# Spectral delay algebraic equation approach to broad area laser diodes

*Authors*

- Pérez-Serrano, Antonio
- Javaloyes, Julien
- Balle, Salvador

*2010 Mathematics Subject Classification*

- 78A60 65P99 65M70

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.Jf 42.60.Mi 42.65.St

*Keywords*

- Semiconductor lasers, Broad area laser diode (BALD), Travelling wave model (TWM)

*DOI*

*Appeared in*

- IEEE J. Select. Topics Quantum Electron., 19 (2013) pp. 1502808.

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# Compact high order finite difference schemes for linear Schrödinger problems on non-uniform meshes

*Authors*

- Radziunas, Mindaugas
- Čiegis, Raimondas
- Mirinavičius, Aleksas

*2010 Mathematics Subject Classification*

- 65M06 65M20 65M99 35Q60 65M12

*Keywords*

- finite-difference schemes, high oder approximation, compact scheme, Schrödinger equation, transparent boundary conditions

*DOI*

*Appeared in*

- Int. J. Numer. Anal. Model., 11 (2014) pp. 303--314.

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# Direct and inverse elastic scattering problems for diffraction gratings

*Authors*

- Elschner, Johannes
- Hu, Guanghui

*2010 Mathematics Subject Classification*

- 74J20 74B05 35B27 35R30 35Q93

*Keywords*

- diffraction gratings, elastic waves, variational formulation, inverse scattering, uniqueness, optimization

*DOI*

*Appeared in*

- J. Elschner, G. Hu, Direct and inverse elastic scattering problems for diffraction gratings, in: Direct and Inverse Problems in Wave Propagation and Applications, I.G. Graham, U. Langer, J.M. Melenk, M. Sini, eds., vol. 14 of Radon Series on Computational and Applied Mathematics, De Gruyter, Berlin, 2014, pp. 101--134

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# Numerical methods for the simulation of an aggregation-driven droplet size distribution

*Authors*

- Bordás, Robert
- John, Volker

ORCID: 0000-0002-2711-4409 - Schmeyer, Ellen
- Thévenin, Dominique

*2010 Mathematics Subject Classification*

- 76F65 76T10

*Keywords*

- population balance systems, TVD-ENO scheme, group FEM-FCT method, numerical computation of coalescence integrals

*DOI*

*Abstract*

A droplet size distribution in a turbulent flow field is considered and modeled by means of a population balance system. This paper studies different numerical methods for the 4D population balance equation and their impact on an output of interest, the time-space-averaged droplet size distribution at the outlet which is known from experiments. These methods include different interpolations of the experimental data at the inlet, various discretizations in time and space, and different schemes for computing the aggregation integrals. It will be shown that notable changes in the output of interest might occur. In addition, the efficiency of the studied methods is discussed.

*Appeared in*

- Theor. Comput. Fluid Dyn., 27 (2013) pp. 253--271.

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# Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints

*Authors*

- Knees, Dorothee
- Schröder, Andreas

*2010 Mathematics Subject Classification*

- 35B65 35J88 74A45, 74M15, 65N30, 65N12

*Keywords*

- global spatial regularity, cracks with selfcontact, Signorini contact, difference quotients, Tresca friction, Finite Elements, a priori error analysis

*DOI*

*Abstract*

A global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D) the displacement fields are B^{ 3/2 }_{ 2,∞} regular. The proof relies on a difference quotient argument for the directions tangential to the crack. In order to obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. Based on Falk's approximation Theorem for variational inequalities, convergence rates for FE-discretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results.

*Appeared in*

- Math. Methods Appl. Sci., 35 (2012) pp. 1859--1884.

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# Multistability of twisted states in non-locally coupled Kuramoto-type models

*Authors*

- Girnyk, Taras
- Hasler, Martin
- Maistrenko, Yuriy

*2010 Mathematics Subject Classification*

- 37C75

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt 02.60.-x

*Keywords*

- Chaos, Nonlinear dynamical systems, Numerical analysis, Oscillators

*DOI*

*Abstract*

A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without limitation of the generality the frequency of the free-running oscillators can be set to zero. The resulting system is of gradient type and therefore all its solutions converge to an equilibrium point. All so-called q-twisted states, where the phase difference between neighboring oscillators on the ring is 2 pi q/N are equilibrium points, where q is an integer. Their stability in the limit N -> inf. is discussed along the line of1. In addition we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N. Note that for smaller N, the same q-twisted states may become unstable and other q-twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2 pi q/N in one sector of the ring, -2 pi q/N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multi-twisted states grows exponentially as N -> inf. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q-twisted states are then fixed points and the multi-twisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N -> 1.

*Appeared in*

- CHAOS Vol. 22, 2012, pp. 013114/1--013114/10

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# A robust SUPG norm a posteriori error estimator for the SUPG finite element approximation of stationary convection-diffusion equations

*Authors*

- John, Volker

ORCID: 0000-0002-2711-4409 - Novo, Julia

*2010 Mathematics Subject Classification*

- 65N30

*Keywords*

- stationary convection-diffusion equations, SUPG finite element method, error in SUPG norm, a posteriori error estimator, adaptive grid refinement

*DOI*

*Abstract*

A robust residual-based a posteriori estimator is proposed for the SUPG finite element method applied to stationary convection-diffusion-reaction equations. The error in the natural SUPG norm is estimated. The main concern of this paper is the consideration of the convection-dominated regime. A global upper bound and a local lower bound for the error are derived, where the global upper estimate relies on some hypotheses. Numerical studies demonstrate the robustness of the estimator and the fulfillment of the hypotheses. A comparison to other residual-based estimators with respect to the adaptive grid refinement is also provided.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 255 (2013) pp. 289--305.

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# Global existence for a strongly coupled Cahn--Hilliard system with viscosity

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Podio-Guidugli, Paolo
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74A15 35K61 35A05 35B40

*Keywords*

- viscous Cahn-Hilliard system, phase field model, nonlinear conductivity, existence of solution

*DOI*

*Abstract*

An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in [CGPS11]. Both systems conform to the general theory developed in [Pod06]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter $rho$ and the chemical potential $mu$. In the system studied in this note, a phase-field equation in $rho$ fairly more general than in [CGPS11] is coupled with a highly nonlinear diffusion equation for $mu$, in which the conductivity coefficient is allowed to depend nonlinearly on both variables.

*Appeared in*

- Boll. Unione Mat. Ital. (9), 5 (2012) pp. 495--513.

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# A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations

*Authors*

- Barrenechea, Gabriel R.
- John, Volker

ORCID: 0000-0002-2711-4409 - Knobloch, Petr

*2010 Mathematics Subject Classification*

- 65N30 65M60

*Keywords*

- finite element method, local projection stabilization, crosswind diffusion, convection-diffusion-reaction equation, well posedness, time dependent problem, stability, error estimates

*DOI*

*Abstract*

An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.

*Appeared in*

- ESAIM Math. Model. Numer. Anal., 47 (2013) pp. 1335--1366.

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# Scattering of time-harmonic electromagnetic plane waves by perfectly conducting diffraction gratings

*Authors*

- Hu, Guanghui
- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 78A45 35J20 35R30 78A46

*Keywords*

- Electromagnetic scattering, diffraction gratings, variational approach, mortar technique, non-uniqueness

*DOI*

*Abstract*

Consider scattering of time-harmonic electromagnetic plane waves by a doubly periodic surface in $R^3$. The medium above the surface is supposed to be homogeneous and isotropic with a constant dielectric coefficient, while below is a perfectly conducting material. This paper is concerned with the existence of quasiperiodic solutions for any frequency of incidence. Based on an equivalent variational formulation established by the mortar technique of Nitsche, we verify the existence of solutions for a broad class of incident waves including plane waves, under the assumption that the grating profile is a Lipschitz biperiodic surface. Our solvability result covers the resonance case where a Rayleigh frequency is allowed. Non-uniqueness examples are also presented in the resonance case and the TE or TM polarization case for classical gratings.

*Appeared in*

- IMA J. Appl. Math., 80 (2015) pp. 508--532.

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# Global existence and uniqueness for a singular/degenerate Cahn--Hilliard system with viscosity

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Podio-Guidugli, Paolo
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K61 35A05 74A15

*Keywords*

- phase field model, nonlinear laws, existence of solutions, new uniqueness proof

*DOI*

*Abstract*

Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic [19]; in the balance equations of microforces and microenergy, the two unknowns are the order parameter $rho$ and the chemical potential $mu$. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phase-field equation for $rho$ is coupled with a genuinely nonlinear diffusion equation for $mu$. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of costant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables.

*Appeared in*

- J. Differential Equations, 254 (2013) pp. 4217--4244.

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# On the parameter choice in grad-div stabilization for incompressible flow problems

*Authors*

- Jenkins, Eleanor
- John, Volker

ORCID: 0000-0002-2711-4409 - Linke, Alexander

ORCID: 0000-0002-0165-2698 - Rebholz, Leo G.

*2010 Mathematics Subject Classification*

- 35Q30 76M10 65L60

*Keywords*

- incompressible Navier-Stokes equations, mixed finite elements, grad-div stabilization, error estimates, parameter choice

*DOI*

*Abstract*

Grad-div stabilization has been proved to be a very useful tool in discretizations of incompressible flow problems. Standard error analysis for inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be O(1). This paper revisits this choice for the Stokes equations on the basis of minimizing the $H^1$ error of the velocity and the $L^2$ error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. Depending on the situation, the optimal stabilization parameter might range from being very small to very large. The analytic results are supported by numerical examples.

*Appeared in*

- Adv. Comput. Math., 40 (2014) pp. 491--516.

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# Hybrid mode-locking in edge-emitting semiconductor lasers: Simulations, analysis and experiments

*Authors*

- Arkhipov, Rostislav
- Pimenov, Alexander
- Radziunas, Mindaugas
- Vladimirov, Andrei G.
- Arsenjević, Dejan
- Rachinskii, Dmitrii
- Schmeckebier, Holger
- Bimberg, Dieter

*2010 Mathematics Subject Classification*

- 78A60 78M35 78-05

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc 42.55.Px 42.65.Sf 85.30.De

*Keywords*

- hybrid mode-locking, semiconductor laser, saturable absorber, voltage modulation, delay differential equations, asymptotic analysis

*DOI*

*Abstract*

Hybrid mode-locking in a two section edge-emitting semiconductor laser is studied numerically and analytically using a set of three delay differential equations. In this set the external RF signal applied to the saturable absorber section is modeled by modulation of the carrier relaxation rate in this section. Estimation of the locking range where the pulse repetition frequency is synchronized with the frequency of the external modulation is performed numerically and the effect of the modulation shape and amplitude on this range is investigated. Asymptotic analysis of the dependence of the locking range width on the laser parameters is carried out in the limit of small signal modulation. Our numerical simulations indicate that hybrid mode-locking can be also achieved in the cases when the frequency of the external modulation is approximately twice larger and twice smaller than the pulse repetition frequency of the free running passively mode-locked laser fP . Finally, we provide an experimental demonstration of hybrid mode-locking in a 20 GHz quantum-dot laser with the modulation frequency of the reverse bias applied to the absorber section close to fP =2.

*Appeared in*

- IEEE J. Select. Topics Quantum Electron., 19 (2013) pp. 1100208/1--1100208/6.

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# Representation of hysteresis operators for vector-valued continuous monotaffine input functions by functions on strings

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603

*2010 Mathematics Subject Classification*

- 47J40

*Keywords*

- hysteresis Operators, vectorial hysteresis, string representation

*DOI*

*Abstract*

In Brokate-Sprekels-1996, it was shown that scalar-valued hysteresis operators for scalar-valued continuous piecewise monotone input functions can be uniquely represented by functionals defined on the set of all finite alternating strings of real numbers. Using this representation, various properties of these hysteresis operators were investigated. In this work, it is shown that a similar representation result can be derived for hysteresis operators dealing with inputs in a general topological linear vector space. Introducing a new class of functions, the so-called emphmonotaffine functions, which can be considered as a vector generalization of monotone scalar functions, and the convexity triple free strings on a vector space as a generalization of the alternating strings allows to formulate the corresponding representation result. As an example for the application of the representation result, a vectorial formulation of the second and third Madelung rule are discussed.

*Appeared in*

- Adv. Math. Sci. Appl., 22 (2012) pp. 471-500, with new title: Representation of hysteresis operators acting on vector-valued monotaffine functions

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# Some inverse problems arising from elastic scattering by rigid obstacles

*Authors*

- Hu, Guanghui
- Kirsch, Andreas
- Sini, Mourad

*2010 Mathematics Subject Classification*

- 35R30 74B05 78A45

*Keywords*

- Linear elasticity, inverse scattering, factorization method, uniqueness

*DOI*

*Abstract*

In the first part, it is proved that a $C^2$-regular rigid scatterer in $R^3$ can be uniquely identified by the shear part (i.e. S-part) of the far-field pattern corresponding to all incident shear waves at any fixed frequency. The proof is short and it is based on a kind of decoupling of the S-part of scattered wave from its pressure part (i.e. P-part) on the boundary of the scatterer. Moreover, uniqueness using the S-part of the far-field pattern corresponding to only one incident plane shear wave holds for a ball or a convex Lipschitz polyhedron. In the second part, we adapt the factorization method to recover the shape of a rigid body from the scattered S-waves (resp. P-waves) corresponding to all incident plane shear (resp. pressure) waves. Numerical examples illustrate the accuracy of our reconstruction in $R^2$. In particular, the factorization method also leads to some uniqueness results for all frequencies excluding possibly a discrete set.

*Appeared in*

- Inverse Problems, 29 (2013) pp. 015009/1--015009/21.

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# A diffuse interface model for quasi-incompressible flows: Sharp interface limits and numerics

*Authors*

- Aki, Gonca
- Daube, Johannes
- Dreyer, Wolfgang
- Giesselmann, Jan
- Kränkel, Mirko
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 35C20 35R35

*Keywords*

- Liquid vapor flow, phase transition, asymptotic analysis, sharp interface limit, free boundary problem

*DOI*

*Abstract*

In this contribution, we investigate a diffuse interface model for quasi-incompressible flows. We determine corresponding sharp interface limits of two different scalings. The sharp interface limit is deduced by matched asymptotic expansions of the fields in powers of the interface. In particular, we study solutions of the derived system of inner equations and discuss the results within the general setting of jump conditions for sharp interface models. Furthermore, we treat, as a subproblem, the convective Cahn-Hilliard equation numerically by a Local Discontinuous Galerkin scheme.

*Appeared in*

- ESAIM Proc., 38 (2012) pp. 54--77.

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# Derivation of an effective damage model with evolving micro-structure

*Authors*

- Hanke, Hauke
- Knees, Dorothee

*2010 Mathematics Subject Classification*

- 74A45 74C05 74R05 74Q15 76M50

*Keywords*

- Two-scale convergence, folding and unfolding operator, rate-independent damage evolution, Γ-convergence, irreversibility, broken Sobolev function

*DOI*

*Abstract*

In this paper rate-independent damage models for elastic materials are considered. The aim is the derivation of an effective damage model by investigating the limit process of damage models with evolving micro-defects. In all presented models the damage is modeled via a unidirectional change of the material tensor. With progressing time this tensor is only allowed to decrease in the sense of quadratic forms. The magnitude of the damage is given by comparing the actual material tensor with two reference configurations, denoting completely undamaged material and maximally damaged material (no complete damage). The starting point is a microscopic model, where the underlying micro-defects, describing the distribution of either undamaged material or maximally damaged material (but nothing in between), are of a given time-dependent shape but of different sizes. Scaling the microstructure of this microscopic model by a parameter ε>0 the limit passage ε→0 is preformed via two-scale convergence techniques. Therefore, a regularization approach for piecewise constant functions is introduced to guaranty enough regularity for identifying the limit model. In the limit model the material tensor depends on a damage variable z:[0,T]→ W^{1,p}(Ω) taking values between 0 and 1 such that, in contrast to the microscopic model, some kind of intermediate damage for a material point x∈Ω is possible. Moreover, this damage variable is connected to the material tensor via an explicit formula, namely, a unit cell formula known from classical homogenization results.

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# Passing from bulk to bulk/surface evolution in the Allen--Cahn equation

*Authors*

- Liero, Matthias

ORCID: 0000-0002-0963-2915

*2010 Mathematics Subject Classification*

- 35K55 35K20 82C26

*Keywords*

- Gradient flow, Allen-Cahn equation, dynamic boundary condition, energy balance, Gamma-convergence

*DOI*

*Abstract*

In this paper we formulate a boundary layer approximation for an Allen-Cahn-type equation involving a small parameter $eps$. Here, $eps$ is related to the thickness of the boundary layer and we are interested in the limit when $eps$ tends to 0 in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2-gradient flow with the corresponding Allen-Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.

*Appeared in*

- NoDEA Nonlinear Differential Equations Appl., 20 (2013) pp. 919--942.

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# Anisotropic surface energy formulations and their effect on stability of a growing thin film

*Authors*

- Korzec, Maciek D.
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 35G20 74S25

*Keywords*

- Anisotropic surface energy, high order partial differential equations, pseudo-spectral methods, surface diffusion

*DOI*

*Abstract*

In this paper we revisit models for the description of the evolution of crystalline films with anisotropic surface energies. We prove equivalences of symmetry properties of anisotropic surface energy models commonly used in the literature. Then we systematically develop a framework for the derivation of surface diffusion models for the self-assembly of quantum dots during Stranski-Krastanov growth that include surface energies also with large anisotropy as well as the effect of wetting energy, elastic energy and a randomly perturbed atomic deposition flux. A linear stability analysis for the resulting sixth-order semilinear evolution equation for the thin film surface shows that that the new model allows for large anisotropy and gives rise to the formation of anisotropic quantum dots. The nonlinear three-dimensional evolution is investigated via numerical solutions. These suggest that increasing anisotropy stabilizes the faceted surfaces and may lead to a dramatic slow-down of the coarsening of the dots.

*Appeared in*

- Interfaces and Free Boundaries, Volume 14(4), (2012), pp. 545--567.

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# Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints

*Authors*

- Krumbiegel, Klaus
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35K20 35B65 47F05 49J20 49K20

*Keywords*

- Parabolic equation, maximal parabolic regularity, optimal control, sufficient optimality conditions

*DOI*

*Abstract*

In this paper we study optimal control problems governed by semilinear parabolic equations where the spatial dimension is two or three. Moreover, we consider pointwise constraints on the control and on the state. We formulate first order necessary and second order sufficient optimality conditions. We make use of recent results regarding elliptic regularity and apply the concept of maximal parabolic regularity to the occurring partial differential equations.

*Appeared in*

- SIAM J. Control Optim., 51 (2013) pp. 301--331.

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# Optimal control of an Allen--Cahn equation with singular potentials and dynamic boundary condition

*Authors*

- Colli, Pierluigi
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74M15 49K15 49K20

*Keywords*

- optimal control, parabolic problems, dynamic boundary conditions, optimality conditions

*DOI*

*Abstract*

In this paper, we investigate optimal control problems for Allen--Cahn equations with singular nonlinearities and a dynamic boundary condition involving singular nonlinearities and the Laplace--Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. Parabolic problems with nonlinear dynamic boundary conditions involving the Laplace--Beltrami operation have recently drawn increasing attention due to their importance in applications, while their optimal control was apparently never studied before. In this paper, we first extend known well-posedness and regularity results for the state equation and then show the existence of optimal controls and that the control-to-state mapping is twice continuously Fréchet differentiable between appropriate function spaces. Based on these results, we establish the first-order necessary optimality conditions in terms of a variational inequality and the adjoint state equation, and we prove second-order sufficient optimality conditions.

*Appeared in*

- SIAM J. Control Optim., 53 (2015) pp. 213--234.

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# Impact of interfacial slip on the stability of liquid two-layer polymer films

*Authors*

- Jachalski, Sebastian
- Peschka, Dirk

ORCID: 0000-0002-3047-1140 - Münch, Andreas
- Wagner, Barbara

*Keywords*

- fluid dynamics, thin-film models, two-phase flow, interfacial slip

*DOI*

*Abstract*

In this study systems of coupled thin-film models for two immiscible liquid polymer layers on a solid substrate that account for interfacial slip and intermolecular forces are derived. On the scale of tens to hundred nanometers such two-layer systems are susceptable to instability and may rupture and dewet. The stability of the two-layer system and its significant dependence on the order of magnitude of slip is investigated via these thin-film models. With no-slip at both, the liquid-liquid and liquid-solid interface and polymer layers of comparable thickness, the dispersion relation typically shows two local maxima, one in the long-wave regime and the other at moderate wavenumbers. The former is associated with perturbations that mainly affect the gas-liquid interface and the latter with higher relative perturbation amplitudes at the liquid-liquid interface. Slip at the liquid-liquid interface generally favors the former perturbations. However, when the liquid-liquid and the liquidsolid interface exhibit large slip, the maxima shift to small wavenumbers for increasing slip and hence may significantly change the spinodal patterns.

*Appeared in*

- J. Engrg. Math., 86, (2014) pp. 9--29.

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# A degenerating Cahn--Hilliard system coupled with complete damage processes

*Authors*

- Heinemann, Christian
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 35K85 35K55 49J40 49S05 35J50 74A45 74G25 34A12 82B26 82C26 35K92 35K65 35K35

*Keywords*

- Cahn-Hilliard system, phase separation, complete damage, elliptic-parabolic systems, elliptic degenerate operators, linear elasticity, energetic solution, weak solution, doubly nonlinear differential inclusions, existence results, rate-dependent systems

*DOI*

*Abstract*

In this work, we analytically investigate a degenerating PDE system for phase separation and complete damage processes considered on a nonsmooth time-dependent domainwith mixed boundary conditions. The evolution of the system is described by a degenerating Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a degenerating quasi-static balance equation for the displacement field. All these equations are highly nonlinearly coupled. Because of the doubly degenerating character of the system, the doubly nonlinear differential inclusion and the nonsmooth domain, the structure of the model is very complex from an analytical point of view. A novel approach is introduced for proving existence of weak solutions for such degenerating coupled system. To this end, we first establish a suitable notion of weak solutions, which consists of weak formulations of the diffusion and the momentum balance equation, a variational inequality for the damage process and a total energy inequality. To show existence of weak solutions, several new ideas come into play. Various results on shrinking sets and its corresponding local Sobolev spaces are used. It turns out that, for instance, on open sets which shrink in time a quite satisfying analysis in Sobolev spaces is possible. The presented analysis can handle highly nonsmooth regions where complete damage takes place. To mention only one difficulty, infinitely many completely damaged regions which are not connected with the Dirichlet boundary may occur in arbitrary small time intervals.

*Appeared in*

- Nonlinear Anal. Real World Appl., 22 (2015) pp. 388--403.

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# Discretization scheme for drift-diffusion equations with a generalized Einstein relation

*Authors*

- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Gärtner, Klaus

*2010 Mathematics Subject Classification*

- 65N08 35K55

*Keywords*

- generalized Einstein relation, generalized Scharfetter-Gummel scheme, drift-diffusion equations, non-Boltzmann statistic distributions, diffusion enhancement

*DOI*

*Abstract*

Inspired by organic semiconductor models based on hopping transport introducing Gauss-Fermi integrals a nonlinear generalization of the classical Scharfetter-Gummel scheme is derived for the distribution function F(η)=1/(exp(-η)+γ). This function provides an approximation of the Fermi-Dirac integrals of different order and restricted argument ranges. The scheme requires the solution of a nonlinear equation per edge and continuity equation to calculate the edge currents. In the current formula the density-dependent diffusion enhancement factor, resulting from the generalized Einstein relation, shows up as a weighting factor. Additionally the current modifies the argument of the Bernoulli functions.

*Appeared in*

- Opt. Quantum Electron., 45 (2013) pp. 791--796.

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# On the structure of the quasiconvex hull in planar elasticity

*Authors*

- Heinz, Sebastian

*2010 Mathematics Subject Classification*

- 26B25 52A30

*Keywords*

- Quasiconvexity, rank-one convexity, frame invariance

*DOI*

*Abstract*

Let K and L be compact sets of real 2x2 matrices with positive determinant. Suppose that both sets are frame invariant, meaning invariant under the left action of the special orthogonal group. Then we give an algebraic characterization for K and L to be incompatible for homogeneous gradient Young measures. This result permits a simplified characterization of the quasiconvex hull and the rank-one convex hull in planar elasticity.

*Appeared in*

- Calc. Var. Partial Differ. Equ., 50 (2014) pp. 481--489.

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# Dissipative quantum mechanics using GENERIC

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 34D20 37N20 47N50 80A99 81Q05 81V19

*Keywords*

- Quantum mechanics, density matrices, Hamiltonian systems, gradient systems, Onsager systems, GENERIC, von Neumann entropy, canonical correlation

*DOI*

*Abstract*

Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework (General Equations for Non-Equilibrium Reversible Irreversible Coupling) to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition. One of our models couples a quantum system to a finite number of heat baths each of which is described by a time-dependent temperature. The dissipation mechanism is modeled via the canonical correlation operator, which is the inverse of the Kubo-Mori metric for density matrices and which is strongly linked to the von Neumann entropy for quantum systems. Thus, one recovers the dissipative double-bracket operators of the Lindblad equations but encounters a correction term for the consistent coupling to the dissipative dynamics. For the finite-dimensional and isothermal case we provide a general existence result and discuss sufficient conditions that guarantee that all solutions converge to the unique thermal equilibrium state. Finally, we compare of our gradient flow formulation for quantum systems with the Wasserstein gradient flow formulation for the Fokker-Planck equation and the entropy gradient flow formulation for reversible Markov chains.

*Appeared in*

- Recent Trends in Dynamical Systems, Proceedings of a Conference in Honor of Jürgen Scheurle, A. Johann, H.-P. Kruse, S. Schmitz, eds., vol. 35 of Proceedings in Mathematics and Statistics, Springer, Heidelberg, 2013, pp. 555--585

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# Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally coupled phase oscillators

*Authors*

- Omel'chenko, Oleh

ORCID: 0000-0003-0526-1878 - Wolfrum, Matthias
- Yanchuk, Serhiy
- Maistrenko, Yuri
- Sudakov, Oleksandr

*2010 Mathematics Subject Classification*

- 34C15 37N20, 37N25

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt 89.75.Kd

*Keywords*

- phase oscillators, partial synchronization, coherence-incoherence pattern, non-local coupling

*DOI*

*Abstract*

Recently it has been shown that large arrays of identical oscillators with non-local coupling can have a remarkable type of solutions that display a stationary macroscopic pattern of coexisting regions with coherent and incoherent motion, often caled chimera states. We present here a detailed numerical study of the appearance of such solutions in two-dimensional arrays of coupled phase oscillators. We discover a variety of stationary patterns, including circular spots, stripe patterns, and patterns of multiple spirals. Here, the stationarity means that for increasing system size the locally averaged phase distributions tend to the stationary profile given by the corresponding thermodynamic limit equation.

*Appeared in*

- Phys. Rev. E (3), 85 (2012) pp. 036210/1--036210/5.

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# Convergence analysis of the FEM coupled with Fourier-mode expansion for the electromagnetic scattering by biperiodic structures

*Authors*

- Hu, Guanghui
- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 78A45 78M10 65N30 35J20

*Keywords*

- electromagnetic scattering, diffraction gratings, convergence analysis, finite element methods, mortar technique

*DOI*

*Abstract*

Scattering of time-harmonic electromagnetic plane waves by a doubly periodic surface structure in R^3 can be simulated by a boundary value problem of the time-harmonic curl-curl equation. For a truncated FEM domain, non-local boundary value conditions are required in order to satisfy the radiation conditions for the upper and lower half spaces. Alternatively to boundary integral formulations, to approximate radiation conditions and absorbing boundary methods, Huber et al. [11] have proposed a coupling method based on an idea of Nitsche. In the case of profile gratings with perfectly conducting substrate, the authors have shown previously that a slightly modified variational equation can be proven to be equivalent to the boundary value problem and to be uniquely solvable. Now it is shown that this result can be used to prove convergence for the FEM coupled by truncated wave mode expansion. This result covers transmission gratings and gratings bounded by additional multi-layer systems.

*Appeared in*

- Electron. Trans. Numer. Anal., 41 (2014) pp. 350--375.

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# Accelerated rogue waves generated by soliton fusion at the advanced stage of supercontinuum formation in photonic crystal fibers

*Authors*

- Driben, Rostislav
- Babushkin, Ihar

*2010 Mathematics Subject Classification*

- 78A60 35Q61 35Q60

*Keywords*

- Optical Solitons, Fiber Optics, Rogue Waves

*DOI*

*Abstract*

Soliton fusion is a fascinating and delicate phenomenon that manifests itself in optical fibers in case of interaction between co-propagating solitons with small temporal and wavelengths separation. We show that the mechanism of acceleration of trailing soliton by dispersive waves radiated from the preceding one provides necessary conditions for soliton fusion at the advanced stage of supercontinuum generation in photonic crystal fibers (PCFs). As a result of fusion large intensity robust light structures arise and propagate over significant distances. In presence of small random noise the delicate condition for the effective fusion between solitons can easily be broken, making the fusion induced giant waves a rare statistical event. Thus oblong-shaped giant accelerated waves become excellent candidates for optical rogue waves.

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# Periodic solutions of isotone hybrid systems

*Authors*

- Seidman, Thomas I.
- Klein, Olaf

ORCID: 0000-0002-4142-3603

*2010 Mathematics Subject Classification*

- 34K34 47J40 34A38 34C55 93C30

*Keywords*

- Periodic, hybrid system, discontinuous, hysteresis, isotone, fixed point, calcium waves

*DOI*

*Abstract*

Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work (Krasnosel'skiĭ-Pokrovskiĭ 1974) of theirs by defining a setting of hybrid systems with isotone switching rules for a partially ordered set of modes and then obtaining a periodicity result in that context. An application is given to a partial differential equation modeling calcium release and diffusion in cardiac cells.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. B, 18 (2013) pp. 483--493.

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# 3D numerical simulations of THz generation by two-color laser filaments

*Authors*

- Bergé, Luc
- Skupin, Stefan
- Köhler, Christian
- Babushkin, Ihar
- Herrmann, Joachim

*2010 Mathematics Subject Classification*

- 78A60 35Q61 35Q60

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Jx 32.80.Fb 52.38.Hb 52.25.Os

*Keywords*

- THz light generation, Photoionization, Ultrashort pulse propagation

*DOI*

*Abstract*

Terahertz (THz) radiation produced by the filamentation of two-color pulses over long distances in argon is numerically investigated using a comprehensive model in full space-time resolved geometry. We show that the dominant physical mechanism for THz generation in the filamentation regime at clamping intensity is based on quasi-dc plasma currents. The calculated THz spectra for different pump pulse energies and pulse durations are in agreement with previously reported experimental observations. For the same pulse parameters, near-infrared pump pulses at 2 $mu$m are shown to generate a more than one order of magnitude larger THz yield than pumps centered at 800 nm.

*Download Documents*

# Complete damage in linear elastic materials -- Modeling, weak formulation and existence results

*Authors*

- Heinemann, Christian
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 35K85 35K55 49J40 49S05 74C10 35J50 74A45 74G25 34A12

*Keywords*

- complete damage, linear elasticity, elliptic-parabolic systems, energetic solution, weak solution, doubly nonlinear differential inclusions, existence results, rate-dependent systems

*DOI*

*Abstract*

The analysis of material models which allow for complete damage is of major interest in material sciences and has received an increasing attraction in the recent years. In this work, we study a degenerating evolution inclusion describing complete damage processes coupled with a quasi-static force balance equation and mixed boundary conditions. For a realistic description, the inclusion is considered on a time-dependent domain and degenerates when the material undergoes maximal damage. We propose a weak formulation where the differential inclusion is translated into a variational inequality in combination with a total energy inequality. The damage variable is proven to be in a suitable SBV-space and the displacement field in a local Sobolev space. We show that the classical differential inclusion and the boundary conditions can be regained from the notion of weak solutions under additional regularity assumptions.

The main aim is to prove global-in-time existence of weak solutions for the degenerating system by performing a degenerate limit. The variational inequality in the limit is recaptured by suitable approximation techniques whereas the energy inequality is gained via Gamma-convergence techniques. To establish a displacement field for the elastic behavior in the limit, a rather technical representation result of nonsmooth domains by Lipschitz domains, which keep track of the Dirichlet boundary, is proven.*Appeared in*

- Calc. Var. Partial Differ. Equ., 54 (2015) pp. 217--250.

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# Sensitivity analysis of 2D photonic band gaps of any rod shape and conductivity using a very fast conical integral equation method

*Authors*

- Goray, Leonid
- Schmidt, Gunther

*2008 Physics and Astronomy Classification Scheme*

- 02.60.Nm 02.70.Pt 71.36.+c

*Keywords*

- Diffraction, multilayer periodic structure, integral method, oblique incidence, photonic crystal grating, S-matrix method

*DOI*

*Abstract*

The conical boundary integral equation method has been proposedto calculate the sensitive optical response of 2D photonic band gaps (PBGs),including dielectric, absorbing, and high-conductive rods of various shapes working in any wavelength range. It is possible to determine the diffracted field by computing the scattering matrices separately for any gratingboundary profile. The computation of the matrices is based on the solution of a 2 x 2system of singular integral equations at each interface between two different materials. The advantage of our integral formulation is that the discretization of the integral equations system and the factorization of the discrete matrices, which takes the major computing time, are carried out only oncefor a boundary. It turned out that a small number of collocation points per boundary combined with a high convergence rate can provide adequate description of the dependence on diffracted energy of very different PBGs illuminated at arbitrary incident and polarization angles. Thenumerical results presented describe the significant impact of rod shape on diffraction in PBGs supporting polariton-plasmon excitation, particularly in the vicinity of resonances and at high fillingratios. The diffracted energy response calculated vs. array cell geometry parameters was found to vary from a few percent up to a few hundred percent. The influence of other types of anomalies (i.e. waveguide anomalies, cavity modes, Fabry-Perot and Bragg resonances, Rayleigh orders, etc), conductivity, and polarization states on the optical response has been demonstrated.

*Appeared in*

- Phys. Rev. E (3), 85 (2012) pp. 036701/1--036701/12 under the new title ``Analysis of two-dimensional photonic band gaps of any rod shape and conductivity using a conical-integral-equation method''.

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# Convergence of stochastic particle systems undergoing advection and coagulation

*Authors*

- Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857

*2010 Mathematics Subject Classification*

- 60K35 65C35 82C22

*Keywords*

- stochastic particle systems, source, outflow, coagulation, advection

*DOI*

*Abstract*

The convergence of stochastic particle systems representing physical advection, inflow, outflow and coagulation is considered. The problem is studied on a bounded spatial domain such that there is a general upper bound on the residence time of a particle. The laws on the appropriate Skorohod path space of the empirical measures of the particle systems are shown to be relatively compact. The paths charged by the limits are characterised as solutions of a weak equation restricted to functions taking the value zero on the outflow boundary. The limit points of the empirical measures are shown to have densities with respect to Lebesgue measure when projected on to physical position space. In the case of a discrete particle type space a strong form of the Smoluchowski coagulation equation with a delocalised coagulation interaction and an inflow boundary condition is derived. As the spatial discretisation is refined in the limit equations, the delocalised coagulation term reduces to the standard local Smoluchowski interaction.

*Appeared in*

- Stoch. Anal. Appl., 31 (2013) pp. 800--829.

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# Weak solutions to lubrication systems describing the evolution of bilayer thin films

*Authors*

- Jachalski, Sebastian
- Kitavtsev, Georgy
- Taranets, Roman

*2010 Mathematics Subject Classification*

- 35D30 76A20 76D08

*Keywords*

- bilayer systems, lubrication equations, weak solutions, slippage

*DOI*

*Abstract*

The existence of global nonnegative weak solutions is proved for coupled one-dimen- sional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown.

*Appeared in*

- Comm. Math. Sci., 12 (2014) pp. 527--544.

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# Some remarks on stability of generalized equations

*Authors*

- Henrion, René
- Kruger, Alexander
- Outrata, Jiří

*2010 Mathematics Subject Classification*

- 49J53 90C31 90C46

*Keywords*

- Parameterized generalized equation, regular and limiting coderivative, constant rank CQ, mathematical program with equilibrium constraint

*DOI*

*Abstract*

The paper concerns the computation of the graphical derivative and the regular (Fréchet) coderivative of the solution map to a class of generalized equations, where the multi-valued term amounts to the regular normal cone to a (possibly nonconvex) set given by $C^2$ inequalities. Instead of the Linear Independence qualification condition, standardly used in this context, one assumes a combination of the Mangasarian-Fromovitz and the Constant Rank qualification conditions. On the basis of the obtained generalized derivatives, new optimality conditions for a class of mathematical programs with equilibrium constrains are derived, and a workable characterization of the isolated calmness of the considered solution map is provided.

*Appeared in*

- J. Optim. Theory Appl., 159 (2013) pp. 681--697.

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# Spatial ``rocking'' for improving the spatial quality of the beam of broad area semiconductor lasers

*Authors*

- Radziunas, Mindaugas
- Staliunas, Kestutis

*2010 Mathematics Subject Classification*

- 78A60 35B36 37M05 78A45

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.65.Pc 42.60.Jf

*Keywords*

- broad area semiconductor laser, traveling wave model, off-axis optical injection, stabilization, focusing, rocking

*DOI*

*Abstract*

The spatial ``rocking'' is a dynamical effect converting a phase-invariant oscillatory system into a phase-bistable one, where the average phase of the system locks to one of two values differing by $pi$. We demonstrate theoretically the spatial rocking in experimentally accessible and practically relevant systems -- the broad area semiconductor lasers. By numerical integration of the laser model equations we show the phase bistability of the optical fields and explore the bistability area in parameter space. We also predict the spatial patterns, such as phase domain walls and phase solitons, which are characteristic for the phase-bistable spatially extended pattern forming systems.

*Appeared in*

- Semiconductor Lasers and Laser Dynamics V, K. Panajotov, M. Sciamanna, A.A. Valle, R. Michalzik, eds., vol. 8432 of Proceedings of SPIE, SPIE, 2012, pp. 84320Q/1--84320Q/9.

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# Reflection of plane waves by rough surfaces in the sense of Born approximation

*Authors*

- Arnold, Thomas
- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 78A45 35J25 35J05 35B40

*Keywords*

- Electromagnetic scattering, rough surfaces, Fourier analysis, Born approximation

*DOI*

*Abstract*

The topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, i.e., by smooth and bounded perturbations of planar faces. Moreover, the contrast between the cover material and the substrate beneath the rough surface is supposed to be low. In this case, a modification of Stearns' formula based on Born approximation and Fourier techniques is derived for a special class of surfaces. This class contains the graphs of functions if the interface function is a radially modulated almost periodic function. For the Born formula to converge, a sufficient and almost necessary condition is given. A further technical condition is defined, which guarantees the existence of the corresponding far field of the Born approximation. This far field contains plane waves, far-field terms like those for bounded scatterers, and, additionally, a new type of terms. The derived formulas can be used for the fast numerical computations of far fields and for the statistics of random rough surfaces.

*Appeared in*

- Math. Methods Appl. Sci., 37 (2014) pp. 2091--2111.

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# Overcoming the shortcomings of the Nernst--Planck model

*Authors*

- Dreyer, Wolfgang
- Guhlke, Clemens
- Müller, Rüdiger

ORCID: 0000-0003-2643-722X

*2010 Mathematics Subject Classification*

- 35Q35 76T30 65N22

*2008 Physics and Astronomy Classification Scheme*

- 82.45.Gj

*Keywords*

- electrolyte, equilibrium, modeling, asymptotic analysis, numerical method

*DOI*

*Abstract*

This is a study on electrolytes that takes a thermodynamically consistent coupling between mechanics and diffusion into account. It removes some inherent deficiencies of the popular Nernst-Planck model. A boundary problem for equilibrium processes is used to illustrate the new features of our model.

*Appeared in*

- Phys. Chem. Chem. Phys., 15 (2013), pp. 7075--7086, DOI 10.1039/C3CP44390F .

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# Continuous dependence for a nonstandard Cahn--Hilliard system with nonlinear atom mobility

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Podio-Guidugli, Paolo
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K61 35A05 74A15

*Keywords*

- phase-field model, nonlinear system of partial differential equations, existence of solutions, new uniqueness proof

*DOI*

*Abstract*

This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice [Podio-Guidugli 2006]; it consists of the balance equations of microforces and microenergy; the two unknowns are the order parameter $rho$ and the chemical potential $mu$. Some recent results obtained for this class of problems is reviewed and, in the case of a nonconstant and nonlinear atom mobility, uniqueness and continuous dependence on the initial data are shown with the help of a new line of argumentation developed in Colli/Gilardi/Podio-Guidugli/Sprekels 2012.

*Appeared in*

- Rend. Semin. Mat. Univ. Politec. Torino, 70 (2012) pp. 27--52.

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# On regular coderivatives in parametric equilibria with non-unique multipliers

*Authors*

- Henrion, René
- Outrata, Jiří
- Surowiec, Thomas

ORCID: 0000-0003-2473-4984

*2010 Mathematics Subject Classification*

- 49J53 90C31 90C46

*Keywords*

- Parameterized generalized equation, regular and limiting coderivative, constant rank CQ, mathematical program with equilibrium constraint

*DOI*

*Abstract*

This paper deals with the computation of regular coderivatives of solution maps associated with a frequently arising class of generalized equations. The constraint sets are given by (not necessarily convex) inequalities, and we do not assume linear independence of gradients to active constraints. The achieved results enable us to state several versions of sharp necessary optimality conditions in optimization problems with equilibria governed by such generalized equations. The advantages are illustrated by means of examples.

*Appeared in*

- Math. Program., 136 (2012) pp. 111--131.

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# Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals

*Authors*

- Dreyer, Wolfgang
- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500 - Klein, Olaf

ORCID: 0000-0002-4142-3603 - Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35Q30 35Q61 45G05 76W05 80A20 80A22

*Keywords*

- Czochralski method, crystal growth, traveling magnetic fields, radiative heat transfer, nonlinear PDE systems, Navier--Stokes equations, MHD equations, Maxwell's equations, well-posedness, optimal control, first-order necessary optimality conditions, numerical simulation

*DOI*

*Abstract*

This paper deals with the mathematical modeling and simulation of crystal growth processes by the so-called Czochralski method and related methods, which are important industrial processes to grow large bulk single crystals of semiconductor materials such as, e.,g., gallium arsenide (GaAs) or silicon (Si) from the melt. In particular, we investigate a recently developed technology in which traveling magnetic fields are applied in order to control the behavior of the turbulent melt flow. Since numerous different physical effects like electromagnetic fields, turbulent melt flows, high temperatures, heat transfer via radiation, etc., play an important role in the process, the corresponding mathematical model leads to an extremely difficult system of initial-boundary value problems for nonlinearly coupled partial differential equations. In this paper, we describe a mathematical model that is under use for the simulation of real-life growth scenarios, and we give an overview of mathematical results and numerical simulations that have been obtained for it in recent years.

*Appeared in*

- Milan J. Math., 80 (2012) pp. 311--332 .

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# A variance-reduced electrothermal Monte Carlo method for semiconductor device simulation

*Authors*

- Muscato, Orazio
- Di Stefano, Vincenza
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 82D37 65C05

*Keywords*

- semiconductor devices, Monte Carlo simulation, electrothermal modelling

*DOI*

*Abstract*

This paper is concerned with electron transport and heat generation in semiconductor devices. An improved version of the electrothermal Monte Carlo method is presented. This modification has better approximation properties due to reduced statistical fluctuations. The corresponding transport equations are provided and results of numerical experiments are presented.

*Appeared in*

- Comput. Math. Appl., 65 (2013) pp. 520--527.

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# Elastic scattering by finitely many point-like obstacles

*Authors*

- Hu, Guanghui
- Sini, Mourad

*2010 Mathematics Subject Classification*

- 74B05 78A45 81Q10

*Keywords*

- linear elasticity, point-like scatterers, Navier equation, Green's tensor, far field pattern

*DOI*

*Abstract*

This paper is concerned with the time-harmonic elastic scattering by a finite number $N$ of point-like obstacles in $R^n (n=2,3)$. We analyze the $N$-point interactions model in elasticity and derive the associated Green's tensor (integral kernel) in terms of the point positions and the scattering coefficients attached to them, following the approach in quantum mechanics for modeling $N$-particle interactions. In particular, explicit expressions are given for the scattered near and far fields corresponding to elastic plane waves or point-source incidences. As a result, we rigorously justify the Foldy method for modeling the multiple scattering by finitely many point-like obstacles for the Lame model. The arguments are based on the Fourier analysis and the Weinstein-Aronszajn inversion formula of the resolvent for the finite rank perturbations of closed operators in Hilbert spaces.

*Appeared in*

- J. Math. Phys., 54 (2013) pp. 042901--16.

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# Multiple scattering of electromagnetic waves by a finite number of point-like obstacles

*Authors*

- Challa, Durga Prasad
- Hu, Guanghui
- Sini, Mourad

*2010 Mathematics Subject Classification*

- 35R30 78A45 78A46

*Keywords*

- electromagnetic scattering, point-like scatterers, multiple scattering, MUSIC algorithm

*DOI*

*Abstract*

This paper is concerned with the time-harmonic electromagnetic scattering problem for a finite number M of point-like obstacles in R^3. First, we give a rigorous justification of the Foldy method and describe the intermediate levels of scattering between the Born and Foldy models. Second, we study the problem of detecting the scatterers and the scattering strengths from the far-field measurements and discuss the effect of multiple scattering related to each of these models.

*Appeared in*

- Math. Models Methods Appl. Sci., 24 (2014) pp. 863--899.

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# Simulation based policy iteration for American style derivatives --- A multilevel approach

*Authors*

- Belomestny, Denis
- Ladkau, Marcel
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 62L15 65C05 91B28

*Keywords*

- Optimal stopping, Multilevel Monte Carlo, Howard policy iteration

*DOI*

*Abstract*

This paper presents a novel approach to reduce the complexity of simulation based policy iteration methods for pricing American options. Typically, Monte Carlo construction of an improved policy gives rise to a nested simulation algorithm for the price of the American product. In this respect our new approach uses the multilevel idea in the context of the inner simulations required, where each level corresponds to a specific number of inner simulations. A thorough analysis of the crucial convergence rates in the respective multilevel policy improvement algorithm is presented. A detailed complexity analysis shows that a significant reduction in computational effort can be achieved in comparison to standard Monte Carlo based policy iteration.

*Appeared in*

- SIAM/ASA Journal on Uncertainty Qualification, 3 (2015) pp. 460--483.

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# A quasi-incompressible diffuse interface model with phase transition

*Authors*

- Aki, Gonca
- Dreyer, Wolfgang
- Giesselmann, Jan
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 35C20 35R35 76T99 35Q30 35Q35 76D05 76D45 80A22

*Keywords*

- Multi-component flow, phase transition, asymptotic analysis, sharp interface limit, free boundary problems, Cahn-Hilliard equation, Allen-Cahn equation, Navier-Stokes-Korteweg system

*DOI*

*Abstract*

This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier-Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier-Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.

*Appeared in*

- Math. Models Methods Appl. Sci., 24 (2014) pp. 827--861.

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# On the efficiency and robustness of the core routine of the quadrature method of moments (QMOM)

*Authors*

- John, Volker

ORCID: 0000-0002-2711-4409 - Thein, Ferdinand

*2010 Mathematics Subject Classification*

- 65D30

*Keywords*

- Quadrature Method of Moments, optimal quadrature rules, Product-Difference Algorithm, Long Quotient-Modified Difference Algorithm, Golub--Welsch Algorithm

*DOI*

*Abstract*

Three methods are reviewed for computing optimal weights and abscissas which can be used in the Quadrature Method of Moments (QMOM): the Product-Difference Algorithm (PDA), the Long Quotient-Modified Difference Algorithm (LQMDA, variants are also called Wheeler algorithm or Chebyshev algorithm), and the Golub--Welsch Algorithm (GWA). The PDA is traditionally used in applications. It is discussed that the PDA fails in certain situations whereas the LQMDA and the GWA are successful. Numerical studies reveal that the LQMDA is also more efficient than the PDA.

*Appeared in*

- Chem. Engng. Sci., 75 (2012) pp. 327--333.

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# From an adhesive to a brittle delamination model in thermo-visco-elasticity

*Authors*

- Rossi, Riccarda
- Thomas, Marita

*2010 Mathematics Subject Classification*

- 35K85 74R10 47J20 49J45 49S05 74F07

*Keywords*

- Rate-independent evolution of adhesive contact, brittle delamination, Kelvin-Voigt visco-elasticity, nonlinear heat equation, Mosco-convergence, functions of bounded variation

*DOI*

*Abstract*

We address the analysis of a model for brittle delamination of two visco-elastic bodies, bonded along a prescribed surface. The model also encompasses thermal effects in the bulk. The related PDE system for the displacements, the absolute temperature, and the delamination variable has a highly nonlinear character. On the contact surface, it features frictionless Signorini conditions and a nonconvex, brittle constraint acting as a transmission condition for the displacements. We prove the existence of (weak/energetic) solutions to the associated Cauchy problem, by approximating it in two steps with suitably regularized problems. We perform the two consecutive passages to the limit via refined variational convergence techniques.

*Appeared in*

- ESAIM Control Optim. Calc. Var., 21 (2015) pp. 1--59.

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# A vanishing diffusion limit in a nonstandard system of phase field equations

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Krejčí, Pavel
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K61 35A05 35B40 74A15

*Keywords*

- nonstandard phase field system, nonlinear partial differential equations, asymptotic limit, convergence of solutions

*DOI*

*Abstract*

We are concerned with a nonstandard phase field model of Cahn--Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011, and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient $sigma$ in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.

*Appeared in*

- Evol. Equ. Control Theory, 3 (2014) pp. 257--275.

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# Influence of the carrier reservoir dimensionality on electron-electron scattering in quantum dot materials

*Authors*

- Wilms, Alexander
- Mathé, Peter

ORCID: 0000-0002-1208-1421 - Schulze, Franz
- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Knorr, Andreas
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 81V65 65C05

*2008 Physics and Astronomy Classification Scheme*

- 73.21.La 73.63.Kv.

*Keywords*

- quantum dots, Coulomb scattering, quasi-Monte Carlo

*DOI*

*Abstract*

We calculated Coulomb scattering rates from quantum dots (QDs) coupled to a 2D carrier reservoir and QDs coupled to a 3D reservoir. For this purpose, we used a microscopic theory in the limit of Born-Markov approximation, in which the numerical evaluation of high dimensional integrals is done via a quasi-Monte Carlo method. Via a comparison of the so determined scattering rates, we investigated the question whether scattering from 2D is generally more efficient than scattering from 3D. In agreement with experimental findings, we did not observe a significant reduction of the scattering efficiency of a QD directly coupled to a 3D reservoir. In turn, we found that 3D scattering benefits from it?s additional degree of freedom in the momentum space.

*Appeared in*

- Phys. Rev. B., 88 (2013) pp. 235421/1--235421/11.

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# On existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations

*Authors*

- Nefedov, Nikolai
- Recke, Lutz
- Schneider, Klaus

*2010 Mathematics Subject Classification*

- 35B10 35B25 35B35 35C20 35K10

*Keywords*

- singulary pertubed parabolic periodic problems, exponential asymptotic stability, Krein-Rutman theorem, lower and upper solutions

*DOI*

*Abstract*

We consider a singularly perturbed parabolic periodic boundary value problem for a reaction-advection-diffusion equation. We construct the interior layer type formal asymptotics and propose a modified procedure to get asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics. Moreover, we are able to establish the asymptotic stability of this solution with interior layer

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# Spontaneous motion of cavity solitons induced by a delayed feedback

*Authors*

- Tlidi, Mustapha
- Averlant, Etienne
- Vladimirov, Andrei G.
- Panajotov, Krassimir

*2010 Mathematics Subject Classification*

- 78A60 37L10

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a 02.30.Ks 42.65.-k

*Keywords*

- semiconductor lasers, delayed feedback, Swift-Hohenberg equation, localized structures, bifurcations

*DOI*

*Abstract*

We consider a broad area Vertical-Cavity Surface Emitting Laser (VCSEL) operating below the lasing threshold and subject to optical injection and time-delayed feedback. We derive a generalized delayed Swift-Hohenberg equation for the VCSEL system which is valid close to the nascent optical bistability. We first characterize the stationary cavity solitons by constructing their snaking bifurcation diagram and by showing clustering behavior within the pinning region of parameters. Then we show that the delayed feedback induces a spontaneous motion of two-dimensional cavity solitons in an arbitrary direction in the transverse plane. We characterize moving cavity solitons by estimating their threshold and calculating their velocity. Numerical 2D solutions of the governing semiconductor laser equations are in close agreement with those obtained from the delayed generalized Swift- Hohenberg equation.

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# Bessel bridges decomposition with varying dimension. Applications to finance

*Authors*

- Faraud, Gabriel
- Goutte, Stéphane

*2010 Mathematics Subject Classification*

- 60G07 60H35 91B70

*Keywords*

- Squared Bessel process, Bessel bridges decomposition, Laplace transform, Lévy Ito representation, Financial applications

*DOI*

*Abstract*

We consider a class of stochastic processes containing the classical and well-studied class of Squared Bessel processes. Our model, however, allows the dimension be a function of the time. We first give some classical results in a larger context where a time-varying drift term can be added. Then in the non-drifted case we extend many results already proven in the case of classical Bessel processes to our context. Our deepest result is a decomposition of the Bridge process associated to this generalized squared Bessel process, much similar to the much celebrated result of J. Pitman and M. Yor. On a more practical point of view, we give a methodology to compute the Laplace transform of additive functionals of our process and the associated bridge. This permits in particular to get directly access to the joint distribution of the value at $t$ of the process and its integral. We finally give some financial applications to illustrate the panel of applications of our results.

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# Ideal mixture approximation of cluster size distributions at low density

*Authors*

- Jansen, Sabine
- König, Wolfgang

ORCID: 0000-0002-4212-0065

*2010 Mathematics Subject Classification*

- 82B21 60F10 82B31 82B05

*Keywords*

- Classical particle system, canonical ensemble, equilibrium statistical mechanics, dilute system, large deviations.

*DOI*

*Abstract*

We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture.

*Appeared in*

- J. Statist. Phys., 147 (2012) pp. 963--980.

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# Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction-diffusion problems

*Authors*

- Fiebach, André
- Glitzky, Annegret
- Linke, Alexander

ORCID: 0000-0002-0165-2698

*2010 Mathematics Subject Classification*

- 35K57 65M08 65M22 80A30

*Keywords*

- reaction-diffusion systems, heterostructures, finite volume method, global bounds, discrete Gagliardo-Nirenberg inequalities, discrete Moser iteration

*DOI*

*Abstract*

We consider discretizations for reaction-diffusion systems with nonlinear diffusion in two space dimensions. The applied model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. We propose an implicit Voronoi finite volume discretization on regular Delaunay meshes that allows to prove uniform, mesh-independent global upper and lower $L^infty$ bounds for the chemical potentials. These bounds provide the main step for a convergence analysis for the full discretized nonlinear evolution problem. The fundamental ideas are energy estimates, a discrete Moser iteration and the use of discrete Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalities we exploit that the discrete Voronoi finite volume gradient norm in $2d$ coincides with the gradient norm of continuous piecewise linear finite elements.

*Appeared in*

- Numer. Math., 128 (2014) pp. 31--72.

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# Parabolic equations with dynamical boundary conditions and source terms on interfaces

*Authors*

- ter Elst, A.F.M.
- Meyries, Martin
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35K20 35K59 35M13 35R05

*Keywords*

- Parabolic equation, quasilinear parabolic problem, mixed boundary condition, dynamical boundary condition, maximal parabolic L
^{p}-regularity, nonsmooth geometry, nonsmooth coefficients

*DOI*

*Abstract*

We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable L^{p}-setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.

*Appeared in*

- Ann. Mat. Pura Appl. IV. Ser., 193 (2014) pp. 1295--1318.

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# Homogenization of elastic waves in fluid-saturated porous media using the Biot model

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Rohan, Eduard

*2010 Mathematics Subject Classification*

- 35B27 74F10 76M50 76S05

*Keywords*

- Two-scale homogenization, porous media, acoustic waves, elastodynamics, Darcy's law, seepage

*DOI*

*Abstract*

We consider periodically heterogeneous fluid-saturated poroelastic media described by the Biot model with inertia effects. The weak and semistrong formulations for displacement, seepage and pressure fields involve three equations expressing the momentum and mass balance and the Darcy law. Using the two-scale homogenization method we obtain the limit two-scale problem and prove the existence and uniqueness of its weak solutions. The Laplace transformation in time is used to decouple the macroscopic and microscopic scales. It is shown that the seepage velocity is eliminated form the macroscopic equations involving strain and pressure fields only. The plane harmonic wave propagation is studied using an example of layered medium. Illustrations show some influence of the orthotropy on the dispersion phenomena.

*Appeared in*

- Math. Models Methods Appl. Sci., 23 (2013) pp. 873--916.

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# A central limit theorem for the effective conductance: I. Linear boundary data and small ellipticity contrasts

*Authors*

- Biskup, Marek
- Salvi, Michele
- Wolff, Tilman

*2010 Mathematics Subject Classification*

- 37H10 60K37 60J60

*Keywords*

- Random conductance model, second order discrete elliptic equations with random coefficients, homogenization theory

*DOI*

*Abstract*

We consider resistor networks on $Z^d$ where each nearest-neighbor edge is assigned a non-negative random conductance. Given a finite set with a prescribed boundary condition, the effective conductance is the minimum of the Dirichlet energy over functions that agree with the boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box is known to converge to a deterministic limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) central limit theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and arbitrary ellipticity contrasts are to be addressed in a subsequent paper.

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# Gradient structures and geodesic convexity for reaction-diffusion systems

*Authors*

- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K57 53C21 53C23 60J60 82B35

*Keywords*

- Geodesic convexity, gradient structures, gradient flow, Onsager operator, reaction-diffusion system, Wasserstein metric, relative entropy

*DOI*

*Abstract*

We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory. We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.

*Appeared in*

- Phil. Trans. R. Soc. A, 371 (2013) pp. 20120346/1--20120346/28.

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# Moment asymptotics for branching random walks in random environment

*Authors*

- Gün, Onur
- König, Wolfgang

ORCID: 0000-0002-4212-0065 - Sekulović, Ozren

*2010 Mathematics Subject Classification*

- 60J80 60J55 60F10 60K37

*Keywords*

- branching random walk, random potential, parabolic Anderson model, Feynman-Kac-type formula, annealed moments, large deviations

*DOI*

*Abstract*

We consider the long-time behaviour of a branching random walk in random environment on the lattice ℤ^{d}. The migration of particles proceeds according to simple random walk in continuous time, while the medium is given as a random potential of spatially dependent killing/branching rates. The main objects of our interest are the annealed moments ⟨ m_{n}^{p} ⟩, i.e., the p-th moments over the medium of the n-th moment over the migration and killing/branching, of the local and global population sizes. For n=1, this is well-understood [GM98], as m_{1} is closely connected with the parabolic Anderson model. For some special distributions, [ABMY00] extended this to n ≥ 2, but only as to the first term of the asymptotics, using (a recursive version of) a Feynman-Kac formula for m_{n}. In this work we derive also the second term of the asymptotics, for a much larger class of distributions. In particular, we show that ⟨ m_{n}^{p} ⟩ and ⟨ m_{1}^{np} ⟩ are asymptotically equal, up to an error e^{o(t)}. The cornerstone of our method is a direct Feynman-Kac-type formula for m_{n}, which we establish using the spine techniques developed in [HR12].

*Appeared in*

- Electron. J. Probab., 18 (2013) pp. 1--18.

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# Efficient maximum likelihood estimation for Lévy-driven Ornstein--Uhlenbeck processes

*Authors*

- Mai, Hilmar

*2010 Mathematics Subject Classification*

- 62F12, 62M05

*Keywords*

- discrete time observations, efficient drift estimation, Lévy process, maximum likelihood, Ornstein-Uhlenbeck process

*DOI*

*Abstract*

We consider the problem of efficient estimation of the drift parameter of an Ornstein-Uhlenbeck type process driven by a Lévy process when high-frequency observations are given. The estimator is constructed from the time-continuous likelihood function that leads to an explicit maximum likelihood estimator and requires knowledge of the continuous martingale part. We use a thresholding technique to approximate the continuous part of the process. Under suitable conditions we prove asymptotic normality and efficiency in the Hájek-Le Cam sense for the resulting drift estimator. To obtain these results we prove an estimate for the Markov generator of a pure jump Lévy process. Finally, we investigate the finite sample behavior of the method and compare our approach to least squares estimation.

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# Some mathematical problems related to the 2nd order optimal shape of a crystallization interface

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 49K20 80A22 53A10 35J25

*Keywords*

- Stefan-Gibbs-Thompson problem, Singularity of mean-curvature type, Optimal control, Pointwise gradient state constraints, First order optimality conditions

*DOI*

*Abstract*

We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.

*Appeared in*

- Discrete Contin. Dyn. Syst., 35 (2015) pp. 2443--2463.

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# Elastic scattering by unbounded rough surfaces

*Authors*

- Elschner, Johannes
- Hu, Guanghui

*2010 Mathematics Subject Classification*

- 74J20 74B05 35J57 35Q74

*Keywords*

- Elastic waves, rough surfaces, Navier equation, variational formulation, radiation condition

*DOI*

*Abstract*

We consider the two-dimensional time-harmonic elastic wave scattering problem for an unbounded rough surface, due to an inhomogeneous source term whose support lies within a finite distance above the surface. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the elastic displacement vanishes. We propose an upward propagating radiation condition (angular spectrum representation) for solutions of the Navier equation in the upper half-space above the rough surface, and establish an equivalent variational formulation. Existence and uniqueness of solutions at arbitrary frequency is proved by applying a priori estimates for the Navier equation and perturbation arguments for semi-Fredholm operators.

*Appeared in*

- SIAM J. Math. Anal., 6 (2012) pp. 4101--4127.

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# Self-heating, bistability, and thermal switching in organic semiconductors

*Authors*

- Fischer, Axel
- Pahner, Paul
- Lüssem, Björn
- Leo, Karl
- Scholz, Reinhard
- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Gärtner, Klaus
- Glitzky, Annegret

*2010 Mathematics Subject Classification*

- 82D37 80A20 34C55

*2008 Physics and Astronomy Classification Scheme*

- 81.05.Fb, 73.61.Wp, 72.80.Le

*Keywords*

- Organic semiconductor, C60, Joule self-heating, S-shaped negative differential resistance (SNDR), thermal switching, bistability, hysteresis, Arrhenius-like conductivity law

*DOI*

*Abstract*

We demonstrate electric bistability induced by the positive feedback of self-heating onto the thermally activated conductivity in a two-terminal device based on the organic semiconductor C60. The central undoped layer with a thickness of 200 nm is embedded between thinner n-doped layers adjacent to the contacts minimizing injection barriers. The observed current-voltage characteristics follow the general theory for thermistors described by an Arrhenius-like conductivity law. Our findings including hysteresis phenomena are of general relevance for the entire material class since most organic semiconductors can be described by a thermally activated conductivity.

*Appeared in*

- Phys. Rev. Lett., 110 (2013) pp. 126601/1--126601/5.

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# Dynamics of particle settling and resuspension in viscous liquids

*Authors*

- Murisic, Nebojsa
- Pausader, Benoit
- Peschka, Dirk

ORCID: 0000-0002-3047-1140 - Bertozzi, Andrea L.

*2010 Mathematics Subject Classification*

- 76D08 76T20 76A20

*Keywords*

- particles, resuspension, lubrication theory

*DOI*

*Abstract*

We derive and study a dynamical model for suspensions of negatively buoyant particles on an incline. Our theoretical model includes the settling/sedimentation due to gravity as well as the resuspension of particles induced by shear-induced migration, leading to disaggregation of the dense sediment layer. Out of the three different regimes observed in the experiments, we focus on the so-called settled case, where the particles settle out of the flow, and two distinct fronts, liquid and particle, form. Using an approach relying on asymptotics, we systematically connect our dynamic model with the previously developed equilibrium theory for particle-laden flows. We show that the resulting transport equations for the liquid and the particles are of hyperbolic type, and study the dilute limit, for which we derive the analytic solution. We also carry out a systematic experimental study of the settled regime, focusing on the motion of the liquid and the particle fronts. Finally, we carry out numerical simulations of our transport equations. We show that the model predictions for small to moderate values of the particle volume fraction and the inclination angle of the solid substrate agree well with the experimental data.

*Appeared in*

- J. Fluid Mech., 717 (2013) pp. 203--231.

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# Libor model with expiry-wise stochastic volatility and displacement

*Authors*

- Ladkau, Marcel
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266 - Zhang, Jianing

*2010 Mathematics Subject Classification*

- 91G30 91G60 60H10

*Keywords*

- Displaced Libor models, stochastic volatility, calibration to cap-strike-maturity matrix, swaption pricing

*DOI*

*Abstract*

We develop a multi-factor stochastic volatility Libor model with displacement, where each individual forward Libor is driven by its own square-root stochastic volatility process. The main advantage of this approach is that, maturity-wise, each square-root process can be calibrated to the corresponding cap(let)vola-strike panel at the market. However, since even after freezing the Libors in the drift of this model, the Libor dynamics are not affine, new affine approximations have to be developed in order to obtain Fourier based (approximate) pricing procedures for caps and swaptions. As a result, we end up with a Libor modeling package that allows for efficient calibration to a complete system of cap/swaption market quotes that performs well even in crises times, where structural breaks in vola-strike-maturity panels are typically observed.

*Appeared in*

- Int. J. Portfolio Analysis and Management, 1 (2013) pp. 224--249.

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# A generalization of Lagrange's algebraic identity and connections with Jensen's inequality

*Authors*

- Niculescu, Constantin P.
- Stephan, Holger

*2010 Mathematics Subject Classification*

- 26B25 26D15

*Keywords*

- Jensen's inequality, Lagrange's algebraic identity, convex function, slope function, barycenter of a measure

*DOI*

*Abstract*

We discuss a generalization of Lagrange's algebraic identity that provides valuable insights into the nature of Jensen's inequality and of many other inequalities of convexity.

*Appeared in*

- Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 56 (104) (2013) pp. 487--496.

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# Spatio-temporal pulse propagation in nonlinear dispersive optical media

*Authors*

- Brée, Carsten
- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 78A60 35Q60 35Q55

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Re 42.65.St 42.65.Tg 02.30.Mw 52.35.Mw

*Keywords*

- Ultrashort pulses, Nonlinear Schrödinger Equation, Filaments

*DOI*

*Abstract*

We discuss state-of-art approaches to modeling of propagation of ultrashort optical pulses in one and three spatial dimensions.We operate with the analytic signal formulation for the electric field rather than using the slowly varying envelope approximation, because the latter becomes questionable for few-cycle pulses. Suitable propagation models are naturally derived in terms of unidirectional approximation.

*Appeared in*

- Opt. Quantum Electron., 45 (2013) pp. 727--733.

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# Analysis and optimal boundary control of a nonstandard system of phase field equations

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74A15 35K55 49K20

*Keywords*

- nonlinear phase field systems, Cahn--Hilliard systems, parabolic systems, optimal boundary control, first-order necessary optimality conditions

*DOI*

*Abstract*

We investigate a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced in Podio-Guidugli (2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in Colli, Gilardi, Podio-Guidugli, and Sprekels (2011a and b) for the case of homogeneous Neumann boundary conditions. In this paper, we investigate the case that the boundary condition for one of the unknowns of the system is of third kind and nonhomogeneous. For the resulting system, we show well-posedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the first-order necessary optimality conditions are derived. Owing to the strong nonlinear couplings in the PDE system, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional will be of standard type.

*Appeared in*

- Milan J. Math., 80 (2012) pp. 119--149.

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# Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry

*Authors*

- Gross, Hermann
- Henn, Mark-Alexander
- Heidenreich, Sebastian
- Rathsfeld, Andreas
- Bär, Markus

*2010 Mathematics Subject Classification*

- 78A46 65N30 62P35

*Keywords*

- diffraction gratings, metrology

*DOI*

*Abstract*

We investigate the impact of line edge and line width roughness (LER, LWR) on the measured diffraction intensities in angular resolved extreme ultraviolet (EUV) scatterometry for a periodic line-space structure designed for EUV lithography. LER and LWR with typical amplitudes of a few nanometers were previously neglected in the course of the profile reconstruction. The 2D rigorous numerical simulations of the diffraction process for periodic structures are carried out with the finite element method (FEM) providing a numerical solution of the two-dimensional Helmholtz equation. To model roughness, multiple calculations are performed for domains with large periods, containing many pairs of line and space with stochastically chosen line and space widths. A systematic decrease of the mean efficiencies for higher diffraction orders along with increasing variances is observed and established for different degrees of roughness. In particular, we obtain simple analytical expressions for the bias in the mean efficiencies and the additional uncertainty contribution stemming from the presence of LER and/or LWR. As a consequence this bias can easily be included into the reconstruction model to provide accurate values for the evaluated profile parameters. We resolve the sensitivity of the reconstruction from this bias by using the LER/LWR perturbed efficiency datasets for multiple reconstructions. If the scattering efficiencies are bias-corrected, significant improvements are found in the reconstructed bottom and top widths toward the nominal values.

*Appeared in*

- Appl. Optics, 51 (2012) pp. 7384--7394.

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# Nonuniversal transitions to synchrony in the Sakaguchi--Kuramoto model

*Authors*

- Omel'chenko, Oleh

ORCID: 0000-0003-0526-1878 - Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 34C15 37N20 37N25

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt 89.75.Kd

*Keywords*

- Coupled phase oscillators, synchronization transition, Sakaguchi-Kuramoto model

*DOI*

*Abstract*

We investigate the transition to synchrony in a system of phase oscillators that are globally coupled with a phase lag (Sakaguchi-Kuramoto model). We show that for certain unimodal frequency distributions there appear unusual types of synchronization transitions, where synchrony can decay with increasing coupling, incoherence can regain stability for increasing coupling, or multistability between partially synchronized states and/or the incoherent state can appear. Our method is a bifurcation analysis based on a frequency dependent version of the Ott-Antonsen method and allows for a universal description of possible synchronization transition scenarios for any given distribution of natural frequencies.

*Appeared in*

- Phys. Rev. Lett., 109 (2012) pp. 164101/1--164101/4.

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# Computation of volume potentials over bounded domains via approximate approximations

*Authors*

- Lanzara, Flavia
- Maz'ya, Vladimir
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 41A30 65D30 41A63 41A25

*Keywords*

- Cubature of volume potentials, multivariate approximation, bounded domain

*DOI*

*Abstract*

We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced to the quadrature of one dimensional integrals over the halfline. We conclude the paper providing numerical tests which show that these formulas give very accurate approximations and confirm the predicted order of convergence.

*Appeared in*

- J. Math. Sci. (N. Y.), 189 (2013) pp. 508--524.

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# Numerical convergence for semilinear parabolic equations

*Authors*

- Huth, Robert

*2010 Mathematics Subject Classification*

- 47N40 65J15

*Keywords*

- numerical analysis, semilinear parabolic equation, sectorial operators

*DOI*

*Abstract*

We present a convergence result for finite element discretisations of semilinear parabolic equations, in which the evaluation of the nonlinearity requires some high order of regularity of the solution. For example a coefficient might depend on derivatives or pointevaluation of the solution. We do not rely on high regularity of the exact solution itself and as a payoff we can not deduce convergence rates. As an example the convergence result is applied to a nonlinear Fokker--Planck type battery model.

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# Eigensolutions of the Wigner--Eisenbud problem for a cylindrical nanowire within finite volume method

*Authors*

- Racec, Paul N.
- Schade, Stanley
- Kaiser, Hans-Christoph

*2010 Mathematics Subject Classification*

- 65N30 65Z05 35P99

*2008 Physics and Astronomy Classification Scheme*

- 62.23.Hj 73.63.-b 71.15.-m

*Keywords*

- Finite element method, Schrödinger operator, cylindrical coordinates, R-matrix formalism, Wigner--Eisenbud problem, nanowire

*DOI*

*Abstract*

We present a finite volume method for computing a representative range of eigenvalues and eigenvectors of the Schrödinger operator on a three dimensional cylindrically symmetric bounded domain with mixed boundary conditions. More specifically, we deal with a semiconductor nanowire which consists of a dominant host material and contains heterostructure features such as double-barriers or quantum dots. The three dimensional Schrödinger operator is reduced to a family of two dimensional Schrödinger operators distinguished by a centrifugal potential. Ultimately, we numerically treat them by means of a finite volume method. We consider a uniform, boundary conforming Delaunay mesh, which additionally conforms to the material interfaces. The 1/r singularity is eliminated by approximating r at the vertexes of the Voronoi boxes. We study how the anisotropy of the effective mass tensor acts on the uniform approximation of the first K eigenvalues and eigenvectors and their sequential arrangement. There exists an optimal uniform Delaunay discretization with matching anisotropy. This anisotropic discretization yields best accuracy also in the presence of a mildly varying scattering potential, shown exemplarily for a nanowire resonant tunneling diode. For potentials with 1/r singularity one retrieves the theoretically established first order convergence, while the second order convergence is recovered only on uniform grids with an anisotropy correction.

*Appeared in*

- J. Comput. Phys., 252 (2013) pp. 52--64.

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# Sasa--Satsuma equation: Soliton on a background and its limiting cases

*Authors*

- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Akhmediev, Nail

*2010 Mathematics Subject Classification*

- 35Q55 35Q60 37K40

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Tg 05.45.Yv 42.81.Dp

*Keywords*

- Generalized nonlinear Schrödinger equations, rogue wave, soliton, Sasa-Satsuma equation

*DOI*

*Abstract*

We present a multi-parameter family of a soliton on a background solutions to the Sasa-Satsuma equation. The solution is controlled by a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits a few nontrivial limiting cases that are considered in detail. Among these special cases is the NLSE limit and the limit of rogue wave solutions.

*Appeared in*

- Phys. Rev. E (3), 86 (2012) pp. 026606/1--026606/8.

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# Persistence of rogue waves in extended nonlinear Schrödinger equations: Integrable Sasa--Satsuma case

*Authors*

- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Akhmediev, Nail

*2010 Mathematics Subject Classification*

- 35Q55 35Q60 37K40

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Tg 05.45.Yv 42.81.Dp

*Keywords*

- Generalized nonlinear Schrödinger equations, Sasa-Satsuma equation, solitons, rogue waves

*DOI*

*Abstract*

We present the lowest order rogue wave solution of the Sasa-Satsuma equation (SSE) which is one of the integrable extensions of the nonlinear Schrödinger equation (NLSE). In contrast to the Peregrine solution of the NLSE, it is significantly more involved and contains polynomials of fourth order rather than second order in the corresponding expressions. The correct limiting case of Peregrine solution appears when the extension parameter of the SSE is reduced to zero.

*Appeared in*

- Phys. Lett. A, 376 (2012) pp. 1558--1561.

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# A criterion for a two-dimensional domain to be Lipschitzian

*Authors*

- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35A01 57N40 57N50

*Keywords*

- Elliptic/parabolic problems, bi-Lipschitzian parametrization

*DOI*

*Abstract*

We prove that a two-dimensional domain is already Lipschitzian if only its boundary admits locally a one-dimensional, bi-Lipschitzian parametrization.

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# A gradient formula for linear chance constraints under Gaussian distribution

*Authors*

- Henrion, René
- Möller, Andris

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Chance constraints, probabilistic constraints, derivative of singular normal distribution, derivative of Gaussian probability for polyhedra

*DOI*

*Abstract*

We provide an explicit gradient formula for linear chance constraints under a (possibly singular) multivariate Gaussian distribution. This formula allows one to reduce the calculus of gradients to the calculus of values of the same type of chance constraints (in smaller dimension and with different distribution parameters). This is an important aspect for the numerical solution of stochastic optimization problems because existing efficient codes for e.g., calculating singular Gaussian distributions or regular Gaussian probabilities of polyhedra can be employed to calculate gradients at the same time. Moreover, the precision of gradients can be controlled by that of function values which is a great advantage over using finite difference approximations. Finally, higher order derivatives are easily derived explicitly. The use of the obtained formula is illustrated for an example of a transportation network with stochastic demands.

*Appeared in*

- Math. Oper. Res., 37 (2012) pp. 475--488.

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# Self-heating effects in organic semiconductor devices enhanced by positive temperature feedback

*Authors*

- Fischer, Axel
- Pahner, Paul
- Lüssem, Björn
- Leo, Karl
- Scholz, Reinhard
- Koprucki, Thomas

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

ORCID: 0000-0003-4432-2434 - Gärtner, Klaus
- Glitzky, Annegret

*2010 Mathematics Subject Classification*

- 80A20 35K05 35C05 65N08

*Keywords*

- heat conductivity, organic semiconductor, C60, crossbar electrodes, Joule heating, device temperature, thermal resistance, break down, analytical solution, heat flow equation, 3D simulation, finite volume method

*DOI*

*Abstract*

We studied the influence of heating effects in an organic device containing a layer sequence of n-doped / intrinsic / n-doped C_{60} between crossbar metal electrodes. A strong positive feedback between current and temperature occurs at high current densities beyond 100 A/cm^{2}, as predicted by the extended Gaussian disorder model (EGDM) applicable to organic semiconductors. These devices give a perfect setting for studying the heat transport at high power densities because C_{60} can withstand temperatures above 200° C. Infrared images of the device and detailed numerical simulations of the heat transport demonstrate that the electrical circuit produces a superposition of a homogeneous power dissipation in the active volume and strong heat sources localized at the contact edges. Hence, close to the contact edges, the current density is significantly enhanced with respect to the central region of the device, demonstrating that three-dimensional effects have a strong impact on a device with seemingly one-dimensional transport.

*Appeared in*

- Organic Electronics, 13 (2012) pp. 2461--2468 under the title ``Self-heating effects in organic semiconductor crossbar structures with small active area''

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# Quasi-phase-matching for third harmonic generation in noble gases employing ultrasound

*Authors*

- Sapaev, Usman
- Babushkin, Ihar
- Herrmann, Joachim

*2010 Mathematics Subject Classification*

- 78A60 35Q61 35Q60

*Keywords*

- Four Wave Mixing, Photoionization, Ultrashort pulse propagation

*DOI*

*Abstract*

We study a novel method of quasi-phase-matching for third harmonic generation in a gas cell using the periodic modulation of the gas pressure and thus of the third order nonlinear coefficient in the axial direction created by an ultrasound wave. Using a comprehensive numerical model we describe the quasi-phase matched third harmonic generation of UV (at 266 nm) and VUV pulses (at 133 nm) by using pump pulses at 800 nm and 400 nm, respectively, with pulse energy in the range from 3 mJ to 1 J. In addition, using chirped pump pulses, the generation of sub-20-fs VUV pulses without the necessity for an external chirp compensation is predicted.

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# On balance laws for mixture theories of disperse vapor bubbles in liquid with phase change

*Authors*

- Dreyer, Wolfgang
- Hantke, Maren
- Warnecke, Gerald

*2010 Mathematics Subject Classification*

- 35Q53 35G25 76E15

*Keywords*

- averaging methods, conservation laws, phase change, bubbles, phase mixtures

*DOI*

*Abstract*

We study averaging methods for the derivation of mixture equations for disperse vapor bubbles in liquids. The carrier liquid is modeled as a continuum, whereas simplified assumptions are made for the disperse bubble phase. An approach due to Petrov and Voinov is extended to derive mixture equations for the case that there is a phase transition between the carrier liquid and the vapor bubbles in water. We end up with a system of balance laws for a multi-phase mixture, which is completely in divergence form. Additional non-differential source terms describe the exchange of mass, momentum and energy between the phases. The sources depend explicitly on evolution laws for the total mass, the radius and the temperature of single bubbles. These evolution laws are derived in a prior article and are used to close the system. Finally numerical examples are presented.

*Appeared in*

- Continuum Mechanics and Thermodynamics, 26 (2014) pp. 521--549.

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# On elliptic and parabolic regularity for mixed boundary value problems

*Authors*

- Haller-Dintelmann, Robert
- Jonsson, Alf
- Knees, Dorothee
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35B65 35J47 35J57 74B05

*Keywords*

- mixed boundary conditions interpolation, elliptic regularity for equations and systems, analytic semigroups

*DOI*

*Abstract*

We study second order equations and systems on non-Lipschitz domains including mixed boundary conditions. The key result is interpolation for suitable function spaces.

*Appeared in*

- Math. Methods Appl. Sci., 39 (2016), pp. 5007--5026, DOI 10.1002/mma.3484/abstract .

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# Variational convergence of gradient flows and rate-independent evolutions in metric spaces

*Authors*

- Mielke, Alexander

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

*2010 Mathematics Subject Classification*

- 49Q20 58E99

*Keywords*

- Doubly nonlinear equations, evolution in metric spaces, generalized gradient flows, viscous regularization, vanishing-viscosity limit, BV solutions, rate-independent systems

*DOI*

*Abstract*

We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials. As a particular application, we show that BV solutions to rate-independent problems arise naturally as a limit of p-gradient flows, p>1, when the exponents p converge to 1.

*Appeared in*

- Milan J. Math., 80 (2012) pp. 381--410.

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# The Turing bifurcation in network systems: Collective patterns and single differentiated nodes

*Authors*

- Wolfrum, Matthias

*2008 Physics and Astronomy Classification Scheme*

- 89.75Fb 89.75Kd

*Keywords*

- Turing instability, Diffusively coupled networks, Localized patterns

*DOI*

*Abstract*

We study the emergence of patterns in a diffusively coupled network that undergoes a Turing instability. Our main focus is the emergence of stable solutions with a single differentiated node in systems with large and possibly irregular network topology. Based on a mean-field approach, we study the bifurcations of such solutions for varying system parameters and varying degree of the differentiated node. Such solutions appear typically before the onset of Turing instability and provide the basis for the complex scenario of multistability and hysteresis that can be observed in such systems. Moreover, we discuss the appearance of stable collective patterns and present a codimension-two bifurcation that organizes the interplay between collective patterns and patterns with single differentiated nodes.

*Appeared in*

- Phys. D, 241 (2012) pp. 1351--1357.

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# Few-cycle optical solitons in dispersive media beyond the envelope approximation

*Authors*

- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Akhmediev, Nail

*2008 Physics and Astronomy Classification Scheme*

- 42.81.Dp, 42.65.Tg, 05.45.Yv, 42.65.Re

*Keywords*

- Optical solitons, Ultrashort pulses, Dispersion

*DOI*

*Abstract*

We study the propagation of few-cycle optical solitons in nonlinear media with an anomalous, but otherwise arbitrary, dispersion and a cubic nonlinearity. Our theory extends beyond the slowly varying envelope approximation. The optical field is derived directly from the Maxwell equations under the assumption that generation of the third harmonic is a non-resonant process or at least cannot destroy the pulse prior to inevitable linear damping. The solitary wave solutions are obtained numerically up to nearly single-cycle duration using a modification of the spectral renormalisation method originally developed for the envelope solitons.

*Appeared in*

- Phys. Rev. A, 87 (2013) pp. 013805/1--013805/8.

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# Dynamical regimes of multi-stripe laser array with external off-axis feedback

*Authors*

- Pimenov, Alexander
- Tronciu, Vasile Z.
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Vladimirov, Andrei G.

*2010 Mathematics Subject Classification*

- 78A60 37M05 78A45 34K60

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.Mi 42.60.Da 42.55.Sf

*Keywords*

- Broad-area semiconductor lasers, multi-stripe laser array, off-axis feedback, delay differential equations, bifurcation analysis

*DOI*

*Abstract*

We study theoretically the dynamics of a multistripe laser array with an external cavity formed by either a single or two off-axis feedback mirrors, which allow to select a single lateral mode with transversely modulated intensity distribution. We derive and analyze a reduced model of such an array based on a set of delay differential equations taking into account transverse carrier grating in the semiconductor medium. With the help of the bifurcation analysis of the reduced model we show the existence of single and multimode instabilities leading to periodic and irregular pulsations of the output intensity. In particular, we observe a multimode instability leading to a periodic regime with anti-phase oscillating intensities of the two counter-propagating waves in the external cavity. This is in agreement with the result obtained earlier with the help of a 2+1 dimensional traveling wave model.

*Appeared in*

- J. Opt. Soc. Amer. B Opt. Phys., 30 (2013) pp. 1606--1613.

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# Multi-channel wavelength conversion using four-wave mixing in semiconductor ring lasers

*Authors*

- Pérez-Serrano, Antonio
- Javaloyes, Julien
- Balle, Salvador

*2010 Mathematics Subject Classification*

- 78A60 65P99

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.65.Hw 42.65.St

*Keywords*

- Semiconductor lasers, Semiconductor ring lasers (SRLs), Four-wave mixing (FWM), All-optical wavelength conversion, Travelling wave model (TWM)

*DOI*

*Abstract*

We theoretically study all-optical simultaneous wavelength conversion of multiple channels by four-wave mixing in semiconductor ring lasers. Locking the semiconductor ring laser to a holding beam allows to achieve large conversion efficiencies with good signal-tonoise ratio in several channels at multi-Gb/s bit rates. Cross-talk between signals, arising from the peculiar four-wave mixing cascade of modes in semiconductor ring lasers and their cross-gain saturation, is studied in detail. We show that it can be controlled by adjusting the intensity of the holding beam, the bias current of the laser and the number, intensity and wavelength of signals that one wants to convert.

*Appeared in*

- IEEE Phot. Tech. Letter, 25 (2013) pp. 476--479.

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