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Workgroup Functional Analysis

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 2.041

Address
Karlsruher Institut für Technologie
Institut für Analysis
Englerstraße 2
76131 Karlsruhe


natascha.katz@kit.edu


Office hours:
9:00 - 11:00

Tel.: +49 721 608 43727

Fax.: +49 721 608 67650

Previous Talks

Talks in the summer term 2019

30.07.2019 TULKKA in Karlsruhe


Die Vorträge finden in Raum 1.067 im Kollegiengebäude Mathematik (20.30) statt.

11:30-12:15 Lucrezia Cossetti (Karlsruhe) Unique continuation for the Zakharov-Kuznetsov equation
In this talk we analyze uniqueness properties of solutions to the (2+1)-Zakharov-Kuznetsov (ZK) equation
 \partial_t u + \partial_{x}^3u + \partial_{x}\partial_{y}^2u + u \partial_x u=0, \qquad (x,y)\in \mathbb{R}^2,\quad t\in [0,1].
Mainly motivated by the very well known PDE's counterpart of the Hardy uncertainty principle, we provide a two times unique continuation result. More precisely, we prove that given u_1, u_2 two solutions to ZK, as soon as the difference u1-u2 decays (spatially) fast enough at two different instants of time, then u1 \equiv u2. As expected, it turns out that the decay rate needed to get uniqueness reflects the asymptotic behavior of the fundamental solution of the associated linear problem. Encouraged by this fact we also prove optimality of the result.
Some recent results concerning the (3+1)- dimensional ZK equation will be also presented.
The seminar is based on a recent paper (CFL) in collaboration with L. Fanelli and F. Linares.
(CFL) L.Cossetti, L.Fanelli and F.Linares, Uniqueness results for Zakharov-Kuznetsov equation, Comm. Partial Differential Equations, DOI:10.1080/03605302.2019.1581803
12:30-14:00 Mittagspause
14:00-14:45 Marius Müller (Ulm) The biharmonic Alt-Caffarelli problem
We discuss a variational free boundary problem of Alt-Caffarelli type. We consider the energy
\mathcal{E}(u):=\int_{\Omega}(\Delta u)^2 dx+\lvert \{x\in\Omega\ :\ u(x)>0\}\rvert,
defined for a membrane u\in W^{2,2}(\Omega) that is spiked at some positive level u_0>0 on \partial\Omega. The two summands impose conflicting interests on minimizers: Little bending versus a large region of nonpositivity. We study regularity of minimizers and the free boundary, which happens to be the nodal set \{u=0\}.
As it will turn out, each minimizer has non-vanishing gradient on its free boundary, which connects the regularity of the two objects. Regularity discussion of the minimizer leads to the study of measure-valued Dirichlet problems and carries a potential theoretic flavor.
15:00-15:45 Karsten Herth (Konstanz) Decay rates for anisotropic Reissner-Mindlin plates
The Reissner-Mindlin plate is a model for thick plates, where the mid-surface normal is not required to remain perpendicular to the mid-surface. We analyze the asymptotic behavior of anisotropic thermoelastic Reissner-Mindlin plate equations in the whole space, using the Fourier transform and the method of stationary phase. This leads to Fresnel-like surfaces, similar to those in anisotropic elasticity, whose points of vanishing curvature are linked with the decay behavior.
16:00-16:45 Kaffeepause (in Raum 1.058)
16:45-17:30 Lahcen Maniar (Marrakesch) Null controllability for a heat equation with dynamic boundary condition and drift terms
We consider the heat equation in a space bounded domain subject to dynamic boundary conditions of surface diffusion type and involving drift terms in the bulk and in the boundary. We prove that the system is null controllable at any time. The results is based on new Carleman estimates for these type of boundary conditions. We conclude by new results on semilinear equations with non linear functions occuring a blow up of the solutions without control.
ab 18:00 Abendessen im Restaurant "Il Caminetto" (Kronenstr. 5)



21.05.2019Lucrezia Cossetti (Karlsruhe) Multipliers method for Spectral Theory.
Originally arisen to understand characterizing properties connected with dispersive phenomena, in the last decades the multipliers method has been recognized as a useful tool in Spectral Theory, in particular in connection with proof of absence of point spectrum for both self-adjoint and non self-adjoint operators.
Starting from recovering very well known facts about the spectrum of the free Laplacian H_0=-\Delta in L^2(\mathbb{R}^d), we will see the developments of the method reviewing some recent results concerning self-adjoint and non self-adjoint perturbations of this Hamiltonian in different settings, specifically both when the configuration space is the whole Euclidean space \mathbb{R}^d and when we restrict to domains with boundary. We will show how this technique allows to detect physically natural repulsive and smallness conditions on the potentials which guarantee the absence of eigenvalues. Some very recent results concerning Pauli and Dirac operators will be presented too.
The talk is based on joint works with L. Fanelli and D. Krejcirik.
28.05.2019Philipp Harms (Freiburg) Smoothness of the functional calculus and applications to variational PDEs.
The functional calculus, which maps operators A to functionals f(A), is holomorphic for a certain class of operators A and holomorphic functions f. In particular, fractional Laplacians depend real analytically on the underlying Riemannian metric in suitable Sobolev topologies. As an application, this can be used to prove local well-posedness of some geometric PDEs, which arise as geodesic equations of fractional order Sobolev metrics.
Joint work with Martins Bruveris, Martin Bauer, and Peter W. Michor.

Talks in the winter term 2018/2019

29.01.2019Fabian Hornung (Karlsruhe)Neural Network Approximation for high dimensional Kolmogorov PDE.
First we introduce some basic concepts in the theory of artificial neural networks (ANNs) and present how Kolmogorov PDEs can be reformulated as a minimization problem using techniques from stochastic analysis. This can serve as foundation of deep learning algorithms to numerically solve these PDEs.
In the second part of the talk, we sketch a proof that artificial neural networks approximate the PDE-solution without curse of dimensionality, i.e. that the number of parameters of the approximating ANN is bounded by a polynomial in the dimension of the problem and the reciprocal of the accuracy.
The talk is based on joint work with Philipp Grohs, Arnulf Jentzen, and Philippe von Wurstemberger.

The talk takes place at 15:45 in room 3.060.
19.02.2019 TULKKA in Ulm


11:30-12:15 Lisa Fischer (Konstanz) Generalized thermoelastic plate: Well-posedness and frequency analysis
12:15-13:45 Lunch break
13:45-14:30 Konstantin Zerulla (Karlsruhe) Ein ADI-Verfahren mit gleichmäßig exponentiell stabilen Approximationen für die Maxwell-Gleichungen
14:45-15:30 Emil Wiedemann (Ulm) Analysis of Turbulent Flows: Compressible and Incompressible
15:30-16:15 Coffee break
16:15-17:00 Patrick Tolksdorf (Darmstadt) A smooth introduction to fluid mechanics in rough domains
ab 17:45 Dinner in the city center of Ulm
More information on Tulkka can be found here.


12.03.2019Boris Baeumer (Dunedin)Boundary conditions for Levy processes on bounded domains and their governing PDEs.
Levy processes are jump processes governed by non-local operators on \mathbb{R}^d and are used to model dispersive systems where the occasional large dispersal event (many standard deviations) is driving the system. In modelling, boundaries appear naturally and in 1D we answer the question of what type of boundary condition for the non-local operator corresponds to what type of boundary behaviour of the process by using numerical approximation schemes.

The talk takes place in room 2.066 from 14:00 to 15:00.
12.03.2019Petru Cioica-Licht (Essen)SPDEs on domains with corner singularities.
Although there exists an almost fully-fledged L_p-theory for (semi-)linear second order stochastic partial differential equations (SPDEs, for short) on smooth domains, very little is known about the regularity of these equations on non-smooth domains with corner singularities. As it is already known from the deterministic theory, corner singularities may have a negative effect on the regularity of the solution. For stochastic equations, this effect comes on top of the already known incompatibility of noise and boundary condition. In this talk I will show how a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary may be used in order to deal with both sources of singularity and their interplay.

The talk takes place in room 2.066 from 15:00 to 16:00.


Talks in the summer term 2018

09.05.2018Pierre Portal (Canberra)An operator theoretic generalisation of pseudo-differential calculus.
The H^{\infty} calculus is an operator theoretic construction that allows one to extend Fourier multiplier theory, towards rough settings in particular. In this talk, we consider a similar functional calculus that extends pseudo-differential operator theory. It involves two group generators satisfying the canonical commutator relations, and thus generalising the usual position and momentum operators. I'll discuss a transference result relating this calculus to twisted convolutions on Bochner spaces, and a formula connecting it to the calculus of abstract harmonic oscillators. The latter allows us, in particular, to show using Kriegler-Weis theory that these harmonic oscillators have a Hormander calculus.

The talk takes place at 10:00 in room 2.067.
17.07.2018 TULKKA in Konstanz


11:45-12:30 Marcel Kreuter (Ulm) Vektorwertige elliptische Randwertprobleme auf rauen Gebieten
12:30-13:45 Lunch break
13:45-14:30 Martin Spitz (Karlsruhe) Lokale Wohlgestelltheit nichtlinearer Maxwell-Gleichungen mit perfekt leitenden Randbedingungen
14:45-15:30 Sita Siewert (Tübingen) Exponentielle Dichotomie und Spektrum dynamischer Banach-Moduln
15:30-16:15 Coffee break
16:15-17:00 Gieri Simonett (Nashville) On the Muskat problem
ab 17:45 Dinner in the restaurant "Hafenhalle"
More information on Tulkka can be found here.


Talks in the winter term 2017/2018

21.11.2017Konstantin Zerulla (Karlsruhe)Stabilitätserhaltende diskretisierte Approximationen gedämpfter Wellengleichungen.
28.11.2017Martin Spitz (Karlsruhe)Nichtlineare Maxwellgleichungen - Blow-up Kriterium und stetige Abhängigkeit.
05.12.2017Yuri Tomilov (IM PAN, Warsaw)Why do circles in the spectrum matter?
We present several results linking the joint numerical ranges of Hilbert space operator tuples to the circle structure of the spectrum of tuples. We will explain how our approach allows us to unify, extend or supplement several results where the circular structure of the spectrum is crucial: Arveson's theorem on almost-wandering vectors of unitary actions, Brown-Chevreau-Pearcy's theorem on invariant subspaces of Hilbert space contractions and Hamdan's recent result on supports of Rajchman measures, to mention a few. Moreover, we will give several applications of the approach to new operator-theoretical constructions inverse in a sense to classical power dilations.
This is joint work with V. Müller (Prague).
30.01.2018Luca Hornung (Karlsruhe)Wohlgestelltheit einer nichtlinearen Maxwell-Gleichung mit retardiertem Materialgesetz.


06.02.2018 TULKKA in Karlsruhe


The talks take place in room 1.067 in the math building.

11:30-12:15 Marie-Luise Hein (Ulm) Das Prinzip der linearisierten Stabilität für parabolische Volterra Gleichungen
In diesem Vortrag werde ich das Prinzip der linearisierten Stabilität für parabolische Volterra Gleichungen für den Spezialfall des Standard-Kerns erläutern. Anschließend werde ich ein Stabilitätsresultat für quasilineare zeit-fraktionelle Evolutionsgleichungen in der Situation der maximalen L_p-Regularität präsentieren.
12:30-14:00 Lunch break
14:00-14:45 Tim Binz (Tübingen) Gleichmäßig elliptische Operatoren mit Wentzell Randbedingung und der Dirichlet-zu-Neumann Operator
Für gleichmäßig elliptische Operatoren und die zugehörigen konormalen Ableitungen auf stetigen Funktionen gelang es J. Escher 1994 zu zeigen, dass der assoziierte Dirichlet-zu-Neumann Operator Generator einer analytischen Halbgruppe ist.
Im ersten Teil des Vortrages werden wir den Zusammenhang zwischen Operatoren mit Wentzell Randbedingungen und Dirichlet-zu-Neumann Operatoren studieren.
Dazu wird ein abstrakter Rahmen eingeführt, der eine Diskussion von Randwertproblemen mit Hinblick auf Generatoreneigenschaften erlaubt. Anschließend geben wir einen alternativen Beweis für die Aussage von Escher, der es uns zusätzlich erlaubt den Winkel zu berechnen. Darüber hinaus verallgemeinern wir die Aussage auf kompakte Mannigfaltigkeiten mit Rand.
15:00-15:45 Felix Kammerlander (Konstanz) Exponentielle Stabilität für ein gekoppeltes System von ungedämpft-gedämpften Plattengleichungen
Wir betrachten ein Transmissionsproblem elastischer Platten in einem Gebiet \Omega, welches aus einem inneren Gebiet \Omega_2 \subset \Omega mit \overline{\Omega_2} \subset \Omega und einem äußeren Gebiet \Omega_1 = \Omega \backslash \overline{\Omega_2}, welches \Omega_2 umschließt, besteht. In \Omega_2 betrachten wir eine ungedämpfte Plattengleichung, in \Omega_1 hingegen eine strukturell gedämpfte Platte. Mithilfe passender Transmissionsbedingungen sind die beiden Gleichungen an der Grenzschicht der beiden Gebiete miteinander gekoppelt.
Unter Verwendung von Halbgruppentheorie zeigt man die Wohlgestelltheit des Problems in einem geeigneten Hilbertraum. Die Energie des Gesamtsystems nimmt ab, wobei der Verlust der Energie einzig und allein durch die Dämpfung in \Omega_1 verursacht wird. Wir zeigen, dass die Dämpfung in \Omega_1 bereits stark genug ist, um exponentielles Abklingen der Energie für das Gesamtsystem zu erhalten. Hierfür wird eine gewisse a-priori Abschätzung für das parabolische System der gedämpften Platte benötigt.
16:00-16:45 Coffee break
16:45-17:30 Amru Hussein (Darmstadt) Beyond maximal L^p-regularity - a case study in spaces of bounded functions
For semilinear equations the maximal L^p-regularity approach gives local well-posedness for initial values in trace spaces. For typical second order parabolic problems these lie between the ground space L^p and H^{2,p}. In particular some differentiability is necessary.
In particular cases one can weaken the assumptions to consider rough initial data without differentiability assumptions by moving to the end point of the L^p scale, i.e. considering L^{\infty}. Here, we illustrate this for the case of the primitive equations. This is a geophysical model derived from Navier-Stokes equations assuming a hydrostatic balance. We prove that the combination of heat semigroup and Riesz transforms is a bounded operator in spaces of bounded functions and that this combination satisfies certain smoothing properties. This is essential to tackle the semilinear problem by an evolution equation approach. The classical maximal L^p-regularity approach gives additional regularity properties, and suitable a priori bounds lead to a global solution even for rough initial data.
ab 18:00 Dinner in the restaurant "Il Caminetto" (Kronenstr. 5)
More information on Tulkka can be found here.



21.02.2018Emiel Lorist (Delft)Vector-valued extrapolation to Banach function spaces.
If an operator T is bounded on L^p(\mathbb{R}^d,w) for some 1<p<\infty and all weights w in the class of Muckenhoupt weights A_p, then T extends to a bounded operator on the Bochner space L^p(\mathbb{R};X) for any Banach function space X with the UMD property, which is a vector-valued extrapolation theorem by Rubio de Francia. In this talk I will discuss several generalizations of this theorem. In particular I will present a multilinear limited range version for vector-valued extrapolation to Banach function spaces and discuss various applications, including vector-valued Littlewood-Paley-Rubio de Francia-type estimates, the L^p(\mathbb{R},w;X)-boundedness of Fourier multipliers and the variational Carleson operator, and boundedness of the vector-valued bilinear Hilbert transform.
This is joint work with Alex Amenta, Bas Nieraeth and Mark Veraar (TU Delft).

See the german page for older talks.