Department 
Seminars 
 

Maxwell Institute for Mathematical Sciences


Heiko Gimperlein
Maxwell Institute for Mathematical Sciences and Department of Mathematics
Edinburgh EH14 4AS
phone: (0044) 131 451 - 8293
email: h.gimperlein at hw.ac.uk


Publications

  • Heiko Gimperlein and Jakub Stocek,
    Space-time adaptive finite elements for nonlocal parabolic variational inequalities.
    preprint, slides of talk.

  • Heiko Gimperlein, Ceyhun Ozdemir, David Stark and Ernst P. Stephan,
    hp-version time domain boundary elements for the wave equation on quasiuniform meshes.
    preprint, slides of talk.

    Solutions to the wave equation in a polyhedral three-dimensional domain exhibit singular behavior near corners and edges. We present quasioptimal hp-explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions on quasiuniform meshes on the boundary. The results are applied to an hp-version of the time domain boundary element method, and an posteriori error estimate is obtained towards adaptive mesh refinements for the Dirichlet problem. Numerical examples confirm the theoretical results.

  • Gissell Estrada-Rodriguez and Heiko Gimperlein,
    Interacting particles with Levy strategies: limits of transport equations for swarm robotic systems.
    preprint, movies: Levy robots with alignment interactions.

    Levy robotic systems combine superdiffusive random movement with emergent collective behaviour from local communication and alignment in order to find rare targets or track objects. In this article we derive macroscopic fractional PDE descriptions from individual robot strategies. Starting from a kinetic equation which describes the movement of robots based on alignment, collisions and occasional long distance runs according to a Levy distribution, we obtain a system of evolution equations for the fractional diffusion for long times. We show that the system allows efficient parameter studies for a target search problem, addressing basic questions like the optimal number of robots needed to cover an area in a certain time. For shorter times, in the hyperbolic limit of the kinetic equation the PDE model is dominated by alignment, irrespective of the long-range movement. This is in agreement with previous results in swarming of self-propelled particles. The article indicates the novel and quantitative modeling opportunities which swarm robotic systems provide for the study of both emergent collective behaviour and anomalous diffusion, on the respective time scales.

  • Heiko Gimperlein, Ceyhun Ozdemir, David Stark and Ernst P. Stephan,
    A residual a posteriori estimate for the time-domain boundary element method.
    preprint, slides of talk.

    This article investigates residual a posteriori error estimates and adaptive mesh refinements for time-dependent boundary element methods for the wave equation. Based on ideas for elliptic problems, we exhibit estimates for Dirichlet, Neumann and acoustic boundary conditions. Numerical examples study the resulting adaptive mesh refinement procedures for a 3-dimensional model problem and compare the error reduction with a hierarchical error indicator.

  • Heiko Gimperlein and Magnus Goffeng,
    On the magnitude function of domains in Euclidean space.
    preprint, blog posts 1, 2.

    We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain X in an odd-dimensional Euclidean space, we find geometric significance in the function M(R) = mag(R X). M(R) extends from the positive half-line to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit M(R) admits an asymptotic expansion. The three leading terms are given by the volume, surface area and integral of the mean curvature. In particular, for convex X the leading terms are given by the intrinsic volumes, and we obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton, with corrected coefficients.

  • Heiko Gimperlein and Magnus Goffeng,
    Commutator estimates on contact manifolds and applications.
    preprint, slides of talk.

    This article studies sharp norm estimates for the commutator of pseudo-differential operators with non-smooth functions on closed sub-Riemannian manifolds. In particular, we obtain a Calderon commutator estimate: If D is a first-order operator in the Heisenberg calculus and f is Lipschitz in the Carnot-Caratheodory metric, then [D,f] extends to an L^2-bounded operator. Using interpolation, it implies sharp weak-Schatten class properties for the commutator between zeroth-order operators and Holder continuous functions. We present applications to sub-Riemannian spectral triples on Heisenberg manifolds as well as to the regularization of a functional studied by Englis-Guo-Zhang.

  • Heiko Gimperlein and Ernst P. Stephan,
    Adaptive FE-BE coupling for strongly nonlinear transmission problems with friction II.
    preprint, slides of talk.

    This article discusses the well-posedness and error analysis of the coupling of finite and boundary elements for transmission or contact problems in nonlinear elasticity. It concerns W^{1,p}-monotone Hencky materials with an unbounded stress-strain relation, as they arise in the modelling of ice sheets, non-Newtonian fluids or porous media. For pseudoplastic materials, where p is smaller than 2, the bilinear form of the boundary element method fails to be continuous in natural function spaces associated to the nonlinear operator. We propose a functional analytic framework for the numerical analysis and obtain a priori and a posteriori error estimates for Galerkin approximations to the resulting boundary/domain variational inequality. The a posteriori estimate complements recent estimates obtained for mixed finite element formulations of friction problems in linear elasticity.

  • Heiko Gimperlein, Matthias Maischak and Ernst P. Stephan,
    FE-BE coupling for a transmission problem involving microstructure,
    preprint.

    In this second article on the theoretical numerical analysis of nonlinear transmission problems, we extend our previously introduced methods to a nonconvex double-well potential coupled to the linear Laplace equation via contact conditions. This problem arises in phenomenological models for phase transitions and does not even admit a weak solution. However, certain macroscopic quantities like the stress, the region of fine oscillations, so-called microstructure, or the displacement outside this region are captured in a Young measure, which may be computed efficiently.

  • Gissell Estrada-Rodriguez, Heiko Gimperlein, Kevin Painter and Jakub Stocek,
    Space-time fractional diffusion in cell movement models with delay.
    preprint, slides of talk.

    The movement of organisms and cells can be governed by long distance runs according to an approximate Levy walk. In the case of T cells in infected brain tissue the runs are further interrupted by long delays. This article claries the form of continuous model equations for such movement: In the biologically relevant regime we derive nonlocal diffusion equations, involving fractional Laplacians in space and fractional time derivatives, from the microscopic velocity jump models used to interpret the experiments. Starting from power-law distributions of run and waiting times, we investigate the relevant parabolic limit of a kinetic equation for the resting and moving individuals. The analysis and numerical experiments shed light on how chemokines shorten the average time taken by T cells to find rare targets, and on the absence of chemotaxis. In addition to immunology, our work has implications for similar Levy diffusion observed in other organisms, such as mussels, marine predators and monkeys.

  • Heiko Gimperlein, Fabian Meyer, Ceyhun Ozdemir, David Stark and Ernst P. Stephan,
    Boundary elements with mesh refinements for the wave equation.
    Numerische Mathematik 139 (2018), 867-912, preprint, slides of talk.

    The solution of the wave equation in a polyhedral domain admits an asymptotic expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation as an equivalent boundary integral equations in time domain and study the regularity properties and numerical approximation of the solution. Guided by the theory for elliptic equations, graded meshes are shown to recover the optimal approximation rates expected for smooth solutions. Numerical experiments illustrate the theory for screen problems. In particular, we discuss the Dirichlet problem, the Dirichlet-to-Neumann operator and applications to the sound emitted by a tire.

  • Gissell Estrada-Rodriguez, Heiko Gimperlein and Kevin Painter,
    Fractional Patlak-Keller-Segel equations for chemotactic superdiffusion.
    SIAM Journal on Applied Mathematics 78 (2018), 1155-1173, preprint, slides of talk..

    The long range movement of certain organisms in the presence of a chemoattractant can be governed by long distance runs, according to an approximate Levy distribution. This article clarifies the form of biologically relevant model equations: We derive Patlak-Keller-Segel-like equations involving nonlocal, fractional Laplacians from a microscopic model for cell movement. Starting from a power-law distribution of run times, we derive a kinetic equation in which the collision term takes into account the long range behaviour of the individuals. A fractional chemotactic equation is obtained in a biologically relevant regime. Apart from chemotaxis, our work has implications for biological diffusion in numerous processes.

  • Heiko Gimperlein and David Stark,
    On a preconditioner for time domain boundary element methods.
    Engineering Analysis with Boundary Elements 96 (2018), 109-114, preprint.

    We propose a time stepping scheme for the space-time systems obtained from Galerkin time-domain boundary element methods. Based on extrapolation, the method proves stable, becomes exact for increasing degrees of freedom and can be used either as a preconditioner, or as an efficient standalone solver for scattering problems with smooth solutions. It also significantly reduces the number of GMRES iterations for screen problems, with less regularity, and we explore its limitations for enriched methods based on non-polynomial approximation spaces.

  • Heiko Gimperlein and David Stark,
    Algorithmic aspects of enriched time domain boundary element methods.
    Engineering Analysis with Boundary Elements (2018), online first, preprint.

  • Heiko Gimperlein, Fabian Meyer, Ceyhun Ozdemir and Ernst P. Stephan,
    Time domain boundary elements for dynamic contact problems.
    Computer Methods in Applied Mechanics and Engineering 333 (2018), 147-175.
    preprint, slides of talk.

    This article considers a unilateral contact problem for the scalar wave equation. The problem is reduced to a variational inequality for the Dirichlet-to-Neumann operator for the wave equation on the boundary, which is solved in a saddle point formulation using boundary elements in the time domain. As a model problem, also a variational inequality for the single layer operator is considered. A priori estimates are obtained for Galerkin approximations in the case of a flat contact area, where the existence of solutions to the continuous problem is known. Numerical experiments demonstrate the performance of the proposed method. They indicate the stability and convergence beyond flat geometries.

  • Heiko Gimperlein, Ceyhun Ozdemir and Ernst P. Stephan,
    Time domain boundary element methods for the Neumann problem: Error estimates and acoustic problems.
    Journal of Computational Mathematics 36 (2018), 70-89, preprint, slides of talk.
    special issue in honor of Hsiao, Nedelec and Wendland

    We investigate time domain boundary element methods for the wave equation, with a view towards sound emission problems in computational acoustics. The Neumann problem is reduced to a time dependent integral equation for the hypersingular operator, and we present a priori and a posteriori error estimates for conforming Galerkin approximations. Numerical experiments validate the convergence of our boundary element scheme and compare it with the numerical approximations obtained from an integral equation of the second kind. Computations in a half-space, for varying acoustic properties of the street, illustrate the role of our approach in the study of traffic noise.

  • Heiko Gimperlein, Quoc Thong Le Gia, Matthias Maischak and Ernst P. Stephan,
    Solving approximate cloaking problems using finite element methods.
    ANZIAM Journal 58 (2017), C162-C174, Proceedings of CTAC 2016, preprint.

    Motivated by approximate cloaking problems, we consider a variable coefficient Helmholtz equation with a fixed wave number. We use finite element methods to discretize the equation. Numerical results investigate the cloaking behaviour in the numerical solutions.

  • Claudio Dappiaggi, Heiko Gimperlein, Simone Murro and Alexander Schenkel,
    Wavefront sets and polarizations on supermanifolds.
    Journal of Mathematical Physics 58 (2017), 023504, 16 pages, preprint.

    We develop the foundations for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable concept of super-wavefront set for superdistributions which generalizes Dencker's polarization sets for vector-valued distributions to supergeometry. In particular, our super-wavefront sets are able to detect polarization information of the singularities of superdistributions. We prove a refined pullback theorem for superdistributions along supermanifold morphisms, which as a special case establishes criteria when two superdistributions may be multiplied. As an application, we study the singularities of distributional solutions of a supersymmetric field theory.

  • Heiko Gimperlein and Magnus Goffeng,
    Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds.
    Forum of Mathematics, Sigma 5 (2017), e3, 57 pages, preprint, slides of talk.

    We study the spectral behavior and noncommutative geometry of commutators [P,f], where P is an operator of order 0 with geometric origin and f a multiplication operator by a function. When f is Holder continuous, the spectral asymptotics is governed by singularities. Even though a Weyl law fails for these operators, and no pseudo-differential calculus is available, variations of Connes' residue trace theorem and related integral formulas continue to hold. On the circle, a large class of non-measurable Hankel operators is obtained from Holder continuous functions f, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

  • Heiko Gimperlein and Alden Waters,
    A deterministic optimal design problem for the heat equation.
    SIAM Journal on Control and Optimization 55 (2017), 51 - 69, preprint.

    In everyday language, this article studies the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015), and we remove both the randomness and geometric assumptions in their article. Analytically, we obtain quantitative estimates for the wellposedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. Using wave packet decompositions from microlocal analysis, we conclude that there exists a unique optimal such subset, that it is semi-analytic and can be approximated by solving a sequence of finite-dimensional optimization problems.

  • Lothar Banz, Heiko Gimperlein, Abderrahman Issaoui and Ernst P. Stephan,
    Stabilized mixed hp-BEM for frictional contact problems in linear elasticity.
    Numerische Mathematik 135 (2017), 217 - 263, preprint, slides of talk.

    We analyze a high-order, adaptive boundary element method for the approximation of a frictional contact problem. As a basic problem of high-order polynomial ansatz functions, the resulting discretized system degenerates as the polynomial degree tends to infinity. In the 90s, Barbosa and Hughes showed that for finite elements the stability (LBB) condition could be restored by adding certain terms, which vanish for the exact solution. We show that their basic approach can be extended to the boundary element method, prove convergence and obtain an a posteriori error estimate. Numerical experiments confirm the theoretical results and explore the dependence on stabilization parameters.

  • Heiko Gimperlein, Matthias Maischak and Ernst P. Stephan,
    Adaptive time-domain boundary element methods and engineering applications.
    invited survey, Journal of Integral Equations and Applications 29 (2017), 75-105, preprint.

    A review article about the recent progress in time-domain boundary element methods for the wave equation.

  • David Stark and Heiko Gimperlein,
    Boundary elements and mesh refinements for the wave equation.
    conference: Proceedings of the 25th UK Conference of the Association for Computational Mechanics in Engineering (2017), 309-312, preprint.

  • Gissell Estrada-Rodriguez and Heiko Gimperlein,
    Generalized finite elements for blow-up solutions to reaction-diffusion equations.
    conference: Proceedings of the 25th UK Conference of the Association for Computational Mechanics in Engineering (2017), 81-84, preprint.

  • Heiko Gimperlein, Bernhard Krötz and Henrik Schlichtkrull,
    Corrigendum - Analytic representation theory of Lie groups: General theory and analytic globalizations of Harish-Chandra modules,
    Compositio Mathematica 153 (2017), 214-217, preprint.

  • Lothar Banz, Heiko Gimperlein, Zouhair Nezhi and Ernst P. Stephan,
    Time domain BEM for sound radiation of tires.
    Computational Mechanics 58 (2016), 45-57, preprint, slides of talk.

    This article presents computational results for certain benchmark problems related to the noise emission of car tires. We use a time domain single layer potential ansatz to solve the acoustic wave equation outside a tire in the halfspace above a street. The resulting integral equation on the boundary is solved approximately by a Galerkin boundary element method using piecewise constant test and trial functions in space and time. After a validation experiment, we present numerical results for the A-weighted sound pressure of a vibrating tyre, for the noise amplification due to the horn-like geometry of the tire in conjunction with the road as well as for the Doppler effect.

  • Heiko Gimperlein and Alden Waters,
    Stability analysis in magnetic resonance elastography II.
    Journal of Mathematical Analysis and Applications 434 (2016), 1801-1812, preprint.

    In the sequel to Ammari, Waters and Zhang (2015), we show how the general Fredholm theory of elliptic boundary problems improves the available stability estimates for an inverse problem for the Stokes equation, with very short proofs. The new estimates prove the convergence of practically relevant numerical reconstruction schemes.

  • David Stark and Heiko Gimperlein,
    A partition of unity boundary element method for transient wave propagation.
    conference: Proceedings of the 24th UK Conference of the Association for Computational Mechanics in Engineering (2016), 347-351, preprint.
    conference (best paper prize): Proceedings of the Infrastructure and Environment Scotland Postgraduate Conference 2015, 6 pages.

    We introduce and study a time-domain boundary element method based on plane-wave basis functions.

  • Muhammad Iqbal, M. Shadi Mohamed, Omar Laghrouche, Heiko Gimperlein, Mohammed Seaid and Jon Trevelyan
    Solution of three dimensional transient heat diffusion problems using an enriched finite element method.
    conference: Proceedings of the 24th UK Conference of the Association for Computational Mechanics in Engineering (2016), 197-200 preprint.

    We present a highly efficient generalized finite element method for the heat equation in three dimensions.

  • Muhammad Iqbal, Heiko Gimperlein, M. Shadi Mohamed and Omar Laghrouche,
    An a posteriori error estimate for the generalized finite element method for transient heat diffusion problems.
    International Journal for Numerical Methods in Engineering 110 (2017), 1103-1118, preprint.
    conference: Proceedings of the 23rd UK Conference of the Association for Computational Mechanics in Engineering (2015), 445-448, preprint.
    conference: Proceedings of the Infrastructure and Environment Scotland Postgraduate Conference 2015, 6 pages.
    conference: Proceedings of the Infrastructure and Environment Scotland Postgraduate Conference 2014, 6 pages.

    We investigate a posteriori error estimates and their relevance to engineering applications for time-dependent partition of unity finite element methods. We prove a simple, reliable a posteriori estimate of residual type for the heat equation. Numerical experiments illustrate its use for heat diffusion problems and engineering applications.

  • Heiko Gimperlein, Zouhair Nezhi and Ernst P. Stephan,
    A priori error estimates for a time-dependent boundary element method for the acoustic wave equation in a half-space.
    Mathematical Methods in the Applied Sciences 40 (2017), 448 - 462,
    special issue in honor of M. Costabel, preprint.

    We investigate a time-domain Galerkin boundary element method for the wave equation outside a Lipschitz obstacle in an absorbing half-space, with application to the sound radiation of tyres. A priori estimates are presented for both closed surfaces and screens, and we discuss the relevant properties of anisotropic Sobolev spaces and the boundary integral operators between them.

  • Heiko Gimperlein and Gerd Grubb,
    Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators,
    Journal of Evolution Equations 14 (2014), 49 - 83, preprint.

    The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. Operators satisfying such estimates are known to have a rich spectral theory, and applications include maximal regularity for the associated evolution equation and a functional calculus in L^p. In the selfadjoint case, extensions to t in C_+ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.

  • Adrian Costea, Heiko Gimperlein and Ernst P. Stephan,
    A Nash-Hörmander iteration and boundary elements for the Molodensky problem,
    Numerische Mathematik 127 (2014), 1 - 34, preprint.

    We investigate the mathematically justified numerical approximation of the nonlinear Molodensky problem, the free boundary problem to determine the unknown surface of the earth from the gravitational potential and the gravity vector on this surface. While the well-posedness of this problem has been shown by Hörmander, from a numerical perspective only the linearized problem has been understood. The solution, based on a smoothed Nash-Hörmander iteration, solves a sequence of exterior oblique Robin problems and uses a regularization based on a higher-order heat equation to overcome the loss of regularity. We obtain a quantitative a priori estimate for the error after k steps, legitimize the choice of heat-equation smoothing in this context and study the accurate evaluation of second derivatives of the gravitational potential on the boundary, using a representation in terms of a hypersingular integral. Numerical results, using a bondary element method for the exterior problem, compare the error between the approximation and the exact solution in a model problem.
    Costea's thesis contains a preliminary, extended discussion.

  • Lothar Banz, Adrian Costea, Heiko Gimperlein and Ernst P. Stephan,
    Numerical simulations of the nonlinear Molodensky problem,
    Proceedings of the European Geosciences Union General Assembly 2013,
    Studia geophysica et geodaetica 58 (2014), 489 - 504, preprint, conference proceedings.

    A complement to the article "A Nash-Hörmander iteration and boundary elements for the Molodensky problem" concerning numerical and geodetic aspects.

  • Heiko Gimperlein, Bernhard Krötz and Christoph Lienau,
    Analytic factorization of Lie group representations,
    Journal of Functional Analysis 262 (2012), 667 - 681, preprint.

    For every moderate growth representation of a real Lie group on a Frechet space E, the article proves a Dixmier-Malliavin factorization theorem for the space of analytic vectors E^w: There exists a natural algebra A of superexponentially decreasing analytic functions on the group such that E^w = A * E^w. This theorem sharpens a classical result by Nelson about the density of analytic vectors. While the rigid structure of analytic functions makes a direct approach difficult, the crucial idea is to refine the functional calculus of Cheeger, Gromov and Taylor to carefully transport selected functions from the real line to the group. The geometric methods also strengthen certain conclusions from Dixmier-Malliavin's result, extending Goodman's characterization that a vector is smooth whenever it is so for the Laplacian to nonunitary representations and to analytic vectors.

  • Heiko Gimperlein, Bernhard Krötz and Henrik Schlichtkrull,
    Analytic representation theory of Lie groups: General theory and analytic globalizations of Harish-Chandra modules,
    Compositio Mathematica 147 (2011), 1581 - 1607, preprint,
    Corrigendum.

    A representation of a real Lie group on a vector space E is called analytic, if every vector in E is analytic (i.e. its orbit map is a real-analytic function on the group) and if the natural inductive limit topology on the space of analytic vectors coincides with the topology of E. The article introduces a coherent general framework to analyze such representations and systematically studies the resulting categories of representations with moderate or arbitrary growth. For reductive groups, the minimal globalization of Kashiwara and Schmid is analyzed and an elementary proof of Schmid's globalization theorem is obtained: Every Harish-Chandra module over the Lie algebra and a maximal compact subgroup has a unique globalization to a moderate growth analytic representation of the group. It is given by a quotient of A^N, where A denotes the convolution algebra of analytic superexponentially decaying functions on the group from our previous work with Krötz and Lienau, and the theorem effectively provides an equivalence of categories.

  • Heiko Gimperlein, Matthias Maischak, Elmar Schrohe and Ernst P. Stephan,
    Adaptive FE-BE coupling for strongly nonlinear transmission problems with Coulomb friction,
    Numerische Mathematik 117 (2011), 307 - 332, preprint.

    This is the first of two articles analyzing adaptive finite element/boundary element procedures for nonlinear transmission problems with contact. Here we study a nonlinear uniformly monotone operator such as the p-Laplacian in a bounded Lipschitz domain coupled to the linear Laplace equation in the exterior complement. While elliptic problems have been studied for many years, the proper framework for these degenerately-nonlinear problems remained unclear. Our analysis shows how mixed L^p-L^2-Sobolev spaces provide a suitable setting to analyze the convergence of the Galerkin approximations. A posteriori error estimates similar to the ones for nonlinear Dirichlet problems allow to localize the error of the approximation and refine the grid accordingly.

  • Heiko Gimperlein, Bernhard Krötz and Henrik Schlichtkrull,
    Analytic globalizations of Harish-Chandra modules,
    Oberwolfach Reports 7 (2010), 3050-3052, preprint, conference proceedings.

    This extended abstract summarizes the topics covered in my talk in Overwolfach, November 2010. It outlines an elementary approach to aspects of Schmid's and Kashiwara's work on analytic globalizations of Harish-Chandra modules.

  • Heiko Gimperlein,
    Introduction to hyperbolic billiards,
    Oberwolfach Reports 7 (2010), 961-964, preprint, conference proceedings.

    This extended abstract summarizes the topics covered in my survey talk on dispersive billiards and Sinai's local ergodicity theorem in Oberwolfach, April 2010.

  • Heiko Gimperlein and Elmar Schrohe,
    Spectral projections of boundary problems and the regularity of the eta function,
    research announcement: preprint (2009).

    In the article (and the announcement), we show that the eta function associated to any selfadjoint elliptic boundary problem is regular in 0. Also, the value in 0 of the zeta function of a possibly nonselfadjoint boundary problem is independent of the spectral cut. Our result extends a subtle topological theorem of Atiyah-Patodi-Singer, Gilkey and Wodzicki for closed manifolds. Technically, we prove a vanishing theorem for the noncommutative residue, an essentially unique trace, on the K-theory of a new "minimal" algebra of general boundary problems. The theorem then follows by showing that the spectral projections of a boundary problem belong to the algebra. This is the crucial result of the paper and, in the context of spectral asymptotics, resolves a conjecture of Ivrii for second-order operators.

  • Heiko Gimperlein, Matthias Maischak, Elmar Schrohe and Ernst P. Stephan,
    A Finite element / boundary element coupling method for strongly nonlinear transmission problems with contact,
    Oberwolfach Reports 5 (2008), 2077-2080, preprint, conference proceedings.

    The announcement gives a numerically efficient variational formulation for the approximation of nonlinear transmission problems with contact. Certain mixed L^p-L^2-Sobolev spaces are shown to provide a convenient framework to analyze Galerkin approximations of the solutions to e.g. the p-Laplacian or a nonconvex double-well potential in a bounded Lipschitz domain coupled to a linear equation in the exterior complement. The analysis applies to systems encountered in nonlinear elasticity as well as to phenomenological models for phase transitions.

  • Heiko Gimperlein, Stefan Wessel, Jörg Schmiedmayer and Luis Santos,
    Random on-site interactions versus random potential in ultracold
    atoms in optical lattices
    ,
    Applied Physics B 82 (2006), 217-224, preprint.

    The article continues our study of bosons with random interactions and provides details which did not fit into the Letter below.

  • Heiko Gimperlein, Stefan Wessel, Jörg Schmiedmayer and Luis Santos,
    Ultracold atoms in optical lattices with random on-site interactions,
    Physical Review Letters 95 (2005), 170401/1-4, preprint.

    The Letter presents the zero-temperature phase diagram for bosons on a lattice with a random repulsive interaction between particles occupying the same site. Characteristic qualitative changes are observed when compared to the nonrandom case or a random chemical potential: The Mott-insulating regions shrink and eventually vanish for any finite disorder strength beyond a sufficiently large filling factor. Furthermore, at low values of the chemical potential both the superfluid and Mott insulator are stable towards formation of a Bose glass leading to a possibly nontrivial tricritical point. We propose a feasible experimental realization of our scenario in the context of ultracold atoms on optical lattices, which circumvents certain difficulties encountered in current experiments with disordered cold gases.

 


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