On the stability of a point charge for the Vlasov-Poisson system

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We consider the dynamics of the repulsive Vlasov-Poisson equations near a point charge, addressing the question of its stability: For solutions which start as small, smooth and suitably localized perturbations of a point charge we capture the precise asymptotic behavior. Building on optimal decay estimates, this is revealed to be a modified scattering dynamic. Our analysis relies on the Hamiltonian/symplectic structure of the equations, and makes use of an exact integration of the linearized equation through angle-action coordinates. This is joint work with Jiaqi Yang (ICERM) and Benoit Pausader (Brown University).

(Semi)continuity of Krasnoselskii eigenvalues under convergence of the underlying spaces

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The general question in this talk is whether eigenvalues of operators defined on geometric spaces converge as these spaces converge to a limit space. Because the operators that we consider are often nonlinear, we need proper generalizations of eigenvalues, such as Krasnoselskii eigenvalues. I'll describe how to obtain continuity for eigenvalues of the Laplace operator defined on CD(K,N) spaces under measured Gromov-Hausdorff convergence, and I will explain how to get semicontinuity of Krasnoselskii eigenvalues under volume flat convergence. The talk contains results from joint works with Luigi Ambrosio, Shouhei Honda, Jeff Jauregui and Raquel Perales.

Stability of planar multiphase mean curvature flow beyond circular topology change

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A classical result of Gage-Hamilton and Grayson asserts that any smooth, closed and simple curve in the plane evolving by mean curvature flow (MCF) shrinks to a point in finite time and becomes circular in the process. We establish a weak-strong stability result beyond the time of such circular topology change: any weak (i.e., BV) solution of planar multiphase MCF starting sufficiently close to a smooth, closed and simple curve evolving by MCF has to stay close to it for all times.

Previous weak-strong stability results of this form are limited to time horizons before the first topology change of the strong solution. The reason is that the time-dependent constant in the associated Gronwall estimate is borderline non-integrable. We overcome this issue by developing a weak-strong stability theory for circular topology change up to dynamic shift, which amounts to dynamically adapting the strong solution to the weak solution to a degree which takes care of the leading-order non-integrable contributions in the Gronwall estimate. This strategy is in a sense reminiscent of the recently developed theory of L^2-stability up to shift for solutions of conservation laws close to shock solutions due to Vasseur and co-authors.

Rigidity of harmonic functions on the supercritical percolation cluster

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We study the behavior of harmonic functions on the infinite cluster in supercritical Bernoulli bond percolation. Most of the results which have been proved on these functions assert that they behave similarly to the harmonic functions on the lattice Zd (a typical example is the Liouville theorem on the percolation cluster). In this talk, we will be interested in the differences between harmonic functions on the lattice and on the percolation cluster through some specific properties. In this direction, we prove that there cannot exist a Lipschitz harmonic function on the percolation cluster. This property is false on the lattice as the function x -> x_1 is both 1-Lipschitz and harmonic. We will review some of the known results on harmonic functions on the percolation cluster and discuss some ideas of the proof. This is joint work with A. Bou-Rabee and W. Cooperman.

Maximal Regularity for Equations with Degenerate Weights

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We study local and global maximal regularity estimates for elliptic equations with degenerate weights.  We show that higher integrability of the gradients can be obtained by imposing a local small oscillation condition on the weight and a local small Lipschitz condition on the boundary of the domain. Our results are new in the linear and non-linear case. We show by example that the relation between the exponent of higher integrability and the smallness parameters is sharp even in the linear or the unweighted case. We study the regularity of the weighted Laplacian and p-Laplacian with degenerate elliptic matrix-valued weights. We establish a novel logarithmic BMO-condition on the weight that allows to transfer higher integrability of the data to the gradient of the solution.The talk is based on joint works with Lars Diening, Raffaella Giova, Antonia Passarelli di Napoli, Sun-Sig Byun  and Ho-Sik Lee.

Energy scaling of singular-perturbation models involving higher-order laminates

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Motivated by the appearance of complex microstructures in the modelling of shape-memory alloys, we study the energy scaling behaviour of some N-well problems with surface energy given by a singular higher-order term. In the case of absence of gauge invariances (e.g. with respect to the action of SO(n) or Skew(n)), we provide an ansatz-free lower bound which relies on a bootstrap argument in Fourier space and gives evidence of the higher order of lamination involved. The upper bound is provided by iterated branching constructions.
In the end, we show how a similar approach can be used in the determination of a lower bound for a more realistic model, namely the geometrically linearized cubic-to-tetragonal phase transition, in which a second order lamination is forced by the presence of affine boundary conditions. This is a joint work with Angkana Rüland.

Homogenization of stiff inclusions through network approximation

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The homogenization of a conductive medium randomly perforated with inclusions of infi-
nite conductivity is a well-known problem thanks to the work of Vassili Zhikov. However,
the existence of an effective model is shown under assumptions on the interparticle dis-
tance, which prevents the study of clusters and dense setting. In this talk of stochastic
homogenization, we will provide a relaxed criterion ensuring homogenization relying on
ideas from network approximation. This is joint work with David Gérard-Varet from Université Paris-Cité.

Homogenization of nonlinear randomly perforated materials under minimal assumptions on the geometry

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In this work we combine and generalize earlier works of Giunti-Höfer-Velazquez (on the homogenization of the Poisson equation in random critically perforated domains) and Ansini-Braides (on a variational approach for the more general nonlinear vectorial problem in the periodic setting), each one in the direction of the other.

Namely, we show that under similar general assumptions on the geometry of the random perforations as the ones posed in the work of Giunti-Höfer and Velazquez, the stochastic analogue of the result of Ansini-Braides holds true, with an average deterministic nonlinear capacitary-term appearing in the Γ-limit.

This is joint, ongoing work with Caterina Zeppieri and Lucia Scardia.

11. Januar 2023 (Beginn 14:15 Uhr in Raum M511)

Effective viscosity of dilute suspensions

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This talk is devoted to the large-scale rheology of systems of suspended particles in a Stokes fluid. After reviewing recent results on the definition of the effective viscosity of such systems in terms of homogenization theory, I will focus on its asymptotic expansion in dilute regime. I will present a new, optimal proof of Einstein's viscosity formula for the first-order expansion and show how to pursue the asymptotic expansion to higher orders. The essential difficulty originates in the long-range nature of hydrodynamic interactions: these require suitable renormalizations, which will be systematically captured by means diagrammatic expansions. This is based on joint work with A.Gloria.

29. November 2022

(Beginn 12:15 Uhr im Banachraum)

Multiplicity and curvature

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A classical inequality of Li and Yau gives a lower bound on the Willmore energy of a surface in terms of its multiplicity at an arbitrary point. We discuss recent extensions to other curvature-dependent bending energies, such as Euler's elastic energy or the Canham-Helfrich functional. Additionally, we illustrate applications to geometric flows and the regularity of minimizers.

16. November 2022

A new paradigm for the spectral theory of systems

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I will present a new approach to the spectral theory of systems of PDEs on closed manifolds, developed in a series of recent papers by Dmitri Vassiliev (UCL) and myself, based on the use of pseudodifferential projections. Emphasis will be placed on ideas and motivation; the talk will include a brief historical overview of the development of the subject area.

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 High-contrast random composites: homogenisation framework and new spectral phenomena

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We study the homogenisation problem for elliptic operators with high-contrast random coefficients describing a two-phase composite with the heterogeneity scale $\epsilon$ made up of small inclusions randomly distributed in a matrix component. It is assumed that the coefficients of the operator are of order one in the matrix component and of order $\epsilon^2$ in the inclusions (so-called double porosity scaling). High-contrast composites receive special attention in scientific community due to the band-gap structure of their spectrum. In this talk I will give an introduction to the high-contrast setting followed by an overview of our recent results in the topic and some ongoing work.

02.November 2022

A connection between homogenisation of compressible Navier-Stokes and fluid-structure problems

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In this talk I will present some recent improvements in the study of homogenisation of compressible viscous fluid in the case of tiny holes and the connection with fluid structure problems. In particular I highlight a situation where the fluid + rigid body problem has more flexibility respect to case of the fluid alone. This flexibility helps us to deduce new ideas to study the case of the fluid alone.

29. September 2022

(Beginn: 14:15 Uhr im Raum M611)

The factorization method in inverse scattering theory

Up to the boundary regularity for BVPs governed by Fully Nonlinear Operators

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We introduce the notion of viscosity solutions for second order fully nonlinear elliptic/parabolic equations. A viscosity solution, u, is defined through the condition that whenever a test function, φ, touches u from below/above at a point then φ should be a classical super/sub-solutions of the equation at this point. Since u is apriori merely continuous, it is natural to study whether the equation finally impose any regularizing effects on u.

We start by giving an overview of the theory that have been developed the last decades on the interior regularity properties of viscosity solutions, introducing the main tools used, namely the ABP and Harnack inequalities. Then we shall concentrate on oblique derivative problems (where the oblique derivative condition is understood only in the viscosity sense as well) and discuss about the up to the boundary (first order) Hölder regularity of the solutions giving quantitative bounds for the corresponding Hölder norms in terms of the data.

Regularity results for Legendre-Hadamard elliptic systems

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I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.

Asymptotic analysis of singularly perturbed elliptic functionals & application to stochastic homogenization.

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In this talk we introduce a general class of singularly-perturbed elliptic functionals Fε and we study their asymptotic behaviour as the perturbation parameter ε > 0 vanishes. Under suitable
assumptions, which in particular allow us to bound Fε by the Ambrosio-Tortorelli functionals, we show that the functionals Fε Γ-converge to a free-discontinuity functional of brittle type. This is a consequence of a volume-surface decoupling effect.
If time permits, in the last part we will discuss the application of the general convergence result to the setting of stochastic homogenization.
This result was obtained in collaboration with Annika Bach & Caterina Ida Zeppieri.

An L^1 based approach to optimal relaxation of gradient flows: bump-like solutions of the 1-D Cahn-Hilliard equation

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In the beginning we will review the classical approaches to obtain exponential or algebraic rates of convergence towards minimizers of the energy in the case of gradient flows in (strictly) convex energy landscapes. We will discuss advantages and disadvantages of this method generalized to mildly non-convex energies and in particular try to understand in which cases this approach yields optimal rates. In the second part of the talk we will consider a more robust, but related method that relies on L^1-type distances instead of the "natural" distance associated to the gradient flow. By means of example we will consider the Cahn-Hilliard equation in one dimension and look at solutions with two transition layers. Here, in contrast to the case of one transition layer, the long-time limit is not fixed by mass conservation and in order to control the dynamics, a detailed understanding of the movement of the transition layers is needed. This talk is based on joint work with Sarah Biesenbach and Maria G. Westdickenberg

01. Februar 2022 (Beginn 14:15 Uhr)

Jakob Fuchs

The Thresholding Scheme for the Mean Curvature Flow of Mean Convex Sets

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Merriman, Bence and Osher's thresholding scheme is a
time discretization of mean curvature flow. I restrict to the two-phase
setting and mean convex initial conditions. In the sense of the
minimizing movements interpretation of Esedoglu and Otto I show the
time-integrated energy of the approximation to converge to the
time-integrated energy of the limit. As a corollary, the conditional
strong convergence results of Laux and Otto become unconditional in this
case. The results are general enough to handle the extension of the
scheme to anisotropic flows for which a non-negative kernel can be chosen.

Oscillatory solutions of coagulation-fragmentation equations

The effective characterization of metamaterials with nonlocal constitutive relations

Network approximation in high contrast homogenization

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We will discuss the homogenization of inclusions of infinite conductivity, randomly stationary distributed inside a homogeneous conducting medium.
A now classical result by Zhikov shows that, under a logarithmic moment bound on the minimal distance between the inclusions, an effective model with finite homogeneous conductivity exists. Relying on ideas from network approximation, we will provide a relaxed criterion ensuring homogenization. Several examples not covered by the previous theory will be presented. This is joint work with A. Girodroux-Lavigne.

GEOMETRIC RIGIDITY IN VARIABLE DOMAINS

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Abstract

A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

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Weak solution concepts for mean curvature flow have been investigated since the seminal work of Brakke in the 70’s. We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow - is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle. This is joint work with Sebastian Hensel (U Bonn).

Optimal matching and large scale regularity theory for optimal transportation

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Abstract: The optimal matching problem is one of the classical random optimization problems. The macroscopic behaviour is well understood since the work of Ajtai, Komlos, Tusnady, Talagrand and others in the 80s and early 90s. A few years ago, Caracciolo et al. proposed a new ansatz, based on a linearisation of the Monge-Ampere equation to the Poisson equation, to get refined estimates for this problem.

I will show how one can justify this approach in a quantitative way leading to a large scale regularity theory for the optimal transport problem. This in turn can then be used to get new information on the matching problem, like convergence to a Gaussian field which we will shortly discuss at the end of the talk.

(16:15 Uhr, Raum M614)

 Computing oscillatory solutions in fluid mechanics
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Oscillatory solutions to models of inviscid fluids may be described by probability distribution of certain phase variables in terms of the Young measure. Using the analytical results of
Banach - Sachs and Komlos, we propose a method how to visualize/compute effectively these objects. Convergence of the approximate/numerical measure-valued solutions is strong in space and time with respect the standard Wasserstein distance.

08. Juli 2021
(16:15 Uhr, Raum E19)

Emergence of rigid polycrystals from atomistic systems

Wenn Sie an der Veranstaltung teilnehmen wollen, dann melden Sie sich bitte bei Fiona Drees (fiona.drees@tu-dortmund.de) an.

30. März 2020
(14:00 Uhr)

3. März 2020
(Dienstag,11:00 Uhr)