19. Oktober 2023

15.15 Uhr, Raum 110     Yassin El Karrouchi (Frankfurt)

Titel:  The universal failure of the pointwise fractional Talenti comparison principle


13. Juli 2023

14:15 Uhr       Huyuan Chen (Jiangxi, China)

Titel: On a nonlocal MEMS equations with variable dielectric permittivity

Abstract: Joint work with Feng Zhou and Dong Ye
In this talk, we study the nonlocal elliptic equation subject to the Dirichlet boundary condi-
tion. This equation models a electrostatic micro-electromechanical system (MEMS) device in
the more realistic case that the capacitance of the actuator is sensitive to the elastic membran-
ce's deformation. This MEMS equation was built by Lin-Yang in 2007. The difficulty to solve
the nonlocal elliptic equation arises from the nonlinearity, since the nonlocal factor breaks the
monotonicity and the traditional analysis fails to get the classification of solutions.
Combining the analysis of the related parabolic elliptic equation we obtain the dichotomy of
the existence for solutions of elliptic equation and by the choice of potential, we can get sufficient
conditions for the pull in voltage. Inversely, the results of elliptic equations is helpful to obtain
the qualitative properties of solutions parabolic problem


15:15 Uhr     Alberto Saldana (Coyoacán, México)

Title:  From competing species to nodal solutions of the Yamabe problem.

Abstract: What is the relationship between several populations fighting for resources and the Poincaré conjecture?
These two concepts seem to have nothing in common, but in this talk we will establish a close link by
relating three mathematical objects: the Yamabe equation, competitive systems
(that model the interaction between two species), and optimal partitions of the sphere.
This is joint work with Mónica Clapp and Andrzej Szulkin.


1. Juni 2023        Vitali Vougalter (Univ. Toronto)

Title: On the solvability of some systems of
integro-differential equations with concentrated sources

Abstract: The article is devoted to the existence of solutions
of a system of integro-differential equations in the case of the normal
diffusion and the influx/efflux terms proportional to the Dirac delta
function. The proof of the existence of solutions relies on a fixed point
technique. We use the solvability conditions for the non-Fredholm elliptic
operators in unbounded domains.


25. Mai 2023       Xueqin Peng (Gießen)

Title: Semiclassical problem for the logarithmic Schrödinger-Poisson-system
Abstract: I will discuss the Schrödinger-Poisson system with logarithmic nonlinearity and a local condition on the potential function.
The energy functional in this system is not well-defined in usual Sobolev space due to the logarithmic term. To address this,
I use a decomposition technique to express the original functional as a sum of a lower semicontinuous convex functional
and a C^1 functional. However, the lack of compactness still poses a challenge. To overcome this issue, I adapt the penalization
method proposed by M. Del Pino and P.L. Felmer to introduce a modified problem. By applying functional analytical methods,
I obtain the solution to the original problem.

This talk is based on the research results due to Alves and de Morais Filho: Z. Angew. Math. Phys. (2018).

27.04.2023     Daniel Hauer (University of Sydney)

Title: Functional Calculus via the extension technique: a first hitting time approach

13.04.2023     Sarah Kistner (Frankfurt)

Titel: Existence and Stability of Shrinkers for the Harmonic Map Heat Flow in Higher Dimensions

Abstract: A broad topic in geometrical analysis and partial differential equations is the study of harmonic maps
between Riemannian manifolds. A well-known tool in this subject is to consider the associated harmonic map heat flow.
As we are interested in blow-up mechanisms we construct for each d ≥ 4 a compact, d-dimensional, rotationally
symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution that represents
a stable blowup mechanism for the corresponding Cauchy problem.
By this we expand the known results on stable self-similar blow-up for the harmonic map heat flow in the supercritical case.
The only known result apart from ours describes this phenomenon for the target being the three-sphere.
This is joint work with Birgit Schörkhuber and Irfan Glogić.


09. März 2023     Matteo Rizzi (Universität Gießen)

Title: Rigidity results for semilinear elliptic PDEs.

Abstract: In the talk I will present some classification results for solutions to semilinear elliptic equations of the form $\Delta u=W'(u)$, where $W$ is a quite general potential with a local maximum and a local minimum. We are particularly interested in proving radial symmetry of a given solution and Liouville-type Theorems, which generalise some facts which are known to be true for the Cahn-Hilliard equation. It is a joint work with Panayotis Smyrnelis (from Athens University).


15.15 Uhr     Ludwig Striet (Freiburg)

Title: Approximation of fractional Operators and fractional PDEs using a sinc-basis

Abstract: We introduce a spectral method to approximate PDEs involving the fractional Laplacian with zero exterior condition. Our approach is based on interpolation by tensor products of sinc-functions, which combine a simple representation in Fourier-space with fast enough decay to suitably approximate the bounded support of solutions to the Dirichlet problem. This yields a numerical complexity of O(N log N) for the application of the operator to a discretization with N degrees of freedom. Iterative methods can then be employed to solve the fractional partial differential equations with exterior Dirichlet condition. We show a number of example applications  and establish rates of convergence that are in line with rates for finite element based approaches. Further, we discuss how the provided strategies can be easily adapted to approximate other operators as well as long es they have a suitable representation as a Fourier multiplier. This includes in particular the logarithmic Laplacian

20. Oktober 2022


Ignace Aristide Minlend (Frankfurt)


Titel: Free boundary value problems and nonlocal partial differential equation


Di, 20.9.2022, 14 ct  Raum 903 (Sondertermin!)

Silvia Cingolani (Universität Bari)

title: The effect of a potential well's topology on a fractional NLS equation

Montag, den 05.09.2022, um 15 Uhr st im Raum 903

Thomas Bartsch (Universität Giessen)

21. Juli 2022,   11:00 Uhr (s.t.) im Raum 903  

Alberto Saldana (Mexico-City)

Titel: The role of the logarithmic Laplacian in the asymptotic analysis of fractional problems.

Abstract: Fractional derivatives are commonly used to model a variety of phenomena, but...
what does it mean to have a logarithmic derivative? And what would it be used for?
In this talk we focus on the logarithmic Laplacian, a pseudodifferential operator that appears
as a first order expansion of the fractional Laplacian as the exponent s goes to zero. This
operator can also be represented as an integrodifferential operator with a zero order kernel.
We will discuss how the logarithmic Laplacian can be used to study the behavior of linear and
nonlinear fractional problems in the small order limit. This analysis will also reveal a deep and
interesting mathematical structure behind the set of solutions of Dirichlet logarithmic problems.

14. Juli 2022

Jing Wu (Universidad de Granada)

Title: Overdetermined elliptic problems in onduloid-type domains with general nonlinearities.

Abstract: In this paper we prove the existence of solutions to a general semilinear elliptic problem with overdetermined
boundary conditions. The proof uses a local bifurcation argument from the straight cylinder, in analogy with the onduloids
and the theory of Constant Mean Curvature surfaces. Such examples have been found already for linear problems or with
nonlinearity f(u)=1. In this work we are able to extend this phenomenon for a large class of functions f(u).

02. Juni 2022  

26. April 13:30 Uhr in Raum 109c (Sondertermin!)

Antonio Greco (University of Cagliari)

Titel: Symmetry and monotonicity results for solutions of semilinear PDEs in sector-like domains

Abstract: We consider semilinear PDEs in a sector-like domain. Using cylindrical coordinates (r,θ,z), we investigate the shape of solutions whose derivative in θ vanishes at the boundary. Assuming that the non-linearity f(r,z,u) is convex in u and does not depend on θ, we prove that any solution with Morse index less than two must be either independent of θ or strictly monotone with respect to θ. In the special case of a planar domain, the result holds in a circular sector as well as in an annular, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as an extension to unbounded domains. The proof is based on a rotating-plane argument: a convenient manifold is introduced in order to avoid overlapping the domain with its reflected image in the case when its opening is larger than π (joint work with Francesca Gladiali on arXiv:2201.11988).

27.5.2021, 14:15 Uhr     Paolo Luzzini (EPFL Lausanne) - Videovortrag

TITLE: Asymptotics of buckling eigenvalues

ABSTRACT: Since the seminal works of Hermann Weyl at the beginning of 19th century, several authors have investigated the spectral asymptotics of partial differential operators. Following this tradition, in this talk I will first present a recent result on the Weyl's law for the buckling eigenvalues on a wide class of domains, that includes bounded Lipschitz domains. The proof does not make use of microlocal analysis and relies on asymptotically sharp lower and upper bounds that we develop for  Riesz means. Moreover, we compute the second term in Weyl's law in the case of balls and bounded intervals. This, together with some formal considerations, leads us to state a conjecture for the second term in general domains. The talk is based on a joint work with Davide Buoso (UPO), Luigi Provenzano (Sapienza Università di Roma), and Joachim Stubbe (EPFL).
 
29.4.2021 um  15:15 Uhr (abweichende Uhrzeit!)

Omar Cabrera Chávez (Frankfurt) -  Videovotrag


Title: Multiple solutions to weakly coupled supercritical
elliptic systems

04.03.3021 14 Uhr (ct): Videovortrag Yassin El Karrouchi (Frankfurt)

Titel: Einführung in die Stochastische Fluid-Dynamik: Stochastische Navier-Stokes-Gleichung

Abstract: Die Navier-Stokes-Gleichung, ein System von partiellen Differentialgleichungen deren  Lösbarkeit und Regularität zu einer der Milleniumsprobleme gehört, beschreibt jegliches Verhalten von newtonschen Fluiden. Es ist klar, dass Moleküle einer Brownschen Bewegung folgen. Dies ist ein zentraler stochastische Prozess, mit dem wir eine stochastische Formulierung obiger Gleichung erhalten. Leider steht die Existenz einer Lösung der deterministischen Gleichung noch offen und damit auch die der stochastischen Navier-Stokes Gleichung. Diese Arbeit befasst sich mit der Untersuchung einer endlichdimensionalen Charakterisierung der Gleichung. Dazu wird eine stochastische Differentialgleichung betrachtet, dessen Existenz und Regularität von Lösungen zugänglicher sind und für wachsende Dimension untersucht. Hauptaussagen, werden Lösbarkeit und Regularität der endlichdimensionalen Charakterisierung sein und weitere dynamische Aspekte hinsichtlich Existenz invarianter Maße.

24.09.2020     Leon von Essen (Frankfurt)

Titel:    Spektraltheorie diskreter Schrödinger- und Jacobioperatoren mit beschränkten reellwertigen Potentialen

28.07. um 16:15  Uhr     Felix Höfer (Videovortrag)

Titel:   "Shape derivatives for minima of integral functions"

Abstract: In this talk we will first establish a definition for the shape derivative for a certain type of integral functions under Dirichlet- and Neumann-boundary conditions. It will be shown that the shape derivative exists under fairly weak assumptions on the corresponding functions. Since by default the shape derivative is not linear in the deformation component, we achieve this by imposing some more assumptions. In this case the shape derivative can be recast as a boundary integral. The main tools used to do this are convex analysis and an application of gamma-convergence.

25.06.2020   14 Uhr ct  (Videovortrag)

Joel Kübler (Frankfurt)

Title: Symmetry breaking for rotating solutions of nonlinear Klein-Gordon equations

Abstract: We study solutions of nonlinear Klein-Gordon equations with power-type nonlinearities, whose time-dependence is characterized by a rotation. This leads to a velocity-dependent elliptic equation where, in order to produce genuine rotating solutions, we investigate the existence of nonradial ground state solutions. We show that, for suitable values of the exponent in the nonlinearity, symmetry breaking occurs in the sense that the ground states transition from radial to nonradial when the velocity is varied. This is based on new degenerate Sobolev inequalities in the disc and half space with a new critical exponent. Finally, using concentration-compactness arguments, we show that the latter inequality possesses optimizers.
This is a joint work with Tobias Weth.

18.06.2020   14 Uhr ct  (Videovortrag)

Lea Ebrahimi (Vortrag zur Masterarbeit)

Titel: Die Mathematik der Pauke.
Abstract: Dieser Vortrag wird sich damit beschäftigen, die Eigenmoden und -frequenzen einer Paukenmembran unter Berücksichtigung des Luftdrucks analytisch herzuleiten. Dazu wird die Pauke als Zylinder modelliert, die Membran in eine unendliche Schallwand eingebettet und äußere Raumakustik vernachlässigt. Dann soll sowohl der Druck außerhalb als auch innerhalb der Pauke die dreidimensionale Helmholtzgleichung unter Neumann-Randbedingungen erfüllen. Der Luftdruck im Inneren des Kessels wird als Greenfunktion nach Eigenfunktionen (Besselfunktionen) entwickelt. Der Luftdruck außerhalb der Pauke wird mithilfe von Greenfunktionen nach der Spiegelmethode konstruiert.   Insgesamt kann so eine Gleichung an die Eigenmoden unter Luftdruck aufgestellt werden, deren Lösung nach den Eigenmoden im Vakuum entwickelt wird. Die Koeffizienten dieser Entwicklung bilden ein nicht eindeutig lösbares unendliches Gleichungssystem, das es numerisch zu lösen gilt


28.5.2020, 14 Uhr ct (Videovortrag)

Nicola Abatangelo (Frankfurt)

Title: Fractional Laplacians on ellipses

Abstract: We will illustrate some very recent findings concerning explicit evaluations of (higher-order) fractional Laplacians on elliptic domains. This will enable us to give the explicit expression of the torsion function and to construct elementary counterexamples to maximum principles. All the results have been obtained in collaboration with S. Jarohs and A. Saldana.


23.04.2020     Tolga Yesil (Frankfurt) - Videovortrag

Title:                 Weighted Fourier-extension estimates for symmetric functions and an application to a nonlinear Helmholtz equation.

Abstract:        In this talk we present a weighted Fourier-extension estimate for symmetric functions that are nonradial.
                     As an application we will prove the existence of nonradial solutions for a nonlinear Helmholtz equation for
                     values below the Stein-Tomas critical exponent 2(N+1)/(N-1). This is a joint work with Tobias Weth"


30.01.2020   14 Uhr ct  Inka Schnieders (Köln)

Titel: Ein Maximumprinzip für ein Dirichlet-Problem vierter Ordnung

23.01.2020    14:00 Uhr     Babak Maboudi

Title:             Detecting Jumps in a Jump-Discontinuous Random Field with Deep Neural Networks

Abstract:      Random fields are important tools in mathematical modeling which help us include our lack of knowledge or our faulty measurements into scientific computing. Such fields appear frequently as parameters in many modern physics and engineering applications. When these fields are not continuous, statistical analysis and approximation of these models become more involved. In this report, we investigate detecting jump-discontinuities in the diffusion-coefficient of an elliptic stochastic partial differential equation. This is formulated as an inverse problem. We take a Bayesian approach to formulate the distribution of the jumps in the random field. We then use a Markov chain Monte Carlo (MCMC) method to explore this distribution. Common MCMC methods provide a slow convergence rate which makes solving the forward problem inefficient. We propose replacing the forward model with a feed-forward fully-connected deep neural network. This dramatically reduces computational costs while providing an accurate forward model estimation. The accuracy and the performance of the method, as well as the implication of using a neural network as a surrogate model, will be discussed.

Sondertermin Fr. 20.09.2019

                      14:15 Pierre Aime Feulefack (Goethe Univ. Frankfurt)

Titel:             A characterization of the eigenvalues and eigenfunctions of the
                  logarithmic Laplacian


Abstract:      In this talk, we present a new insight into the characterization
                  of the eigenvalues and eigenfunctions of the logarithmic
                  Laplacianand a new technique (delta-decomposition) to obtain the
                  regularity bound of eigenfunctions.
                  The fractional Laplacianand its corresponding eigenvalues and
                  eigenfunctions is the cornerstone of our study.
                  As an application of our results, using the delta-decomposition
                  technique, we prove a uniform regularity bound of eigenfunctions
                  of the fractional Laplacian.
              

                      15:15 Sidy Moctar Djitte (Goethe Univ. Frankfurt)

Title:           NONLOCAL HADAMARD FORMULA AND ITS CONSEQUENCES

12.09.2019  - 2 Vorträge:  

14:15: Jean Louis Woukeng (Université de Dschang, derzeit Univ. Heidelberg)

Title: A general approach to deterministic homogenization theory.

Abstract. We present a comprehensive approach to study deterministic homogenization problems beyond the classical periodic setting. The approach is based on a combination of tools arising from Banach algebras theory associated to the so-called concept of sigma-convergence which generalizes the well-known two-scale convergence method.

15:15 Remi Yvant Temgoua (Goethe Univ. Frankfurt)

Title: The Poisson Problem for the Regional Logarithmic Laplacian


22.08.2019             Sorin Olaru (Centrale Supelec, Paris)

Titel:                        Positive invariant sets for time-delay dynamical systems

Abstract:                 The aim of this talk is to discuss the set invariance concepts for linear time-delay systems. When described in discrete-time, these dynamics allow different set-invariance formulations according to the state space representation. A series of classical or novel existence conditions will be discussed in this framework. On a broader scope it will be shown that set-factorization represents a generalized framework for the characterization of these families of invariant sets. The links with the stability, robustness and applications to mode detection and constrained control design will be also mentioned.


13.06.2019     Antonio Fernandez Sanchez (Universite de Franche Comte/Frankreich)

Titel:                On a class of elliptic problems with quadratic growth in the gradient


Montag, den 27.05.2019 - 2 Vorträge:

11 Uhr c.t., Raum 109c

Dr. Mousomi Bhakta

Titel:      Fractional hardy equations with critical and supercritical exponents

16:30   

Dr. Anup Biswas

Titel:      Simple probabilistic method in the analysis of fractional Laplacian

Abstract: It is well-known that fractional Laplacian operators are the generators of symmetric
stable Lévy processes. In this talk we would see that how some simple probabilistic techniques
could be used to answer questions that are not really probabilistic in nature.
In particular, we would consider a generalization of the Lieb's inequality for
eigenvalues of fractional Laplacian and Liouville type results for systems of equations
involving fractional Laplacian.


12.02.2019    
14:00 - 16:00    Takashi Furuya
Title:     A modification of the factorization method for scatterers with different physical properties
Abstract:    We study an inverse acoustic scattering problem by the Factoriza- tion Method when the unknown scatterer consists of two objects with different physical properties. Especially, we consider the following two cases: One is the case when each object has the different boundary condition, and the other one is when different penetrability. Our idea here is to modify the far field operator depending on the cases to avoid unnecessary a priori assumptions.

07.02.2019

14:00 Uhr                Olga Bondarev
Titel:                        Finanzmathematische Behandlung von Contingent Convertible Bonds
Abstract:                 In diesem Vortrag setzten wir uns zunächst mit der Funktionsweise und den preistreibenden Risiken von Contingent Convertible (CoCo) Bonds auseinander. Auf dieser Grundlage stellen wir ein Modell für den Preis von CoCo Bonds auf, dass das Volatilitäts-Smile und das Kreditrisiko berücksichtigt. Das Grundgerüst für dieses Modell bildet das Equity Derivatives Modell von J. De Spiegeleer und W. Schoutens, welches den CoCo Bond in eine Anleihe und Barriere-Optionen zerlegt. Die Modellparameter kalibrieren wir an Marktpreise von Plain-Vanilla Optionen und an CDS Quotierungen. Den Preis nähern wir mit einem Monte-Carlo-Verfahren ergänzt um die Brownsche Brücke an und untersuchen an einem Beispiel die Konvergenz dieses Verfahrens.

15:15 Uhr                Julian Fels

Titel:                        Particle Swarm Optimization

Abstract:                 Particle Swarm Optimization oder Partikelschwarmoptimierung ist ein globales, schwarmbasiertes Optimierungsverfahren, das im weiten Sinne zum Bereich der künstlichen Intelligenz gezählt wird. Inspiriert durch das natürliche Schwarmverhalten von Vögeln durchsuchen Individuen ein großes Suchgebiet in Abhängigkeit sowohl der eigenen besten Position als auch der besten Position der anderen Schwarmmitglieder. Angewandt auf restringierte Optimierungsprobleme erzielt diese Methode vielversprechende Ergebnisse. Die vorliegende Arbeit dient als Einführung in die Funktionsweise des Algorithmus und bietet einen Überblick über den aktuellen Forschungsstand in diesem Gebiet. Ein Fokus dieser Arbeit liegt dabei auf der einfachen Implementierung in MATLAB und dem breitgefächerten Einsatzgebiet des Algorithmus. Neben restringierten Optimierungsproblemen wird auch in die Anwendung des Algorithmus auf dem Gebiet der Clusteranalyse eingeführt. Als Ausblick auf zukünftige Forschungen soll zusätzlich ein Einblick in die Programmierung auf Grafikkarten mit MATLAB und CUDA gegeben werden. Dazu wird die grundlegende Architektur der GPU-Programmierung erläutert und Kriterien für eine sinnvollen Einsatz herausgearbeitet. Das hohe Potential der parallelen Implementierung und weitere Erkenntnisse dieser Arbeit werden abschließend durch eine experimentelle Analyse gestützt.

22.11.2018   

14:00 - 16:00          Dr. Martin Simon (Head of Equity and Equity Derivatives Valuation, Deka Investment GmbH)

Titel: "Aktienpreisblasen - Ein Indikator auf Basis börsengehandelter Optionen"

Abstract: In diesem Vortrag diskutieren wir einen mathematischen Indikator für Aktienpreisblasen. Der erste Teil hat einführenden Charakter und fasst die Theorie strikter lokaler Martingale zur Modellierung von Preisblasen sowie die Implikationen für die Bewertung derivativer Finanzinstrumente zusammen. Im zweiten Teil wird ein neuartiger, vorwärts blickender Indikator vorgestellt, welcher den Informationsgehalt von Geld- und Briefkursen börsengehandelter Plain Vanilla Optionen verwendet. Diese Konstruktion ist motiviert durch ein neues theoretisches Resultat von A. Jacquier and M. Keller-Ressel, welches zeigt dass Preisblasen anhand des asymptotischen Verhaltens der impliziten Volatilitätsfläche identifiziert werden können. In der Praxis führt dies auf ein schlecht-gestelltes inverses Parameteridentifikationsproblem; wir schlagen einen statistischen Ansatz vor, um die daraus resultierende Unsicherheit zu quantifizieren. Im dritten Teil des Vortrags werden reale Beispiele präsentiert welche sich mit der aktuell viel diskutierten Sorge vor einer Tech Blase 2.0 befassen.

01.11.2018 - zwei Vorträge

14:00 Uhr              Hugo Tavares (Lissabon/Portugal)

Title: Variational Problems with long-range interactions

Abstract: In this talk we consider a class of variational shape optimisation problems for densities
that repel each other at a certain distance. Typical examples are given by the Dirichlet functional
and the Rayleigh functional minimised in the class of functions that attain some H^1 boundary
conditions, subject to the constraint that the supports of different densities are at distance at least one.
For these problems, we show a connection with solutions to variational elliptic systems with nonlocal
competing interactions, we investigate the optimal regularity of the solutions, prove a free-boundary
condition, and derive some preliminary results characterising the free boundary. This is a joint work
with N. Soave, S. Terracini and A. Zilio


15:15 Uhr              Johannes Wagner (Frankfurt)

Titel:                      "Extrapolation für Runge-Kutta-Verfahren"

Abstract: In der Numerik gewöhnlicher Differentialgleichungen spielen Einschrittverfahren eine
zentrale Rolle. Verfahren höherer Konvergenzordnung liefern in der Praxis einen kleineren
Gesamtfehler, mithilfe von Extrapolation kann diese erhöht werden. Wir geben eine Konstruktion
von Runge-Kutta-Verfahren beliebig hoher Ordnung an und untersuchen die konstruierten
Verfahren auf ihre Stabilitätseigenschaften.


27.09.2018    14.15 Uhr      Katharina Jung (Frankfurt)


Titel: Symmetriebrechung bei Lösungen semilinearer Dirichletprobleme

Abstract:  Wir betrachten semilineare Dirichletprobleme in der Einheitskugel bzw. in einem Annulus.
Dabei untersuchen wir die Eigenschaften von Lösungen mit Vorzeichenwechsel anhand des Morse
Index und zeigen, unter welchen Bedingungen die Knotenlinie den Rand berührt. Insbesondere
zeigen wir, dass jede nodale Lösung minimaler Energie nichtradial ist und ihre Knotenlinie den Rand berührt.


30.08.2018       

14:00              Ali Suri (Bu-Ali Sina University, Hamedan, zur Zeit Frankfurt)

Titel: "On the Geometry of the Symplectomorphism Group"                                        

15:30              James Benn (University of Notre Dame, zur Zeit Frankfurt)

Titel: "Geometric Hydrodynamics and Recurrence Properties of Rossby-Hurwitz Waves"


28.06.2018    Tolga Yesil

Titel: Stein-Tomas inequality for symmetric functions

Abstract:  In this talk we study the Stein-Tomas inequality for functions with specific symmetries. This inequality yields (p,q)-estimates for the so called Fourier-restriction operator, i.e. the operator that restricts the Fourier transform of an admissible function to a hypersurface in Euclidean space. By considering the sphere, we will present how additional symmetries on the corresponding functions interact with the possible range of exponents.


21.06.2018    Joel Kübler

Titel: Nonradial bifurcation for the Henon equation

Abstract:  We study the nonlinear Henon equation on a ball and show that, as a parameter goes to infinity, nonradial solutions bifurcate from radial solutions. Our main tool is a thorough analysis of the eigenvalues of associated linearized operators which is based on a suitable rescaling of the equation. This allows us to identify a limit problem and yields asymptotic estimates for these eigenvalues.

14.06.2018    Rebekka Pech

Titel: Image Inpainting

Abstract:  Image Inpainting ist eine bereits fest etablierte Methode in der digitalen Bildbearbeitung, um beschädigte oder entfernte Teile eines Bildes adäquat zu füllen. In dieser Arbeit werden einige diffusionsbasierte (basierend auf der Wärmeleitungsgleichung, dem Variationsansatz oder Eulers Elastica) und texturbasierte Herangehensweisen an dieses Problem studiert und anschließend hinsichtlich ihrer Effizienz miteinander verglichen.

17.05.2018 Daniel Roth

Titel: Monte Carlo pfadweise Ableitungen für Barriere Optionen

Abstract: Das pfadweise Schätzen der Ableitungen bei Monte Carlo Verfahren ist für glatte
Auszahlungsfunktionen fest etabliert. In dieser Arbeit präsentieren wir einen neuen Monte Carlo
Algorithmus, welcher die pfadweisen Ableitungen von unstetigen Funktionen berechnen kann.
Unser Ansatz besteht darin die Idee des one-step survivals von Glasserman und Staum mit der
Methode zum stabilen Ableiten von Alm, Harrach, Harrach und Keller zu verbinden.
Als Anwendung werden wir die hergeleiteten Resultate verwenden, um eine zwei-dimensionale
Kalibrierung eines Coco-Bonds durchzuführen, welchen wir mit verschiedenen Arten von,
an diskreten Zeitpunkten überwachten, Barriere Optionen modellieren.


06.04.2018, Dr. Sarah Eberle
Titel: Wave Propagation and Post-Processing

05.04.2018, Michael Ertel
Titel: Das regularisierte Halley-Verfahren zur Lösung eines inversen nichtlinearen Gravimetrie-Problems

20.12.2017     Remi Yvant Temgoua (AIMS Kamerun)

Titel:                Schauder type estimates for a class of linear partial differential equations

06.07.2017      Huyuan Chen (Nanjing University/China, z Zt. Frankfurt am Main)

Titel:   Liouville theorem for fractional semilinear equations in unbounded domains

29.06.2017       Vortrag von Dr. Janosch Rieger, Monash University

Title: Recent advances in domain reconstruction from electrical impedance
tomography data

Abstract:
Electrical impedance tomography is an emerging budget-priced,
non-invasive medical imaging technique that is very likely to
complement computerised tomography in important applications
such as pulmonary function control and breast cancer screening
in the future. The main difficulty associated with this technology
is that the arising inverse problem is strongly ill-posed.


22.06.2017     Tolga Yesil (Frankfurt am Main)

Titel: Dual ground state solutions for the critical nonlinear Helmholtz equation

Abstract: Using a dual variational approach, we obtain real-valued solutions of a
weighted critical nonlinear Helmholtz equation. The weight function is
assumed to be bounded, positive, asymptotically periodic and to satisfy
a certain flatness condition at one of its maximum points. The solutions
obtained are so-called dual ground states, i.e. solutions arising from
critical points of the dual functional with the property of having
minimal energy among all nontrivial critical points. This is a joint
work with Gilles Evequoz.


Sondertermin: Dienstag, 04.04.2017, 11:00 Uhr (c.t.) in Raum 110 (RMS 10)

Vortrag von Minh Nguyet Mach, Ph.d. (University of Helsinki)

Title:"Convergence of the current-to-voltage measurements of the Shunt Electrode Model to those of the Continuum Model"


09.02.2017    Sven Jarohs (Frankfurt am Main)

Titel: Starshape of superlevel sets of solutions to equation involving the fractional Laplacian

Abstract:  In this talk, I will present a general framework to analyze the geometry of solutions to equations
involving the fractional Laplacian in starshaped rings. By analyzing the difference of the solution with a scaled
version of this solution, using the scaling properties of the fractional Laplacian and different versions of
maximum principles, we show that under rather general assumptions on the right-hand side the solution
has starshaped superlevel sets. I will also present some examples in which this result can be applied.
The talk is based on a joint work with Tadeusz Kulczycki and Paolo Salani.

02.02.2017    Sebastian Becker (Goethe-Universität Frankfurt)

Titel: Nicht-autonome und zufällige dynamische Systeme, die durch deterministische und stochastische
Differentialgleichungen generiert sind (Masterarbeit)

Abstract: Im Vortrag werden autonome, nicht autonome und zufällige dynamische Systeme vorgestellt,
miteinander verglichen und erweitert. Anhand von zunächst skalaren Differentialgleichungen werden die
induzierten dynamischen Systeme mit den zuvor vorgestellten Definitionen in Verbindung gebracht, um
so den nicht autonomen Einfluss besser untersuchen zu können. Des weiteren werden gemeinsam
wirkender nicht-autonomer deterministischer und stochastischer Einfluss betrachtet und so generierte
Systeme analysiert.

19.1.2017    Oscar Agudelo  (University of West Bohemia, Pilsen)

Titel: Boundary concentration phenomena for the higher-dimensional Keller-Segel system

Abstract: We study the existence of steady states to the Keller-Segel system with linear chemotactical
sensitivity function on a smooth bounded domain in RN, N ≥ 3, having rotational symmetry. We find three
different types of chemoattractant concentration which concentrate along suitable (N−2)−dimensional
minimal submanifolds of the boundary. The corresponding
density of the cellular slime molds exhibit in the limit one or more Dirac measures supported on
those boundary submanifolds.
This is a joint work with Angela Pistoia from La Sapienza, Università degli Studi di Roma.

8.12.2016    Nils Ackermann (UNAM, Mexico Stadt, derzeit Goethe-Universität Frankfurt)

Titel: Ground states for irregular and indefinite superlinear Schrödinger equations

Abstract:  We consider the existence of a ground state for the subcritical stationary
semilinear Schrödinger equation −∆u + u = a(x)|u| p−2 u in H 1 , where a ∈
L ∞ (R N ) may change sign. Our focus is on the case where loss of compactness
occurs at the ground state energy. By providing a new variant of the Splitting
Lemma we do not need to assume the existence of a limit problem at infinity, be
it in the form of a pointwise limit for a as |x| → ∞ or of asymptotic periodicity.
That is, our problem may be irregular at infinity. In addition, we allow a to
change sign near infinity, a case that has never been treated before.

27.10.2016     Marcello di Biase (Goethe-Universität Frankfurt)

Titel: Über Stochastische Bifurkation

Abstract: Große Anstrengungen galten in den letzten Jahrzehnten den Versuchen, eine Bifurkationstheorie für zufällige dynamische Systeme zu schaffen, die als Verallgemeinerung der gut verstandenen Bifurkationsszenarien deterministischer Dynamik verstanden werden kann. Die Ergebnisse sind so vielfältig wie die Klassen der untersuchten zufälligen dynamischen Systeme selbst. Einige ausgewählte Ansätze werden umrissen, Beispiele diskutiert und auf ansatzspezifische Eigenheiten aufmerksam gemacht.

20.10.2016    Maximilian Engel (Imperial College London)

Titel: "Bifurcation analysis of stochastically driven limit cycles"

Abstract: We investigate bifurcations from an attractive random equilibrium to shear-induced chaos for stochastically driven limit cycles, indicated by a change of sign of the first Lyapunov exponent. This addresses an open problem posed by Lai-Sang Young and co-workers, extending results on periodically kicked limit cycles to the stochastic context. We also apply concepts from ergodic theory, like entropy and the SRB property of the invariant random measure, to describe the random attractors in the chaotic case.

6.10.2016   14 ct, Raum 110     

Nicola Abatangelo (Brüssel)

Title: A very weak theory for fractional Dirichlet problems

Abstract: It is long since known that s-harmonic functions (namely, functions whose fractional Laplacian is zero) on a domain can show an explosive behaviour at the boundary, in sharp contrast with classical harmonic functions. We will present a theory of Stampacchia's sort for solutions to fractional elliptic Dirichlet problems which can deal with such singular behaviour. In order to gain uniqueness and, more generally, well-posedness, these boundary problems are set by prescribing two different types of Dirichlet datum at the same time, one of which is describing the asymptotic explosive profile of the solution at the boundary.

27.07.2016   11:00 Uhr st, Raum 110     

Sven Jarohs (Goethe-Universität Frankfurt)

Titel:  On the maximum principle for the fractional polylaplacian

Abstract: It is well-known that in general operators of order four do not satisfy
a maximum principle for supersolutions. Since maximum principles are an
important tool in the analysis of partial differential equations, the
question arises why and when this property is lost for operators of
order between 2 and 4. In this talk we will analyze real, positive
powers of the Laplacian and show that whenever the power is in an
Intervall starting with an odd number, then the maximum principle fails.
By the structure of the explicit counterexample it follows that such
powers of the fractional Laplacian may satisfy a maximum principle only
for solutions in certain connected sets. One of such sets is given by
the ball, where the maximum principle follows from an explicit solution
formula given by Boggio's formula. The talk is based on a joint work
with Nicola Abatangelo and Alberto Saldana.

14.07.2016   14 Uhr ct    Rohit Kumar Mishra (Bangalore/Indien)

Titel: On Inversion of Some Integral Transforms in R^n

07.07.2016   14 Uhr c.t.      Andreas Hauptmann (Universität Helsinki/Finnland)

Titel: Direct reconstructions from partial-boundary data in electrical impedance tomography

Abstrakt: In electrical impedance tomography a body is probed with an electrical
current to obtain information about the inner conductivity distribution. In this
application full-boundary measurements are not always possible. Therefore, we
present the partial-boundary inverse conductivity problem in a realistic setting
and analyze the error that partial-boundary measurements introduce. Computational
convergence results and reconstructions are presented for medical motivated simulated data.


07.07.2016, 15 Uhr c.t.     Daniel Roth (Frankfurt am Main)

Titel: Monte-Carlo Verfahren für die Elektrische Impedanz-Tomografie

Abstract: Das direkte Problem der elektrischen Impedanz-Tomografie
beschäftigt sich damit, für eine bekannte Leitfähigkeit durch angelegten
Strom das zugehörige Potential zu berechnen. Dieses Problem kann
beispielsweise mit einem Finite-Differenzen Verfahren gelöst werden. Der
Vortrag widmet sich einem alternativen Verfahren zum Lösen des direkten
Problems. Dieses Verfahren, welches auf auf einem Monte-Carlo
Algorithmus basiert, löst mittels Zufallszahlen das Gleichungssystem der
Finiten Differenzen, ohne das Gleichungssystem explizit aufzustellen.


02.06.2016   14 Uhr ct     Elias Polak (Frankfurt)

Titel: Die Grundzustandsenergie eines N-Fermionen-Systems (Bachelorarbeit)

04.02. 2016   Fabian Rücker (Frankfurt)

Titel: Anwendung des Minimax-Theorems von Nikaido auf Probleme der
Kontrolltheorie

Sondertermin, Mittwoch, den 03.02.2016  16-18 Uhr, Raum 404

16:15 Uhr   Prof. Susanna Terracini (Turin)

Titel: Entire solutions and spiralling asymptotic profiles of competition diffusion sytems


17:15 Uhr   Prof. Gianmaria Verzini (Milano)

Titel: Strong competition versus fractional diffusion

21.01.2016     Dr. Alberto Saldana (Brüssel)

Titel: On the extended Allen-Cahn equation.

Abstract: Nonlinear fourth-order PDEs usually have a richer and more complex set of solutions
when compared to its second-order counterpart. In this sense, many models exhibit behaviors
that could be better described with fourth-order equations, like ocean and atmosphere dynamics,
bridges, and pattern formation, just to mention some of them. The theory for higher-order nonlinear
problems, however, is far less developed than its second-order analogue and many basic questions
remain open. Lack of maximum principles, oscillatory behavior of solutions, and regularity issues
are some of the main difficulties in the study of such problems.
I this talk I consider a fourth-order extension of the Allen-Cahn model with mixed-diffusion and
Navier boundary conditions. I present results on existence, uniqueness, positivity, stability, a priori
estimates, and symmetry. As an application, we construction a saddle solution in the whole space.
The proofs rely on variational and bifurcation methods. Some numerical approximations of solutions
will also be discussed.

14.01.2016     Linda Lintz (Frankfurt)

Titel: Der fraktionale p-Laplace-Operator