Past Events
Summer semester 2024
08.05.2024
Leon Happ Title: Compensated compactness and the role of oscillations in weak convergence
Abstract: At the core of the compensated compactness method stands the insight that differential constraints along a sequence of weakly converging vector valued functions encode information about directions in which oscillations might appear. This knowledge can be used to find at least partial answers to several interesting questions connected to weakly converging sequences, e.g. when functions of possibly oscillating sequences are weakly (lower semi-)continuous. A special choice of differential constraints leads to the celebrated div-curl lemma.
In this talk I will cover the origins of the compensated compactness method that arose from a series of papers and lectures by F. Murat and L. Tartar at the end of the 1970's and the start of the 1980's. Emphasis will be put on understanding the importance of Young measures in this approach. I will also point to the bridge connecting compensated compactness (introduced first to study hyperbolic PDEs) and A-quasiconvexity (emerging in the calculus of variations), a notion introduced by F. Murat and then further developed by I. Fonseca and S. Müller in 1999. This will also reveal that a central idea in both areas is to study oscillations from the viewpoint of harmonic analysis.
Starting from the observation of the prominent role that oscillating sequences play in compensated compactness - and of course in other fields of applied mathematics - I will try to give a more general account of the obstruction such sequences pose for strong convergence results. In this context I will comment on a nice result by A. Visintin on how to improve weak to strong convergence. Along the way we will see how V. Sverák's famous counterexample already appears in a seminal work of L. Tartar on compensated compactness.
24.04.2024
Stefanos Georgiadis Title: Non-uniqueness for weak solutions to the incompressible Euler equations
Abstract: In this talk we consider the Cauchy problem for the incompressible Euler equations. These fundamental equations were derived by Euler more than 250 years ago and have since played a pivotal role in fluid dynamics. Over the years, several open problems have emerged alongside an extensive literature about them. In three space dimensions little is known about smooth solutions apart from classical short-time existence and uniqueness. On the other hand, weak solutions are known to be ill-posed according to Hadamard. Nevertheless, weak solutions have been studied for their anticipated relevance to turbulence. During the talk, I will provide an overview of non-uniqueness results, culminating in the celebrated result of De Lellis and Szekelyhidi using the method of convex integration.
Winter semester 2023/24
22.02.2024
Tùng Nguyen Title: Drift Diffusion Equations for Semiconductors coupled to a Network
Abstract: The memristor is a novel semiconductor device that is equipped with a memory by the change of its electrical resistance. Therefore, it may mimic the behavior of a neuron in the human brain. We analyze a model of memristors that are coupled with an electric network consisting of many different electronic devices. While drift-diffusion equations model the motion of charged particles within the memristor, the node potentials in the network obey Kirchhoff’s laws, i.e. ordinary differential equations and algebraic constraints. The coupling between the memristor and the network takes place via the displacement current flowing through the memristor’s interface which leads to a system of partial differential-algebraic equations. By a fixed-point argument, we aim to establish the short-time existence of weak solutions as a preliminary result.
18.01.2024
Rossella Giorgio Title: Integral functionals involving non-local gradients and an introduction to their variational analysis
Abstract: Fractional and nonlocal gradients operators are the main topic of this PhD Discussion Group meeting. Based on works by [Shieh and Spector, 2015], [Bellido, Cueto and Mora-Corral, 2022] and [Cueto, Kreisbeck and Schönberger, 2023], their properties and tools for switching between different settings are presented. Additionally, integral functionals involving nonlocal gradients and their variational analysis are briefly discussed.
07.12.2023
Samuele Riccò Title: Regularity results for minima of variational integrals and solutions to non-linear systems of PDEs
Abstract: Regularity problems today are not as popular as they were once. Regularity methods are sometimes not very intuitive, and often overburdened by a lot of technical complications, eventually hiding the main, basic ideas. Moreover, very often no room for partial results is given: either the whole problem is solved, or really nothing comes up. That is why I am presenting a (non-exhaustive) collection of regularity results for both minima of variational integrals and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations, striving for casting a relatively general panorama in the unconstrained minimization problem case. This talk is heavily inspired by the paper "Regularity of minima: an invitation to the dark side of the calculus of variations" by G. Mingione.
09.11.2023
Andrea Chiesa Title: Existence or non-existence of ground states for NLS on doubly periodic metric graphs: a dimensional crossover
Abstract: The existence of ground states for the nonlinear Schrödinger equation (NLS) is closely related to Sobolev inequalities, and thus to the dimension of the space. In this regard, an interesting behaviour appears in the case of NLS on doubly periodic graphs, e.g. the square lattice. There, a continuous transition from the two-dimensional to the one-dimensional regime can be observed, i.e., from the regime where the global structure of the graph prevails to the one where the local structure is predominant. In this talk, I aim to give an overview of the state of the art for undefected and defected double periodic graphs.
12.10.2023
Marco Bresciani Title: Surface energies and cavitation in nonlinear elasticity
Abstract: Cavitation is defined as the abrupt formation of voids inside materials in response to mechanical stresses. The formation of cavities and their coalescence has long been recognized as the main cause of fracture and similar failure phenomena in solids. In this talk, we will review some literature on the existence of minimizers for variational models in nonlinear elasticity. We will start by recalling classical results in pure elasticity, then moving towards their more recent refinements. Eventually, we will discuss existence theories allowing for cavitation. Central questions within our discussion will be the weak continuity of Jacobian determinants, the injectivity constraint, and the concept of surface created by deformations.
Summer semester 2023
20.07.2023
Manuel Seitz Title: Evolutionary Gamma-convergence
Abstract: Evolutionary Gamma-convergence is a neat tool to tackle the following problem: Given a family of Gamma-converging energies, do the solutions to the associated gradient flows also converge to the solution of the gradient flow associated to the limiting energy? This technique is based on reformulating the gradient flow as an equivalent Energy-Dissipation-balance and then passing to the limit via certain liminf-inequalities. In this talk, we will introduce the method of evolutionary Gamma-convergence for (generalized) gradient flows, discuss its applicability and present some examples.
15.06.2023
Jakob Deutsch Title: Korn inequalities in fracture mechanics
Abstract: Korn inequalities play a crucial role in the theory of elastic materials. The general objective of this talk is to provide an overview of the classical Korn inequalities in the L^p spaces and then introduce you to Korn inequalities for functions with bounded deformation. The primary focus will be on a soft introduction of functions of generalized bounded deformation, which have been developed to model phenomena in fracture mechanics. Moreover, we will discuss recent advancements in the theory of Korn inequalities for functions with relatively small jump sets.
01.06.2023
Leon Happ Title: Mapping degree theory in analysis
Abstract: The seminal works of mapping degree theory date back to the start of the 20th century and many great matematicians - Cauchy, Poincaré, Hadamard, Brouwer, Hopf, Leray, and Schauder, only to name a few - have since contributed to its development. In this talk, I aim on giving a short introduction to the concept of the degree of a mapping, the different equivalent formulations that exist for it, and its most important properties (as homotopy invariance and the boundary theorem). Furthermore, I will shed some light on how some of the many important results and elegant proofs emerge from this rich theory (e.g. Brouwer's famous Fixed Point Theorem or the Fundamental Theorem of Algebra). I will also cover how a notion of a mapping degree, once introduced only for continuous functions, can also be assigned to special classes of Sobolev mappings.