May 21-23, 2014
We show that two related combinatorial problems have continuum limits that
correspond to solving Hamilton-Jacobi equations. The first problem is
non-dominated sorting, which is fundamental in multi-objective
optimization, and the second is directed last passage percolation (DLPP),
which is an important stochastic growth model closely related to directed
polymers and the totally asymmetric simple exclusion process (TASEP). We
give convergent numerical schemes for both Hamilton-Jacobi equations and
explore some applications.
Dislocations are topological singularities in crystals, which may be described
by lines to which a lattice-valued vector, called Burgers vector, is associated.
They may be identified with divergence-free matrix-valued measures
supported on curves or with 1-currents with multiplicity in a lattice.
In the modeling of dislocations one is thus often lead to energies concentrated
on lines, where the integrand depends on the orientation and on the Burgers
vector of the dislocation.
In this talk I will present the theory of relaxation for such energies and
I will show how they may arise in a multiscale analysis of dislocations,
starting from discrete and semi-discrete models.
Keywords of the presentation: Crystalline curvature flow, viscosity solution, comparison principle.
It is by now standard that a level-set approach provides a global unique generalized solution (up to fattening) for mean curvature flow equations [G]. Even from the early stage of the theory, it is known that the method is very flexible to apply anisotropic curvature flow equations which correspond to anisotropic interfacial energy.
The anisotropy is very important in materials sciences. However, if the anisotropy is very singular for example a crystalline curvature flow corresponding to crystalline interfacial energy a level set approach was not available except evolution of curves to which a foundation of the theory was established by M.-H. Giga and Y. Giga more than ten years ago.
In this talk we push forward a level-set approach to surface evolution by crystalline curvature. The main difficulty is that crystalline curvature is a nonlocal quantity and it may not be a constant on each flat potion of a surface. (In the case of curve evolution it is always a constant over a segment.) We overcome this difficulty by introducing a suitable notion of viscosity solutions so that a comparison principle holds. We further construct a global-in-time solution as a limit of smoother problem. A delicate analysis is necessary to achieve the goal. A similar but a simpler problem was studied in [MGP1], [MGP2]. We elaborate these approaches for our purpose.
[G] Y. Giga, Surface evolution equations. A level set approach. Monographs in Mathematics, 99. Birkhäuser Verlag, Basel, 2006. xii+264 pp.
[MGP1] M.-H. Giga, Y. Giga and N. Požár, Periodic total variation flow of non-divergence type in $R^n$, J. Math. Pures Appl., to appear.
[MGP2] M.-H. Giga, Y. Giga and N. Požár, Anisotropic total variation flow of non-divergence type on a higher dimensional torus. Adv. Math. Sci. Appl. 23 (2013), no. 1, 235–266.
The classical formula for the buckling load of axially compressed
cylindrical shells predicts 4-5 times higher critical load than observed in
experiment. The discrepancy is explained by high sensitivity of the buckling
load to imperfections. However, the exact mechanism of this sensitivity
remains elusive. Our rigorous theory of "near-flip" buckling of slender
bodies reveals a scaling instability of the critical load caused by
imperfections of load. A possible mechanism of sensitivity to imperfections of
shape can also be seen in our analysis. This is a joint work with Davit Harutyunyan.
Keywords of the presentation: Materials science, structure determination, x-rays, isometry groups, objective structures, Maxwell's equations
A central problem of materials science is to determine atomic structure from macroscopic measurements. Von Laue developed a theoretical method that was put into practice and popularized by Bragg, based on the scattering of plane waves by a crystal lattice. Recently, new structures have emerged like buckyballs (Nobel Prize, Chemistry, 1996) and graphene (Nobel Prize, Physics, 2010), and the third fascinating form of carbon, the carbon nanotube (no Nobel prize yet). These have a regular structure but are not crystalline. Regular but noncrystalline structures are also quite common in biology: examples of medical interest include many parts of viruses, and amyloid protein fibrils that cause diseases like Alzheimer's, Parkinson's and Creutzfeldt-Jakob disease. Structures like buckyballs, graphene and carbon nanotubes were not discovered — they are believed to have been present on earth since nearly its beginning — but the recent interest lies in the fact that they were isolated and studied, and exhibited interesting properties. There may well be related, maybe equally interesting but perhaps less common, structures around us. Thus, the determination of the atomic structure of noncrystalline structures by macroscopic methods is a central problem. After clearing up some fallacies about ordinary x-ray crystallography (Bragg’s incorrect picture, the intensity of scattered radiation is not the squared norm of the Fourier transform of the lattice), we propose a new method, which involves exploiting the relationship between structure and the invariance group of Maxwell’s equations. We work out the details for helical structures like carbon nanotubes and amyloid protein fibrils. This is joint work with Dominik Juestel and Gero Friesecke, TU Munich.
We formulate an atomistic-to-continuum coupling method based on blending
atomistic and continuum forces. We present a comprehensive error analysis
that is valid in two and three dimensions, for finite many-body
interactions (e.g., EAM type), and in the presence of lattice defects
(point defects and dislocations). Based on a precise choice of blending
mechanism, the error estimates are considered in terms of degrees of
freedom. The numerical experiments confirm and extend the theoretical
predictions, and demonstrate a superior accuracy of B-QCF over energy-based
Keywords of the presentation: Hyperbolic media, Superlensing, Sub-wavelength Resolution
The study of hyperbolic media, where the dielectric tensor has eigenvalues of mixed signs, has attracted considerable attention as a way of achieving subwavelength resolution. The quasistatic field around a circular hole in a two-dimensional hyperbolic medium is studied. As the loss parameter goes to zero, it is found that the electric field diverges along four lines each tangent to the hole. In this limit, the power dissipated by the field in the vicinity of these lines, per unit length of the line, goes to zero but extends further and further out so that the net power dissipated remains finite. Additionally the interaction between polarizable dipoles in a hyperbolic medium is studied. It is shown that a dipole with small polarizability can dramatically influence the dipole moment of a distant polarizable dipole, if it is appropriately placed. We call this the searchlight effect, as the enhancement depends on the orientation of the line joining the polarizable dipoles and can be varied by changing the frequency. For some particular polarizabilities the enhancement can actually increase the further the polarizable dipoles are apart, a bit like the interaction between quarks.
Keywords of the presentation: Nonlocal Fokker Planck equation, double well potential, time-dependent consrtaint, rate independent model for hysteresis, phase transitions
We discuss a nonlocal Fokker-Planck equation that describes energy minimisation in a double well-potential and is driven by a time-dependent constraint. Via formal asymptotic analysis we identify different small parameter regimes that correspond to hysteretic and non-hysteretic phase transitions respectively. For the fast reaction regime that is related to Kramers-type phase transitions we also indicate how can rigorously derive a rate-independent evolution equation in a small parameter limit.
This is joint work with Michael Herrmann and Juan Velazquez.
Keywords of the presentation: Ginzburg-Landau, Ohta-Kawasaki, Coulomb gases, Abrikosov lattices, crystallization
I will review some results concerning vortices in the Ginzburg-Landau
model, droplets in the two-dimensional Ohta-Kawasaki model, and Coulomb
gases, which all have in common that they reduce to systems of points with
Coulomb interaction. I will discuss the derivation of a "renormalized
energy" for the limits of such systems, and the question of
This is based on joint works with Etienne Sandier, Nicolas Rougerie,
Dorian Goldman, Cyrill Muratov, Simona Rota-Nodari.
Keywords of the presentation: stability, minimizer, liquid crystal, defect
We use the framework of Landau-de Gennes theory to investigate various defect profiles and their stability in 2D and 3D liquid crystal systems. In 2D we show that a defect profile can be characterized by a system of ODEs. We show that in deep nematic regime this defect profile is a ground state of the liquid crystal system. In 3D we investigate the stability of radially symmetric profile ("melting hedgehog") corresponding to the point defect. We show its stability in the mathematically challenging temperature regime near the supercooling temperature.
We present a method developed jointly with Felix Otto to capture optimal convergence rates for a gradient flow via natural algebraic and differential relationships among distance, energy, and dissipation. The method is developed and applied in the context of relaxation to a kink profile in the one-dimensional Cahn-Hilliard equation on the line. Application to other models is discussed.