*Authors*

- Donati, Luca
- Heida, Martin
- Weber, Marcus
- Keller, Bettina

*2008 Physics and Astronomy Classification Scheme*

- 02.50.Ga 05.10Gg

*Keywords*

- molecular simulation, Markov state models, transfer operator, molecular kinetics

*DOI*

*Abstract*

For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approximated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the geometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correlation between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide.

*Appeared in*

- J. Phys.: Condens. Matter, 30 (2018), pp. 425201/1--425201/14, DOI 10.1088/1361-648X/aadfc8 .

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# Robust risk management. Accounting for nonstationarity and heavy tails

*Authors*

- Chen, Ying
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05 62P20

*Keywords*

- exponential smoothing, spatial aggregation, heavy-tailed distribution

*DOI*

*Abstract*

In the ideal Black-Scholes world, financial time series are assumed 1) stationary (time homogeneous) or can be modelled globally by a stationary process and 2) having conditionally normal distribution given the past. These two assumptions have been widely-used in many methods such as the RiskMetrics, one risk management method considered as industry standard. However these assumptions are unrealistic. The primary aim of the paper is to account for nonstationarity and heavy tails in time series by presenting a local exponential smoothing approach, by which the smoothing parameter is adaptively selected at every time point and the heavy-tailedness of the process is considered. A complete theory addresses both issues. In our study, we demonstrate the implementation of the proposed method in volatility estimation and risk management given simulated and real data. Numerical results show the proposed method delivers accurate and sensitive estimates.

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# The universal Airy$_1$ and Airy$_2$ processes in the totally asymmetric simple exclusion process

*Authors*

- Ferrari, Patrik

*2010 Mathematics Subject Classification*

- 82C22 60K35 15A52

*Keywords*

- Simple exclusion process, universality, Airy process, random matrices

*DOI*

*Abstract*

In the totally asymmetric simple exclusion process (TASEP) two processes arise in the large time limit: the Airy$_1$ and Airy$_2$ processes. The Airy$_2$ process is an universal limit process occurring also in other models: in a stochastic growth model on $1+1$-dimensions, 2d last passage percolation, equilibrium crystals, and in random matrix diffusion. The Airy$_1$ and Airy$_2$ processes are defined and discussed in the context of the TASEP. We also explain a geometric representation of the TASEP from which the connection to growth models and directed last passage percolation is immediate.

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# Sensitivities for Bermudan options by regression methods

*Authors*

- Belomestny, Denis
- Milstein, Grigori N.
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60H30 65C05 91B28

*Keywords*

- Monte Carlo simulation, regression method, conditional probabilistic representations, optimal stopping times, American and Bermudan options, deltas

*DOI*

*Abstract*

In this article we propose several pathwise and finite difference based methods for calculating sensitivities of Bermudan options using regression methods and Monte Carlo simulation. These methods rely on conditional probabilistic representations which allows, in combination with a regression approach, an efficient simultaneous computation of sensitivities at all initial positions. Assuming that the price of a Bermudan option can be evaluated sufficiently accurate, we develop a method for constructing deltas based on least squares. We finally propose a testing procedure for assessing the performance of the developed methods.

*Appeared in*

- Decis. Econ. Finance, 33 (2010) pp. 117--138.

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# Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity

*Authors*

- Knees, Dorothee
- Neff, Patrizio

*2010 Mathematics Subject Classification*

- 35B65 74C05 49N60 74A35 74G40

*Keywords*

- Polar materials, perfect plasticity, higher global regularity, quasilinear elliptic systems, error estimates, time-increments

*DOI*

*Abstract*

In this note we investigate the question of higher regularity up to the boundary for quasilinear elliptic systems which origin from the time-discretization of models from infinitesimal elasto-plasticity. Our main focus lies on an elasto-plastic Cosserat model. More specifically we show that the time discretization renders $H^2$-regularity of the displacement and $H^1$-regularity for the symmetric plastic strain $varepsilon_p$ up to the boundary provided the plastic strain of the previous time step is in $H^1$, as well. This result contrasts with classical Hencky and Prandtl-Reuss formulations where it is known not to hold due to the occurrence of slip lines and shear bands. Similar regularity statements are obtained for other regularizations of ideal plasticity like viscosity or isotropic hardening. In the first part we recall the time continuous Cosserat elasto-plasticity problem, provide the update functional for one time step and show various preliminary results for the update functional (Legendre-Hadamard/monotonicity). Using non standard difference quotient techniques we are able to show the higher global regularity. Higher regularity is crucial for qualitative statements of finite element convergence. As a result we may obtain estimates linear in the mesh-width $h$ in error estimates.

*Appeared in*

- SIAM J. Math. Anal., 40 (2008) pp. 21--43.

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# On the Landau--Levich problem for non-Newtonian liquids

*Authors*

- Afanasiev, Konstantin
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 34B15 35G25 35K55 35Q35

*2008 Physics and Astronomy Classification Scheme*

- 68.15.+e

*Keywords*

- Lubrication models, non-Newtonian flow, fluid dynamics, phase plane analysis

*DOI*

*Abstract*

In this paper the drag-out problem for shear-thinning liquids at variable inclination angle is considered. For this free boundary problem dimension-reduced lubrication equations are derived for the most commonly used viscosity models, namely, the power-law, Ellis and Carreau model. For the resulting lubrication models a system of ordinary differential equation governing the steady state solutions is obtained. Phase plane analysis is used to characterize the type of possible steady state solutions and their dependence on the rheological parameters.

*Appeared in*

- Phys. Rev. E, 76 (2007) pp. 036307/1--036307/12.

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# On a class of partial differential equations with hysteresis arising in magnetohydrodynamics

*Authors*

- Eleuteri, Michela

*2010 Mathematics Subject Classification*

- 35K55 47J40 76W05

*Keywords*

- partial differential equations, hysteresis, Preisach operator, magnetohydrodynamics

*DOI*

*Abstract*

In this paper we deal with a class of parabolic partial differential equations containing a continuous hysteresis operator. We get an existence result by means of a technique based on an implicit time discretization scheme and we also analyse the dependence of the solution on the data. This model equation appears in the context of magnetohydrodynamics.

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# Expansion of random boundary excitations for elliptic PDEs

*Authors*

- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 65Z05

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Pt

*Keywords*

- White noise, generalized random processes, Karhunen-Loève expansion, Poisson integral formula, random boundary excitations, Laplace equations, biharmonic equations, Lamé equations

*DOI*

*Abstract*

In this paper we deal with elliptic boundary value problems with random boundary conditions. Solutions to these problems are inhomogeneous random fields which can be represented as series expansions involving a complete set of deterministic functions with corresponding random coefficients. We construct the Karhunen-Loève (K-L) series expansion which is based on the eigen-decomposition of the covariance operator. It can be applied to simulate both homogeneous and inhomogeneous random fields. We study the correlation structure of solutions to some classical elliptic equations in respond to random excitations of functions prescribed on the boundary. We analyze the stochastic solutions for Dirichlet and Neumann boundary conditions to Laplace equation, biharmonic equation, and to the Lamé system of elasticity equations. Explicit formulae for the correlation tensors of the generalized solutions are obtained when the boundary function is a white noise, or a homogeneous random field on a circle, a sphere, and a half-space. These exact results may serve as an excellent benchmark for developing numerical methods, e.g., Monte Carlo simulations, stochastic volume and boundary element methods.

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# Monte Carlo Greeks for financial products via approximative Greenian kernels

*Authors*

- Kampen, Jörg
- Kolodko, Anastasia
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60H10 62G07 65C05

*Keywords*

- American options, Sensitivities, Monte-Carlo methods, WKB expansions

*DOI*

*Abstract*

In this paper we introduce efficient Monte Carlo estimators for the valuation of high-dimensional derivatives and their sensitivities (''Greeks''). These estimators are based on an analytical, usually approximative representation of the underlying density. We study approximative densities obtained by the WKB method. The results are applied in the context of a Libor market model.

*Appeared in*

- SIAM J. Sci. Comput. Vol. 31, 1, pp. 1-22, 2008 under new title: Monte Carlo Greeks for financial products via approximative transition densities

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# GEOMS: A software package for the numerical integration of general model equations of multibody systems

*Authors*

- Steinbrecher, Andreas

*2010 Mathematics Subject Classification*

- 70E55 65L80

*Keywords*

- differential-algebraic equations, equations of motion, multibody system, numerical integration, simulation

*DOI*

*Abstract*

In this paper we present the new numerical algorithm GEOMS for the numerical integration of the most general form of the equations of motion of multibody systems, including nonholonomic constraints and possible redundancies in the constraints, as they may appear in industrial applications. Besides the numerical integration it offers some additional features like stabilization of the model equations, use of different decomposition strategies, or checking and correction of the initial values with respect to their consistency. Furthermore, GEOMS preserves hidden constraints and (possibly) existing solution invariants if they are provided as equations. We will also demonstrate the performance and the applicability of GEOMS for two mechanical examples of different degrees of complexity.

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# A stochastic volatility Libor model and its robust calibration

*Authors*

- Belomestny, Denis
- Mathew, Stanley
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60G51 62G20 60H05 60H10 90A09 91B28

*Keywords*

- Libor modelling, stochastic volatility, CIR processes, calibration

*DOI*

*Abstract*

In this paper we propose a Libor model with a high-dimensional specially structured system of driving CIR volatility processes. A stable calibration procedure which takes into account a given local correlation structure is presented. The calibration algorithm is FFT based, so fast and easy to implement.

*Appeared in*

- Monte Carlo Methods Appl., 15 (2009) pp. 285-310 as "Multiple stochastic volatility extension of the Libor market model and its implementation".

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# An optimal control approach to curved rods

*Authors*

- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 49J20 74K10

*Keywords*

- Curved rods, control variational methods, generalized Naghdi model

*DOI*

*Abstract*

In this paper, a new approach to the generalized Naghdi model for the deformation of three-dimensional curved rods is studied. The method is based on optimal control theory.

*Appeared in*

- SIAM J. Control. Optim, 47 (2009), pp. 3220-3236, in extended form as "The control variational approach for differential systems"

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# Slow motion of quasi-stationary multi-pulse solutions by semistrong interaction in reaction-diffusion systems

*Authors*

- Wolfrum, Matthias
- Ehrt, Julia

*2010 Mathematics Subject Classification*

- 35B25 34C30 35K57

*Keywords*

- Pulse interaction, singular perturbation theory

*DOI*

*Abstract*

In this paper, we study a class of singularly perturbed reaction-diffusion systems, which exhibit under certain conditions slowly varying multi-pulse solutions. This class contains among others the Gray-Scott and several versions of the Gierer-Meinhardt model. We first use a classical singular perturbation approach for the stationary problem and determine in this way a manifold of quasi-stationary $N$-pulse solutions. Then, in the context of the time-dependent problem, we derive an equation for the leading order approximation of the slow motion along this manifold. We apply this technique to study 1-pulse and 2-pulse solutions for classical and modified Gierer-Meinhardt system. In particular, we are able to treat different types of boundary conditions, calculate folds of the slow manifold, leading to slow-fast motion, and to identify symmetry breaking singularities in the manifold of 2-pulse solutions.

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# Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and right-hand side in $L^p$ with $pge 1$

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35D05 35K05 35K15 35K55

*Keywords*

- Nonlinear parabolic equation, nonlocal boundary condition, right-hand side in L-p with p>=1

*DOI*

*Abstract*

It is known that the time-dependent heat equation with nonlocal radiation boundary conditions possesses a unique weak solution if the heat sources are in L-2. In this paper, we generalize the known existence and uniqueness results to the case that the right-hand side belongs to an arbitrary L-p space (p >= 1). This is the continuation of the results that we recently proved for the stationary problem. The purpose of both papers is to obtain energy estimates that involve only the L-p norm of the heat sources for some exponent p close to one. Such estimates are important for the investigation of models in which the heat equation is coupled to Maxwell's equations or to the Navier-Stokes equations (dissipative heating).

*Appeared in*

- Appl. Math., 55 (2010) pp. 111--149.

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# Local limit theorems for ladder moments

*Authors*

- Vatutin, Vladimir
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60G50 60G40

*Keywords*

- Random walk, ladder moment, Spitzer condition

*DOI*

*Abstract*

Let $S_0=0,S_n_ngeq1$ be a random walk generated by a sequence of i.i.d. random variables $X_1,X_2,...$ and let $tau^-:=minleft ngeq1: S_nleq0right $ and $tau^+:=minleft ngeq 1: S_n>0right $. Assuming that the distribution of $X_1$ belongs to the domain of attraction of an $alpha$-stable law$,alphaneq1,$ we study the asymptotic behavior of $mathbbP(tau^pm=n)$ as $nrightarrowinfty.$

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# Convergence of Fourier-wavelet models for Gaussian random processes

*Authors*

- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 60G15

*2008 Physics and Astronomy Classification Scheme*

- 05.10.Ln

*Keywords*

- Fourier-Wavelet model, stationary Gaussian random process, Meyer's wavelets, Nikolskiui-Besov space, convergence in probability, convergence in mean square

*DOI*

*Abstract*

Mean square convergence and convergence in probability of Fourier-Wavelet Models (FWM) of stationary Gaussian Random processes in the metric of Banach space of continuously differentiable functions and in Sobolev space are studied. Sufficient conditions for the convergence formulated in the frame of spectral functions are given. It is shown that the given rates of convergence of FWM in the mean square obtained in the Nikolskiui-Besov classes cannot be improved.

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# Inhomogeneous dependence modelling with time varying copulae

*Authors*

- Giacomini, Enzo
- Härdle, Wolfgang
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05 62P20

*Keywords*

- adaptive estimation, nonparametric estimation, Value-at-Risk

*DOI*

*Abstract*

Measuring dependence in a multivariate time series is tantamount to modelling its dynamic structure in space and time. In the context of a multivariate normally distributed time series, the evolution of the covariance (or correlation) matrix over time describes this dynamic. A wide variety of applications, though, requires a modelling framework different from the multivariate normal. In risk management the non-normal behaviour of most financial time series calls for non-Gaussian dependence. The correct modelling of non-Gaussian dependences is therefore a key issue in the analysis of multivariate time series. In this paper we use copulae functions with adaptively estimated time varying parameters for modelling the distribution of returns, free from the usual normality assumptions. Further, we apply copulae to estimation of Value-at-Risk (VaR) of portfolios and show their better performance over the RiskMetrics approach, a widely used methodology for VaR estimation.

*Appeared in*

- J. Bus. Econom. Statist., 27 (2009) pp. 224--234.

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# On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling

*Authors*

- Henrion, René
- Römisch, Werner

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Electricity markets, bidding, noncooperative games, equilibrium constraint, EPEC, optimality condition, co-derivative, random demand

*DOI*

*Abstract*

Modeling several competitive leaders and followers acting in an electricity market leads to coupled systems of mathematical programs with equilibrium constraints, called equilibrium problems with equilibrium constraints (EPECs). We consider a simplified model for competition in electricity markets under uncertainty of demand in an electricity network as a (stochastic) multi-leader-follower game. First order necessary conditions are developed for the corresponding stochastic EPEC based on a result of Outrata [17]. For applying the general result an explicit representation of the co-derivative of the normal cone mapping to a polyhedron is derived (Proposition 3.2). Later the co-derivative formula is used for verifying constraint qualifications and for identifying M-stationary solutions of the stochastic EPEC if the demand is represented by a finite number of scenarios.

*Appeared in*

- Appl. Math., 522 (2007) pp. 473--494.

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# Stationary solutions of driven fourth- and sixth-order Cahn--Hilliard type equations

*Authors*

- Korzec, Maciek D.
- Evans, Peter L.
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 34E05 74K35 65P99

*Keywords*

- convective Cahn-Hilliard equation, quantum dots, exponential asymptotics, matching, dynamical systems

*DOI*

*Abstract*

New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function. Using phase space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.

*Appeared in*

- SIAM J. Appl. Math., 69 (2008) pp. 348-374.

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# Computing likelihoods for coalescents with multiple collisions in the infinitely-many-sites model

*Authors*

- Birkner, Matthias
- Blath, Jochen

*2010 Mathematics Subject Classification*

- 92D15 60G09 60G52 60J75 60J85

*Keywords*

- $Lambda$-coalescent, likelihood-based inference, infinitely-many-sites, population genetics, Monte-Carlo method

*DOI*

*Abstract*

One of the central problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of the classical Fleming-Viot process, such as the Wright-Fisher- or the Moran model, then the standard means to describe its genealogy is Kingman's coalescent. For this coalescent process, powerful inference methods are well-established. An important feature of the above class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size, and hence all ancestral collisions are binary only. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised $Lambda$-Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called $Lambda$-coalescents, which allow multiple collisions. Moreover, Eldon and Wakeley (2006) provide evidence that such more general coalescents might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier and Griffiths (1987) and Griffiths and Tavaré (1994, 1995) to obtain a likelihood based inference method for general $Lambda$-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of $Lambda$-coalescents, the parametrisable sub-family of Beta$(2-alpha, alpha)$-coalescents, where $alpha in (1,2]$, are of particular relevance. We illustrate our method using simulated datasets, thus obtaining maximum-likelihood estimators of mutation and demographic parameters.

*Appeared in*

- J. Math. Biol., 57 (2008) pp. 435--465.

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# Inference for $Lambda$-coalescents

*Authors*

- Birkner, Matthias
- Blath, Jochen

*2010 Mathematics Subject Classification*

- 92D15 60G09 60G52 60J75 60J85

*Keywords*

- Lambda-coalescent, inference, infinitely-many-sites model, mathematical population genetics, Fleming-Viot process, multiple collisions, frequency spectrum, Monte-Carlo simulation

*DOI*

*Abstract*

One of the main problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of a classical Fleming-Viot process, then the standard means to describe the corresponding genealogy is Kingman's coalescent. For this process, powerful inference methods are well-established. An important feature of this class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised Lambda Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called Lambda-coalescents. Moreover, Eldon & Wakeley (2006) have provided evidence that such more general coalescents, which allow m ultiple collisions, might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier & Griffiths (1987) and Griffiths & Tavaré (1994) to obtain a likelihood based inference method for general Lambda-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of Lambda-coalescents, the parametrisable sub-family of Beta$(2-alpha,alpha)$-coalescents, where $alpha in (1,2]$, are of particular biological relevance. We apply our method in this case to simulated and real data (taken from Árnason (2004)). We conclude that for populations with extreme reproductive behaviour, the Kingman-coalescent as standard model might have to be replaced by more general coalescents, in particular by Beta$(2-alpha,alpha)$-coalescents.

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# Bildsegmentation zur Untersuchung von Streulichtbildern bei der laseroptischen Diagnose von rheumatoider Arthritis

*Authors*

- Gajewski, Herbert
- Griepentrog, Jens André
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Beuthan, J.
- Zabarylo, U.
- Minet, O.

*2010 Mathematics Subject Classification*

- 90C26 82B26 94A08

*2008 Physics and Astronomy Classification Scheme*

- 42.62.Be 87.63.Lk 42.30.Va 07.05.Pj 02.30.Sa 02.60.Lj

*Keywords*

- Image segmentation, Rheumatoid arthritis, Optical diagnostics, Laser transillumination, Non-convex energy functionals

*DOI*

*Abstract*

Optical imaging in biomedicine is governed by the light absorption and scattering interaction on microscopic and macroscopic constituents in the medium. Therefore, light scattering characteristics of human tissue correlates with the stage of some diseases. In the near infrared range the scattering event with the coefficient approximately two orders of magnitude greater than absorption plays a dominant role. The potential of an experimental laser diode based setup for the transillumination of rheumatoid finger joints and the pattern of the stray light detection are demonstrated. For evaluating the scattering light images a new non-local image segmentation method is presented. Regarding a noisy picture as a multicomponent mixture of gray scaled particles, this method minimizes a non-convex free energy functional under the constraint of mass conservation of the components. Contrary to constructing equilibrium distributions as steady states of an adequate evolution equation, a direct descent method for the free energy is used to separate the components of the image.

*Appeared in*

- Mathematics -- Key Technology for the Future, W. JÄGER, H.-J. KREBS, eds., Springer, Heidelberg, 2008, english version ``Image Segmentation for the Investigation of Scattered-Light Images when Laser-Optically Diagnosing Rheumatoid Arthritis'', pp. 149--161.

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# Optimal control problems with delays in state and control and mixed control-state constraints

*Authors*

- Göllmann, Laurenz
- Kern, Daniela
- Maurer, Helmut

*2010 Mathematics Subject Classification*

- 49K15 49K25

*Keywords*

- Retarded optimal control problems, delays in state and control, mixed control-state inequality constraints, Pontryagin's minimum principle, discretization methods, optimal control of a CSTR reactor, optimal fishing

*DOI*

*Abstract*

Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control-state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods for the delayed control problem are discussed which amount to solving a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and two numerical examples from chemical engineering and economics illustrate the results.

*Appeared in*

- Optimal Control Appl. Methods, 30 (2009) pp. 341--365.

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# Discrepancy distances and scenario reduction in two-stage stochastic integer programming

*Authors*

- Henrion, René
- Küchler, Christian
- Römisch, Werner

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Stochastic programming, two-stage, mixed-integer, chance constraints,, scenario reduction, discrepancy, Kolmogorov metric

*DOI*

*Abstract*

Polyhedral discrepancies are relevant for the quantitative stability of mixed-integer two-stage and chance constrained stochastic programs. We study the problem of optimal scenario reduction for a discrete probability distribution with respect to certain polyhedral discrepancies and develop algorithms for determining the optimally reduced distribution approximately. Encouraging numerical experience for optimal scenario reduction is provided.

*Appeared in*

- J. Indust. Management Optim., 4 (2008) pp. 363--384.

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# Control and removing of modulational instabilities in low dispersion photonic crystal fiber cavities

*Authors*

- Tlidi, Mustapha
- Mussot, Arnaud
- Louvergneaux, Eric
- Kozyreff, Gregory
- Vladimirov, Andrei G.
- Taki, Abdelmajid

*2010 Mathematics Subject Classification*

- 78A60 37K45

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k, 42.65.Wi, 42.65.Sf

*Keywords*

- Photonic crystal fibers, modulational instability, forth order dispersion

*DOI*

*Abstract*

Taking up to fourth order dispersion effects into account, we show that fiber resonators become stable for large intensity regime. The range of pump intensities leading to modulational instability becomes finite and controllable. Moreover, by computing analytically the thresholds and frequencies of these instabilities, we demonstrate the existence of a new unstable frequency at the primary threshold. This frequency exists for arbitrary small but nonzero fourth order dispersion coefficient. Numerical simulations for a low and flattened dispersion photonic crystal fiber resonator confirm analytical predictions and opens the way to experimental implementation.

*Appeared in*

- Optics Letters, 32 (2007) pp. 662-664.

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# Transport behaviour of a Bose--Einstein condensate in a bichromatic optical lattice

*Authors*

- Bhattacharjee, Aranya
- Pietrzyk, Monika

*2010 Mathematics Subject Classification*

- 78A60

*2008 Physics and Astronomy Classification Scheme*

- 03.75.Lm 03.75.Kk 32.80.Lg

*Keywords*

- 0:0:optical lattice, Bose Einstein condensate, Gross-Pitaevskii equation

*DOI*

*Abstract*

The Bloch and dipole oscillations of a Bose Einstein condensate (BEC) in an optical superlattice is investigated. We show that the effective mass increases in an optical superlattice, which leads to localization of the BEC, in accordance with recent experimental observations [17]. In addition, we find that the secondary optical lattice is a useful additional tool to manipulate the dynamics of the atoms.

*Appeared in*

- Cent. Eur. J. Math., 6 (2008) pp. 26-32.

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# Rate independent Kurzweil processes

*Authors*

- Krejčí, Pavel
- Liero, Matthias

ORCID: 0000-0002-0963-2915

*2010 Mathematics Subject Classification*

- 49J40 49J53 74C15

*Keywords*

- Rate independent process, Kurzweil integral, variational inequality

*DOI*

*Abstract*

The Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in $BV$ spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree one.

*Appeared in*

- Appl. Math., 54 (2009) pp. 117--145.

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# Existence of bounded discrete steady state solutions of the van Roosbroeck system on boundary conforming Delaunay grids

*Authors*

- Gärtner, Klaus

*2010 Mathematics Subject Classification*

- 65M12 35K65

*Keywords*

- Reaction-diffusion systems, discrete bounded solutions, Delaunay grids, discrete weak maximum principle

*DOI*

*Abstract*

The classic van Roosbroeck system describes the carrier transport in semiconductors in a drift diffusion approximation. Its analytic steady state solutions fulfill bounds for some mobility and recombination/generation models. The main goal of this paper is to establish the identical bounds for discrete in space, steady state solutions on 3d boundary conforming Delaunay grids and the classical Scharfetter-Gummel-scheme. Together with a uniqueness proof for small applied voltages and the known dissipativity (continuous as well as space and time discrete) these discretization techniques carry over the essential analytic properties to the discrete case. The proofs are of interest for deriving averaging schemes for space or state dependent material parameters, which are preserving these qualitative properties, too. To illustrate the properties of the scheme 1, 4, 16 elementary cells of a modified CoolMOS like structure are depleted by increasing the applied voltage until steady state avalanche breakdown occurs.

*Appeared in*

- SIAM J. Sci. Comput., 31 (2009) pp. 1347--1362.

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# A complete-damage problem at small strains

*Authors*

- Bouchitté, Guy
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Roubíček, Tomáš

*2010 Mathematics Subject Classification*

- 35K65 35K85 49S05 74C05 74R05

*Keywords*

- Inelastic damage, small strain, variational inequality, energetic formulation

*DOI*

*Abstract*

The complete damage of a linearly-responding material that can thus completely disintegrate is addressed at small strains under time-varying Dirichlet boundary conditions as a rate-independent evolution problem in multidimensional situations. The stored energy involves the gradient of the damage variable. This variable as well as the stress and energies are shown to be well defined even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular, by detailed investigating the $Gamma$-limit of the stored energy and its dependence on boundary conditions. Eventually, the theory is illustrated on a one-dimensional example.

*Appeared in*

- Z. Angew. Math. Phys., 60 (2009) pp. 205--236.

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# On Eisenbud's and Wigner's R-matrix: A general approach

*Authors*

- Behrndt, Jussi
- Neidhardt, Hagen
- Racec, Roxana
- Racec, Paul N.
- Wulf, Ulrich

*2010 Mathematics Subject Classification*

- 47A40 34L25 81U20

*Keywords*

- Scattering, scattering matrix, R-matrix, symmetric and selfadjoint operators, extension theory, boundary triplets, Weyl function, ordinary differential operators

*DOI*

*Abstract*

The main objective of this paper is to give a rigorous treatment of Wigner's and Eisenbud's R-matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices. In the framework of boundary triplets and associated Weyl functions an abstract generalization of the R-matrix method is developed and the results are applied to Schrödinger operators on the real axis.

*Appeared in*

- J. Differential Equations, 244 (2008) pp. 2545--2577.

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# Multisymplectic analysis of the short pulse equation

*Authors*

- Pietrzyk, Monika
- Kanattšikow, Igor

*2010 Mathematics Subject Classification*

- 37K10 78A60 35Q60 35Q51

*2008 Physics and Astronomy Classification Scheme*

- 02.30.lk, 42.65.-k, 42.81.Gs

*Keywords*

- Multisymplectic formalism, multisymplectic integrator, Short Pulse Equation, ultrashort pulses, nonlinear optics

*DOI*

*Abstract*

The multisymplectic analysis of the Short Pulse Equation known in nonlinear optics is used in order to construct a geometric multisymplectic integrator of it. A brief comparison of its effectiveness relative to the pseudo-spectral integration scheme is presented.

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# Structural adaptive dimension reduction

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Sperlich, Stefan

*2010 Mathematics Subject Classification*

- 62G05

*Keywords*

- Dimension-reduction, multi-index model, index space, structural adaptation, R

*DOI*

*Abstract*

The paper introduces and discusses different estimation methods for multi index models where the indices are parametric and the link function is nonparametric. More specific, the here introduced methods follow the idea of Hristache et al. (2001), modify and try to improve it. Moreover, they constitute alternatives to the so called MAVE-based methods (Xia et al, 2002). We concentrate on an intuitive presentation of what each procedure is doing to the data and its implementation. All methods considered here we have made freely available in R. We conclude with a comparative simulation study based on the provided package EDR.

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# Linear non-autonomous Cauchy problems and evolution semigroups

*Authors*

- Neidhardt, Hagen
- Zagrebnov, Valentin A.

*2010 Mathematics Subject Classification*

- 35L90 34G10 47D06

*Keywords*

- linear evolution equations, evolution semigroups, perturbation theory, time-dependent Schrödinger operators, moving potentials

*DOI*

*Abstract*

The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear non-autonomous evolution equations in Hilbert spaces where the assumption is made that the domains of the quadratic forms associated with the generators are independent of time. Finally, these results are applied to time-dependent Schrödinger operators with moving point interactions in 1D.

*Appeared in*

- Adv. Differential Equations, 14 (2009) pp. 289--340.

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# Exponential bounds for the minimum contrast with some applications

*Authors*

- Golubev, Yuri
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62F10 62F12 62F25

*Keywords*

- risk bound, quasi maximum likelihood, smooth contrast

*DOI*

*Abstract*

The paper studies parametric minimum contrast estimates under rather general conditions. The quality if estimation is measured by the rate function related to the contrast which allows for stating the results without specifying the particular parametric structure of the model. This approach permits also to go far beyond the classical i.i.d. case and to obtain nonasymptotic upper bounds for the risk. These bounds apply even for small or moderate samples. They also cover the case of misspecified parametric models. Another important feature of the approach is that it works well in the case when the parametric set can be unbounded and non-compact. In the case of a smooth contrast, the obtained exponential bounds do not rely on the covering numbers and can be easily computed. We also illustrate how these bound can be used for statistical inference: bounding the estimation risk, constructing the confidence sets for the underlying parameters, establishing the concentration properties of the minimum contrast estimate. The general results are specified to the case of a Gaussian contrast and of an i.i.d. sample. We also illustrate the approach by several popular examples including least squares and least absolute deviation contrasts and the problem of estimating the location of the change point. What we obtain in these examples slightly differs from usual asymptotic results known in the classical literature. This difference is due to the unboundness of the parameter set and a possible model misspecification.

*Appeared in*

- Electron. J. Stat., 3 (2009) pp. 712--746.

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# Impact of size, shape and composition on piezoelectric effects and the electronic properties of InGaAs/GaAs quantum dots

*Authors*

- Schliwa, Andrei
- Winkelnkemper, Momme
- Bimberg, Dieter

*Keywords*

- Electronic properties, Quantum dots, Piezolectricity

*DOI*

*Abstract*

The strain fields in and around self-organized In(Ga)As/GaAs quantum dots (QD) sensitively depend on QD geometry, average InGaAs composition and the In/Ga distribution profile. Piezoelectric fields of varying size are one result of these strain fields. We study systematically a large variety of realistic QD geometries and composition profiles, and calculate the linear and quadratic parts of the piezoelectric field. The balance of the two orders depends strongly on the QD shape and composition. For pyramidal InAs QDs with sharp interfaces a strong dominance of the second order fields is found. Upon annealing the first order terms become dominant, resulting in a reordering of the electron p- and d-states and a reorientation of the hole wavefunctions.

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# Linear stability analysis of a sharp-interface model for dewetting thin films

*Authors*

- King, John R.
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 76M45 34B15 65M06

*2008 Physics and Astronomy Classification Scheme*

- 68.15.+e

*Keywords*

- sharp-interface model, slippage, stability, dewetting, fingering, Lubrication models, rim

*DOI*

*Abstract*

The topic of this study concerns the stability of the three-phase contact-line of a dewetting thin liquid film on a hydrophobised substrate driven by van der Waals forces. The role of slippage in the emerging instability at the three-phase contact-line is studied by deriving a sharp-interface model for the dewetting thin film via matched asymptotic expansions. This allows for a derivation of travelling waves and their linear stability via eigenmode analysis. In contrast to the dispersion relations typically encountered for the finger-instabilty, where the dependence of the growth rate on the wave number is quadratic, here it is linear. Using the separation of time scales of the slowly growing rim of the dewetting film and time scale on which the contact line destabilises, the sharp-interface results are compared to earlier results for the full lubrication model and good agreement for the most unstable modes is obtained.

*Appeared in*

- J. Engrg. Math., 63 (2009) pp. 177--195.

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# Small strain oscillations of an elastoplastic Kirchhoff plate

*Authors*

- Guenther, Ronald B.
- Krejčí, Pavel
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74C05 35Q72 47J40

*Keywords*

- elastoplastic plate, hysteresis operators, vector Prandtl-Ishlinskii model, von Mises model

*DOI*

*Abstract*

The two dimensional equation for transversal vibrations of an elastoplastic plate is derived from a general three dimensional system with a single yield tensorial von Mises plasticity model in the five dimensional deviatoric space. It leads after dimensional reduction to a multiyield three dimensional Prandtl-Ishlinskii hysteresis model whose weight function is explicitly given. The resulting partial differential equation with hysteresis is solved by means of viscous approximations and a monotonicity argument.

*Appeared in*

- ZAMM Z. Angew. Math. Mech., 88 (2008) pp. 199--217.

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# Thermally driven phase transformation in shape-memory alloys

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Petrov, Adrien

*2010 Mathematics Subject Classification*

- 49J40 74C05 74F05 74M05 74N30

*Keywords*

- Shape-memory materials, doubly nonlinear differential inclusion, rate-independent processes, energetic formulation, temperature-induced phase transformation

*DOI*

*Abstract*

This paper analyzes a model for phase transformation in shape-memory alloys induced by temperature changes and by mechanical loading. We assume that the temperature is prescribed and formulate the problem within the framework of the energetic theory of rate-independent processes. Existence and uniqueness results are proved.

*Appeared in*

- Adv. Math. Sci. Appl., 17 (2007) pp. 667--685.

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# A metric approach to a class of doubly nonlinear evolution equations and applications

*Authors*

- Rossi, Riccarda
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Savaré, Giuseppe

*2010 Mathematics Subject Classification*

- 35K55 49Q20 58E99

*Keywords*

- Doubly nonlinear equations, analysis in metric spaces, existence and approximation results

*DOI*

*Abstract*

This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L^1$ spaces.

*Appeared in*

- Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), VII (2008) pp. 97--169.

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# On the stability of elastic-plastic systems with hardening

*Authors*

- Martins, João A.C.
- Monteiro Marques, Manuel D.P.
- Petrov, Adrien

*2010 Mathematics Subject Classification*

- 34A60 47H06 73H99

*Keywords*

- Differential inclusions, plasticity, hardening, existence, stability

*DOI*

*Abstract*

This paper discusses the stability of quasi-static paths for a continuous elastic-plastic system with hardening in a one-dimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasi-static problems involving elastic-plastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasi-static paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasi-static ones, when (as in Lyapunov stability) the size of initial perturbations is decreased and the rate of application of the forces (which plays the role of the small parameter in singular perturbation problems) is also decreased to zero. The stability of the quasi-static paths of these elastic-plastic systems is the main result proved in the paper.

*Appeared in*

- J. Math. Anal. Appl., 343 (2008) pp. 1007--1021.

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# Quantifying hydrodynamic slip: A comprehensive analysis of dewetting profiles

*Authors*

- Fetzer, Renate
- Münch, Andreas
- Wagner, Barbara
- Rauscher, Markus
- Jacobs, Karin

*2010 Mathematics Subject Classification*

- 76D08 76E17 74A55

*Keywords*

- Polymer melts, slip boundary effects, interfacial and free surface flows, lubrication models, Stokes model

*DOI*

*Abstract*

To characterize non-trivial boundary conditions of a liquid flowing past a solid, the slip length is commonly used as a measure. From the profile of a retracting liquid front as measured, e.g., with atomic force microscopy, the slip length as well as the capillary number can be extracted by the help of the Stokes model for a thin liquid film dewetting from a solid substrate. Specifically, we use a lubrication model derived from the Stokes model for strong slippage and linearize the film profile around the flat, unperturbed film, and, for small slip lengths a Taylor approximation of the linearisation for the full Stokes model. Furthermore, from the capillary number and the knowledge of the liquid front velocity and the surface tension, we can obtain the viscosity of the fluid film. We compare theoretical and experimental results, test the consistency and the validity of the models/approximations, and give an easy-to-follow manual of how they can be used to analyze experiments.

*Appeared in*

- Langmuir, 23 (2007) pp. 10559-10566.

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# On the left tail asymptotics for the limit law of supercritical Galton--Watson processes in the Böttcher case

*Authors*

- Fleischmann, Klaus
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60J80 60F10

*Keywords*

- Lower deviation probabilities, Schröder case, Böttcher case, logarithmic asymptotics, fine asymptotics, precise asymptotics, tiny oscillations

*DOI*

*Abstract*

Under a well-known scaling, supercritical Galton-Watson processes $Z$ converge to a non-degenerate non-negative random limit variable $W.$ We are dealing with the left tail (i.e. lose to the origin) asymptotics of its law. In the Bötcher case (i.e. if always at least two offspring are born), we describe the precise asymptotics exposing tiny oscillations (Theorem 1). Under a reasonable additional assumption, the oscillations disappear (Corollary 2). Also in the Böttcher case, we improve a recent lower deviation probability result by describing the precise asymptotics under a logarithmic scaling (Theorem 3). Under additional assumptions, we even get the fine (i.e. without log-scaling) asymptotics (Theorem 4).

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# Stability and sensitivity of optimization problems with first order stochastic dominance constraints

*Authors*

- Dentcheva, Darinka
- Henrion, René
- Ruszczynski, Andrzej

*2010 Mathematics Subject Classification*

- 90C15 90C34 90C48

*Keywords*

- stochastic programming, stochastic ordering, semi-infinite optimization, chance constraints, Lipschitz stability, metric regularity, directional differentiability

*DOI*

*Abstract*

We analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type. Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the dominance constraints.

*Appeared in*

- SIAM J. Optim., 18 (2007) pp. 322--337.

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# Non-nested multi-grid solvers for mixed divergence-free Scott--Vogelius discretizations

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Matthies, Gunar
- Tobiska, Lutz

*2010 Mathematics Subject Classification*

- 76D05 65F10

*2008 Physics and Astronomy Classification Scheme*

- 47.11.-j

*Keywords*

- Non-Nested Multi-Grid, Stabilized Finite Elements, Navier-Stokes Equations, LBB-Stability

*DOI*

*Abstract*

We apply the general framework developed by John et al. in V. John, P. Knobloch, G. Matthies, L. Tobiska: Non-nested multi-level solvers for finite element discretisations of mixed problems, Computing 2002, to analyze the convergence of multi-level methods for mixed finite element discretizations of the generalized Stokes problem using the Scott-Vogelius element. Having in mind that semi-implicit operator splitting schemes for the Navier-Stokes equations lead to this class of problems, we take symmetric stabilization operators into account. The use of the class of Scott-Vogelius elements seems to be promising since discretely divergence-free functions are pointwise divergence-free. However, to satisfy the Ladyzhenskaya-Babuška-Brezzi stability condition, we have to deal in the multi-grid analysis with non-nested families of meshes which are derived from nested macro element triangulations.

*Appeared in*

- Computing, 83 (2008) pp. 87--107.

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# Weak-convergence methods for Hamiltonian multiscale problems

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35B27 74Q10 37Kxx 37K60 70Hxx

*Keywords*

- Homogenization, infinite-dimensional Hamiltonian and Lagrangian, effective Hamiltonian, wave equation, oscillator chain, Gamma convergence, recovery operators

*DOI*

*Abstract*

We consider Hamiltonian problems depending on a small parameter like in wave equations with rapidly oscillating coefficients or the embedding of an infinite atomic chain into a continuum by letting the atomic distance tend to $0$. For general semilinear Hamiltonian systems we provide abstract convergence results in terms of the existence of a family of joint recovery operators which guarantee that the effective equation is obtained by taking the $Gamma$-limit of the Hamiltonian. The convergence is in the weak sense with respect to the energy norm. Exploiting the well-developed theory of $Gamma$-convergence, we are able to generalize the admissible coefficients for homogenization in the wave equations. Moreover, we treat the passage from a discrete oscillator chain to a wave equation with general $rmL^infty$ coefficients

*Appeared in*

- Discrete Contin. Dyn. Syst., 20 (2008) pp. 53--79.

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# State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions

*Authors*

- Meyer, Christian
- Yousept, Irwin

*2010 Mathematics Subject Classification*

- 35J60 49K20 49M05 65K10

*Keywords*

- Nonlinear optimal control, nonlocal radiation interface conditions, state constraints, first-order necessary conditions, second-order sufficient conditions

*DOI*

*Abstract*

We consider a control- and state-constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. The nonlocal radiation interface condition and the pointwise state-constraints represent the particular features of this problem. To deal with the state-constraints, continuity of the state is shown which allows to derive first-order necessary conditions. Afterwards, we establish second-order sufficient conditions that account for strongly active sets and ensure local optimality in an $L^2$-neighborhood.

*Appeared in*

- SIAM J. Control Optim., 48 (2009) pp. 734--755

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# Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

*Authors*

- Giannoulis, Johannes
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Sparber, Christof

*2010 Mathematics Subject Classification*

- 81Q20 34E13 34E20 35Q55

*Keywords*

- Nonlinear Schrödinger equation, Bloch eigenvalue problem, two scale asymptotics, modulation equations, four-wave interaction

*DOI*

*Abstract*

We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.

*Appeared in*

- J. Differential Equations, 245 (2008) pp. 939--963.

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# Positivity and time behavior of a general linear evolution system, non-local in space and time

*Authors*

- Stephan, Holger
- Khrabustovskyi, Andrii

*2010 Mathematics Subject Classification*

- 35B27 35K60

*Keywords*

- diffusion-reaction systems, positive solutions, maximum principle, homogenization, Riemannian manifold

*DOI*

*Abstract*

We consider a general linear reaction-diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle, and positivity of the solution, and investigate its asymptotic behavior. Moreover, we give an explicite expression of the limit of the solution for large times. In order to obtain these results we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution of a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction-diffusion system what allows us to investigate its properties.

*Appeared in*

- Math. Methods Appl. Sci., 31 (2008) pp. 1809--1834.

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# Large time asymptotics of growth models on space-like paths I: PushASEP

*Authors*

- Borodin, Alexei
- Ferrari, Patrik

*2010 Mathematics Subject Classification*

- 82C22 60K35 15A52

*Keywords*

- Simple exclusion process, space-like universality, KPZ class, Airy processes

*DOI*

*Abstract*

We consider a new interacting particle system on the one-dimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any space-like path are described by the Airy$_1$ and Airy$_2$ processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.

*Appeared in*

- Electron. J. Probab., 13 (2008) pp. 1380--1418.

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# Universality of the REM for dynamics of mean-field spin glasses

*Authors*

- Ben Arous, Gérard
- Bovier, Anton
- Černý, Jiři

*2010 Mathematics Subject Classification*

- 82C44 60K35 60G70

*Keywords*

- aging, universality, spin glasses, SK model, random walk

*DOI*

*Abstract*

We consider a version of a Glauber dynamics for a $p$-spin Sherrington--Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the $N$-dimensional hypercube. We show that, for any $p geq 3$ and any inverse temperature $beta>0$, there exist constants $g_0>0$, such that for all exponential time scales, $exp(gamma N)$, with $gleq g_0$, the properly rescaled emphclock process (time-change process), converges to an $a$-stable subordinator where $a=g/b^2<1$. Moreover, the dynamics exhibits aging at these time scales with time-time correlation function converging to the arcsine law of this hbox$alpha$-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system), the dynamics of $p$-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case $p=2$) seems to belong to a different universality class.

*Appeared in*

- J. Stat. Mech. Theory Exp., 4 (2008) pp. L04003/1--L04003/8.

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# Temporal decorrelation for branching random walks with state dependent branching rate

*Authors*

- Birkner, Matthias

*2010 Mathematics Subject Classification*

- 60K35

*Keywords*

- State dependent branching, temporal decorrelation

*DOI*

*Abstract*

We consider branching random walks in $d ge 3$ with a Lipschitz branching rate function and show that the system, starting either in a Poisson field or in equilibrium, decorrelates over long time horizons, and employ this to obtain an ergodic theorem. We use coupling and a stochastic representation of the Palm distribution.

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# Energy estimates for continuous and discretized electro-reaction-diffusion systems

*Authors*

- Glitzky, Annegret
- Gärtner, Klaus

*2010 Mathematics Subject Classification*

- 35B40 35K57 78A35 35R05 65M12

*Keywords*

- Reaction-diffusion systems, drift-diffusion processes, motion of charged particles, energy estimates, thermodynamic equilibria, asymptotic behaviour, time and space discretization

*DOI*

*Abstract*

We consider electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistic relations. We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly. The same properties are shown for an implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electro-reaction-diffusion system which is dissipative (the free energy decays monotonously). On a fixed grid we use for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species.

*Appeared in*

- Nonlinear Anal., 70 (2009) pp. 788--805.

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# Scaling limit and aging for directed trap models

*Authors*

- Zindy, Olivier

*2010 Mathematics Subject Classification*

- 60K37 60G50 60G52 60F17 82D30

*Keywords*

- Directed trap model, random walk, scaling limit, subordinator, aging

*DOI*

*Abstract*

We consider one-dimensional directed trap models and suppose that the trapping times are heavy-tailed. We obtain the inverse of a stable subordinator as scaling limit and prove an aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality described by Ben Arous and Černý for a large class of graphs.

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# On the approximation of the limit cycles function

*Authors*

- Cherkas, Leonid
- Grin, Alexander
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 34C05 34C07 65L99

*Keywords*

- Family of limit cycles, multiple limit cycle, Liénard system

*DOI*

*Abstract*

We consider planar vector fields depending on a real parameter. It is assumed that this vector field has a family of limit cycles which can be described by means of the limit cycles function $l$. We prove a relationship between the multiplicity of a limit cycle of this family and the order of a zero of the limit cycles function. Moreover, we present a procedure to approximate $l(x)$, which is based on the Newton scheme applied to the Poincaré function and represents a continuation method. Finally, we demonstrate the effectiveness of the proposed procedure by means of a Liénard system. The obtained result supports a conjecture by Lins, de Melo and Pugh.

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# Regular polynomial interpolation and approximation of global solutions of linear partial differential equations

*Authors*

- Kampen, Jörg

*2010 Mathematics Subject Classification*

- 65D05 35G05

*Keywords*

- extended Newtonian interpolation, linear systems of partial differential equations, error estimates

*DOI*

*Abstract*

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the "limit" of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.

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# Convergence of a finite volume scheme to the eigenvalues of a Schrödinger operator

*Authors*

- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Eymard, Robert
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434

*2010 Mathematics Subject Classification*

- 65N25 65N30 81Q10

*Keywords*

- Schrödinger operator, eigenvalues, finite volume schemes.

*DOI*

*Abstract*

We consider the approximation of a Schrödinger eigenvalue problem arising from the modeling of semiconductor nanostructures by a finite volume method in a bounded domain $OmegasubsetR^d$. In order to prove its convergence, a framework for finite dimensional approximations to inner products in the Sobolev space $H^1_0(Omega)$ is introduced which allows to apply well known results from spectral approximation theory. This approach is used to obtain convergence results for a classical finite volume scheme for isotropic problems based on two point fluxes, and for a finite volume scheme for anisotropic problems based on the consistent reconstruction of nodal fluxes. In both cases, for two- and three-dimensional domains we are able to prove first order convergence of the eigenvalues if the corresponding eigenfunctions belong to $H^2(Omega)$. The construction of admissible meshes for finite volume schemes using the Delaunay-Voronoï method is discussed. As numerical examples, a number of one-, two- and three-dimensional problems relevant to the modeling of semiconductor nanostructures is presented. In order to obtain analytical eigenvalues for these problems, a matching approach is used. To these eigenvalues, and to recently published highly accurate eigenvalues for the Laplacian in the L-shape domain, the results of the implemented numerical method are compared. In general, for piecewise $H^2$ regular eigenfunctions, second order convergence is observed experimentally.

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# Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP

*Authors*

- Borodin, Alexei
- Ferrari, Patrik
- Sasamoto, Tomohiro

*2010 Mathematics Subject Classification*

- 82C22 60K35 15A52

*Keywords*

- Simple exclusion process, space-like universality, KPZ class, Airy processes

*DOI*

*Abstract*

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy$_1$ process. This result holds more generally for the observation points located along any space-like path in the space-time plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update.

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# Higher integrability of the Lorentz force for weak solutions to Maxwell's equations in complex geometries

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35D10 35J55 35Q60

*Keywords*

- Maxwell's equations, natural interface conditions, Lorentz force, higher integrability

*DOI*

*Abstract*

We consider the stationary Maxwell system in a domain filled with different materials. The magnetic permeability being only piecewise smooth, we have to take into account the natural interface conditions for the electromagnetic fields. We present two sets of hypotheses under which we can prove the existence of weak solutions to the Maxwell system such that the Lorentz force jxB is integrable to a power larger than 6/5. This property is important for the investigation of problems in magnetohydrodynamics, with many industrial applications such as crystal growth.

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# Transition between Airy$_1$ and Airy$_2$ processes and TASEP fluctuations

*Authors*

- Borodin, Alexei
- Ferrari, Patrik
- Sasamoto, Tomohiro

*2010 Mathematics Subject Classification*

- 82C22 60K35 15A52

*Keywords*

- Simple exclusion process, universality, KPZ class, Airy process, random matrices

*DOI*

*Abstract*

We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. F or large time $t$, one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy$_1$ and Airy$_2$ processes, whose one-point distributions are the GOE and GUE Tracy-Widom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its one-point distribution is a new interpolati on between GOE and GUE edge distributions.

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# Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

*Authors*

- Enriquez, N.
- Sabot, C.
- Zindy, O.

*2010 Mathematics Subject Classification*

- 60K37 60G50 60J45 82D30

*Keywords*

- Random walk in random environment; aging; quenched, localization

*DOI*

*Abstract*

We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of 'valleys' of height log $t$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.

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# Destabilization patterns in large regular networks

*Authors*

- Yanchuk, Serhiy
- Wolfrum, Matthias

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt, 02.30.Oz, 89.75.Fb, 82.40.Ck

*Keywords*

- Networks, coupled oscillators, bifurcations, Eckhaus instability

*DOI*

*Abstract*

We describe a generic mechanism for the destabilization in large regular networks of identical coupled oscillators. Based on a reduction method for the spectral problem, we first present a criterion for this type of destabilization. Then, we investigate the related bifurcation scenario, showing the existence of a large number of coexisting periodic solutions with different frequencies, spatial patterns, and stability properties. Even for unidirectional coupling this can be understood in analogy to the well-known Eckhaus scenario for diffusive systems.

*Appeared in*

- Phys. Rev. E, 77 (2008) pp. 026212/1-026212/7.

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# Stochastic simulation method for a 2D elasticity problem with random loads

*Authors*

- Sabelfeld, Karl
- Shalimova, Irina
- Levykin, Alexander I.

*2010 Mathematics Subject Classification*

- 65C05 65C20 65Z05

*Keywords*

- Isotropic Random Fields, Spectral Tensor, Poisson integral formula, Random Walk on Fixed Spheres, Lamé equation, Successive Over Relaxation Method, Transverse and Longitudinal Correlations

*DOI*

*Abstract*

We develop a stochastic simulation method for a numerical solution of the Lamé equation with random loads. To treat the general case of large intensity of random loads, we use the Random Walk on Fixed Spheres (RWFS) method described in our paper citesab-lev-shal-2006. The vector random field of loads which stands in the right-hand-side of the system of elasticity equations is simulated by the Randomization Spectral method presented in citesab-1991 and recently revised and generalized in citekurb-sab-2006. Comparative analysis of RWFS method and an alternative direct evaluation of the correlation tensor of the solution is made. We derive also a closed boundary value problem for the correlation tensor of the solution which is applicable in the case of inhomogeneous random loads. Calculations of the longitudinal and transverse correlations are presented for a domain which is a union of two arbitrarily overlapped discs. We also discuss a possibility to solve an inverse problem of determination of the elastic constants from the known longitudinal and transverse correlations of the loads.

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# On a thermomechanical model of phase transitions in steel

*Authors*

- Chełminski, Krzysztof
- Hömberg, Dietmar
- Kern, Daniela

*2010 Mathematics Subject Classification*

- 35Q72 74A15 74F05

*Keywords*

- Heat treatment, coupled partial differential equation, existence and uniqueness, phase transitions, linear thermoelasticity, thermomechanics, distortion

*DOI*

*Abstract*

We investigate a thermomechanical model of phase transitions in steel. The strain is assumed to be additively decomposed into an elastic and a thermal part as well as a contribution from transformation induced plasticity. The resulting model can be viewed as an extension of quasistatic linear thermoelasticity. We prove existence of a unique solution and conclude with some numerical simulations.

*Appeared in*

- Adv. Math. Sci. Appl., 18 (2008), pp. 119--140

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# Convexity of chance constraints with independent random variables

*Authors*

- Henrion, René
- Strugarek, Cyrille

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Chance constraints, probabilistic constraints, stochastic programming, convexity, random matrix

*DOI*

*Abstract*

We investigate the convexity of chance constraints with independent random variables. It will be shown, how concavity properties of the mapping related to the decision vector have to be combined with a suitable property of decrease for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels. It turns out that the required decrease can be verified for most prominent density functions. The results are applied then, to derive convexity of linear chance constraints with normally distributed stochastic coefficients when assuming independence of the rows of the coefficient matrix.

*Appeared in*

- Computational Optimization and Applications 41 (2008) 263-276.

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# Elliptic model problems including mixed boundary conditions and material heterogeneities

*Authors*

- Haller-Dintelmann, Robert
- Kaiser, Hans-Christoph
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35B65 35J25 35R05

*Keywords*

- Elliptic transmission problems, mixed boundary problems, $W^1,p$ regularity

*DOI*

*Abstract*

We present model problems in three dimensions, where the operator $-nabla cdot mu nabla$ maps the Sobolev space $W^1,p_Gamma(Omega)$ isomorphically onto $W^-1,p_Gamma(Omega)$ for a $p>3$. The emphasis is here on the case where different boundary conditions meet material heterogeneities.

*Appeared in*

- J. Math. Pures Appl., 89 (2008) pp. 25--48.

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# Phase transition and hysteresis in a rechargeable lithium battery

*Authors*

- Dreyer, Wolfgang
- Gaberšček, Miran
- Jamnik, Janko

*2010 Mathematics Subject Classification*

- 74A15 74A50 74F25 74N20 74N25 74N30

*2008 Physics and Astronomy Classification Scheme*

- 81.30.Mh 82.56.Lz 82.45.Fk 82.60.Qr

*Keywords*

- Thermodynamics, Structured surfaces and interfaces, Solid-phase precipitation,, coexistent phases, Chemical and reactive effects, Dynamics of phase, boundaries, Transformations involving diffusion, Problems involving, hysteresis

*DOI*

*Abstract*

We represent a model which describes the evolution of a phase transition that occurs in some part of a rechargeable lithium battery during the process of charging/discharging. The model is capable to simulate the hysteretic behavior of the voltage - charge characteristics. During discharging of the battery, the interstitial lattice sites of a small crystalline host system are filled up with lithium atoms and these are released again during charging. We show within the context of a sharp interface model that two mechanical phenomena go along with a phase transition that appears in the host system during supply and removal of lithium. At first the lithium atoms need more space than it is available by the interstitial lattice sites, which leads to a maximal relative change of the crystal volume of about $6%$. Furthermore there is an interface between two adjacent phases that has very large curvature of the order of magnitude 100 m, which evoke here a discontinuity of the normal component of the stress. In order to simulate the dynamics of the phase transitions and in particular the observed hysteresis we establish a new initial and boundary value problem for a nonlinear PDE system that can be reduced in some limiting case to an ODE system.

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# Measure-valued diffusions, general coalescents and population genetic inference

*Authors*

- Birkner, Matthias
- Blath, Jochen

*2010 Mathematics Subject Classification*

- 92D15 60J70 60G09 60G57

*Keywords*

- Generalised Fleming-Viot process, $Lambda$-coalescent, lookdown construction, mathematical population genetics, Monte-Carlo simulation

*DOI*

*Abstract*

We review recent progress in the understanding of the interplay between population models, measure-valued diffusions, general coalescent processes and inference methods for evolutionary parameters in population genetics. Along the way, we will discuss the powerful and intuitive (modified) lookdown construction of Donnelly and Kurtz, Pitman's and Sagitov's $Lambda$-coalescents as well as recursions and Monte Carlo schemes for likelihood-based inference of evolutionary parameters based on observed genetic types

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# Rescaled stable generalised Fleming--Viot processes: Flickering random measures

*Authors*

- Birkner, Matthias
- Blath, Jochen

*2010 Mathematics Subject Classification*

- 60G57 60G17

*Keywords*

- Generalised Fleming-Viot process, measure-valued diffusion,, tightness, Skorohod topology, lookdown construction, wandering random measure, path properties,

*DOI*

*Abstract*

We show how Donnelly and Kurtz' (modified) lookdown construction for measure-valued processes can be used to analyse the longterm- and scaling properties of spatially stable generalised $Lambda$-Fleming Viot processes, exhibiting a rare ``natural'' example of a scaling family converging in f.d.d. sense, but not in any of Skorohod's topologies on path space. This completes results of Fleischmann and Wachtel (2004) about the spatial Neveu process and complements results of Dawson and Hochberg (1982) about the classical Fleming Viot process. The lookdown construction provides an elegant machinery and clear intuition to describe the path properties of the family in terms of a ``flicker effect'', clarifying ``what can go wrong.''

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# Chaotic soliton walk in periodically modulated media

*Authors*

- Turaev, Dmitry
- Radziunas, Mindaugas
- Vladimirov, Andrei G.

*2010 Mathematics Subject Classification*

- 78A60 37K45

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Sf, 05.45.-a, 42.65.Tg

*Keywords*

- nonlinear Shrödinger equation, soliton, chaotic motion

*DOI*

*Abstract*

We show that a weak transverse spatial modulation in (2+1) nonlinear Schrödinger equation with saturable nonlinearity can result in nontrivial dynamics of radially symmetric solitons. In particular, in the case of hexagonal profile of the modulation the soliton moves chaotically.

*Appeared in*

- Phys. Rev. E, 77 (2008) pp. 06520/1--06520/4.

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# Ostwald ripening of faceted two-dimensional islands

*Authors*

- Kaganer, Vladimir
- Braun, Wolfgang
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20

*2008 Physics and Astronomy Classification Scheme*

- 81.10.Aj,05.10.Ln,68.43.Jk,81.15.-z

*Keywords*

- Ostwald ripening, Lifshitz--Slyozov--Wagner theory, Becker--Döring equations, faceted crystalline droplets, Gibbs--Thomson formula, kinetic Monte Carlo simulations

*DOI*

*Abstract*

We study Ostwald ripening of two-dimensional adatom and advacancy islands on a crystal surface by means of kinetic Monte Carlo simulations. At large bond energies the islands are square-shaped, which qualitatively changes the coarsening kinetics. The Gibbs--Thomson chemical potential is violated: the coarsening proceeds through a sequence of `magic' sizes corresponding to square or rectangular islands. The coarsening becomes attachment-limited, but Wagner's asymptotic law is reached only after a very long transient time. The unusual coarsening kinetics obtained in the Monte Carlo simulations are well described by the Becker--Döring equations of nucleation kinetics. These equations can be applied to a wide range of coarsening problems.

*Appeared in*

- Phys. Rev. B., 76 (2007) pp. 075415 (11 pages).

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# Glauber dynamics on hyperbolic graphs: Boundary conditions and mixing time

*Authors*

- Bianchi, Alessandra

ORCID: 0000-0003-1566-6000

*2010 Mathematics Subject Classification*

- 82B20 82B43 82C80 60K35

*Keywords*

- Ising model, Glauber dynamics, hyperbolic graphs, Dirichlet form, spectral gap, mixing time

*DOI*

*Abstract*

We study a continuous time Glauber dynamics reversible with respect to the Ising model on hyperbolic graphs and analyze the effect of boundary conditions on the mixing time. Specifically, we consider the dynamics on an $n$-vertex ball of the hyperbolic graph $H(v,s)$, where $v$ is the number of neighbors of each vertex and $s$ is the number of sides of each face, conditioned on having $(+)$-boundary. If $v>4$, $s>3$ and for all low enough temperatures (phase coexistence region) we prove that the spectral gap of this dynamics is bounded below by a constant independent of $n$. This implies that the mixing time grows at most linearly in $n$, in contrast to the free boundary case where it is polynomial with exponent growing with the inverse temperature $b$. Such a result extends to hyperbolic graphs the work done by Martinelli, Sinclair and Weitz for the analogous system on regular tree graphs, and provides a further example of influence of the boundary condition on the mixing time.

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# Mathematical results on existence for viscoelastodynamic problems with unilateral constraints

*Authors*

- Petrov, Adrien
- Schatzman, M.

*2010 Mathematics Subject Classification*

- 35L85 49J40 73D99 73V25

*Keywords*

- Viscoelasticity, Signorini conditions, penalty method, traces, variational inequality, convolution

*DOI*

*Abstract*

We study a damped wave equation and the evolution of a Kelvin-Voigt material, both problems have unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, both problems possess a weak solution which is obtained as the limit of a sequence of penalized problems; the functional properties of all the traces are precisely identified through Fourier analysis, and this enables us to infer the existence of a strong solution.

*Appeared in*

- SIAM J. Math. Anal., 40 (2009) pp. 1882--1904.

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# The equilibria of vapour-liquid systems revisited

*Authors*

- Dreyer, Wolfgang
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 82B26 49Q20

*2008 Physics and Astronomy Classification Scheme*

- 64.70.Fx

*Keywords*

- two-phase fluid, mechanical and phase equilibria, surface tension, mean curvature, contact angle

*DOI*

*Abstract*

We study equilibrium conditions of liquid-vapour phase transitions for a single substance at constant temperature. The phase transitions are modelled by a classical sharp interface model with boundary contact energy. We revisit this old problem mainly for the following reasons. Equilibria in a two-phase system can be established either under fixed external pressure or under fixed total volume. These two different settings lead to distinct equilibria, a fact that is usually ignored in the literature. In nature and in most technical processes, the approach of a two-phase system to equilibrium runs at constant pressure, whereas mathematicians prefer to study processes in constant domains, i.e. at constant volume. Furthermore, in the literature the sharp interface of the liquid and the vapour phase is usually described by a surface with high symmetry like a plane interface or a radially symmetric interface which has the shape of the boundary of a ball. In this paper we establish equilibrium conditions for pressure control as well as for volume control with arbitrary shapes of the interface. The results are derived by methods of differential geometry. Further, the common features and differences of pressure and volume control are worked out for some simple cases.

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# Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion

*Authors*

- Yulin, Alexey
- Skryabin, Dmitry
- Vladimirov, Andrei G.

*2010 Mathematics Subject Classification*

- 78A60 37K45 37K40

*2008 Physics and Astronomy Classification Scheme*

- 2.65.-k,42.65.Tg,42.65.Sf,42.81.Qb,42.81.Dp

*Keywords*

- Discrete solitons, modulational instability, waveguide arrays

*DOI*

*Abstract*

We study temporal modulational instability of spatial discrete solitons in waveguide arrays with group velocity dispersion (GVD). For normal GVD we report existence of the strong 'neck'-type instability specific for the discrete solitons. For anomalous GVD the instability leads to formation of the mixed discrete-continuous spatio-temporal quasi-solitons. Feasibility of experimental observation of these effects in the arrays of silicon-on-insulator waveguides is discussed.

*Appeared in*

- Optics Express, 14 (2006) pp. 12347--12352.

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# On the inviscid limit of a model for crack propagation

*Authors*

- Knees, Dorothee
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Zanini, Chiara

*2010 Mathematics Subject Classification*

- 49J40 74R10 49L25 35K90 74B20 74G65

*Keywords*

- rate-indepentent problems, energetic formulation, energy release rate, Griffith fracture criterion, vanishing viscosity method

*DOI*

*Abstract*

We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.

*Appeared in*

- Math. Models Methods Appl. Sci., 18 (2008) pp. 1529--1569.

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# Non-Raman redshift by pulse splitting in the normal dispersion regime

*Authors*

- Demircan, Ayhan
- Kroh, Marcel
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 35Q55 35Q60 78A60

*Keywords*

- Nonlinear Schrödinger equation, optical Fiber.

*DOI*

*Abstract*

While usually the generation of a Stokes component is attributed to Raman scattering, we present here experimentally and numerically a more fundamental mechanism which can be explained by the nonlinear Schrödinger equation alone. It can be employed to excite new frequency components on the red side, by using pulse splitting in the normal dispersion regime.

*Appeared in*

- Proceedings of the 7th International Conference on Numerical Simulation of Optoelectronic Devices, NUSOD '07, 24--27 September 2007, J. Piprek, D. Prather, eds., IEEE, Piscataway, NJ, 2007, pp. 99--100

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