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Spring Semester 2018

Date / Time Speaker Title Location
6 March 2018
15:15-16:15
Prof. Dr. Italo Capuzzo Dolcetta
Università Roma La Sapienza
Event Details

Analysis Seminar

Title On the weak maximum principle in unbounded domains
Speaker, Affiliation Prof. Dr. Italo Capuzzo Dolcetta, Università Roma La Sapienza
Date, Time 6 March 2018, 15:15-16:15
Location HG G 43
Abstract I will discuss some recent results concerning the validity of the weak maximum principle u≤0 on∂ Ω, F(x,u,Du,D2u)≥0 in Ω in an unbounded domain Ω satisfying either measure-type or geometric conditions related to the directions of ellipticity of the (possibly degenerate) fully nonlinear mapping F.
On the weak maximum principle in unbounded domainsread_more
HG G 43
13 March 2018
15:15-16:15
Dr. Bozhidar Velichkov
Université Grenoble Alpes
Event Details

Analysis Seminar

Title Variational approach to the regularity of the singular free boundaries
Speaker, Affiliation Dr. Bozhidar Velichkov, Université Grenoble Alpes
Date, Time 13 March 2018, 15:15-16:15
Location HG G 43
Abstract This talk is based on a joint work with Luca Spolaor (Princeton) and Max Engelstein (MIT). In this talk we will present some recent results on the structure of the free boundaries of the (local) minimizers of the Bernoulli free boundary problem in higher dimension. In 1981 Alt and Caffarelli proved that for the minimizers of the Bernoulli problem, the free boundary can be decomposed into a regular part (Reg) and a singular part (Sing), where Reg is locally the graph of a smooth function and Sing is a small (possibly empty) set. Recently, De Silva and Jerison proved that, starting from dimension d=7, there are minimal cones with isolated singularities in zero. In particular, the set of singular points might not be empty. The aim of this talk is to describe the structure of the free boundary around a singular point. We will show that if X_0 is a point of the free boundary and there exists one blow-up limit, which has an isolated singularity in zero, then the free boundary is a C^1 graph over the cone determined by the blow-up limit. In particular, the blow-ups at isolated singularities are unique. Our approach is based on the so called logarithmic epiperimetric inequality, which is a purely variational tool introduced in the framework of the obstacle problem in a series of works in collaboration with Maria Colombo and Luca Spolaor.
Variational approach to the regularity of the singular free boundariesread_more
HG G 43
20 March 2018
15:15-16:15
Prof. Dr. Mircea Petrache
Pontificia Universidad Catolica de Chile (Santiago)
Event Details

Analysis Seminar

Title When does optimal transport branch?
Speaker, Affiliation Prof. Dr. Mircea Petrache, Pontificia Universidad Catolica de Chile (Santiago)
Date, Time 20 March 2018, 15:15-16:15
Location HG G 43
Abstract Consider the problem of transporting some objects between N distinct locations. Depending on how we package together different objects and on how the transport cost (per unit of distance traveled) depends on the package that we are moving, we may cook up a minimum-cost transport strategy. Is it always the best option to let our objects travel independently of each other, or is it sometimes more cost-efficient to merge/split packages along the way, following a branched, tree-like, global network? We consider the model in which the "packaging arithmetics" and the transport cost are quantified via a normed Abelian group G, and we extract a purely intrinsic condition on G that guarantees that the optimal transport is not branching. This seems to initiate a new geometric classification of certain normed groups. In the nonbranching case we also provide global calibrations, i.e. a generalization of Monge-Kantorovich duality. (Joint work with R. Zust)
When does optimal transport branch?read_more
HG G 43
27 March 2018
15:15-16:15
Dr. Marco Méndez Guaraco
University of Chicago
Event Details

Analysis Seminar

Title Geometric aspects of semilinear elliptic PDEs and minimal hypersurfaces on closed manifolds
Speaker, Affiliation Dr. Marco Méndez Guaraco, University of Chicago
Date, Time 27 March 2018, 15:15-16:15
Location HG G 43
Abstract In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation, arising from the theory of phase transitions, has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding the analogy between both theories, focusing on min-max constructions. In particular, new insights into both Almgren-Pitts and Marques-Neves existence theories of minimal hypersurfaces will be discussed.
Geometric aspects of semilinear elliptic PDEs and minimal hypersurfaces on closed manifoldsread_more
HG G 43
10 April 2018
15:15-16:15
Dr. Swarnendu Sil
EPF Lausanne
Event Details

Analysis Seminar

Title Calculus of Variations: A Differential Form perspective
Speaker, Affiliation Dr. Swarnendu Sil, EPF Lausanne
Date, Time 10 April 2018, 15:15-16:15
Location HG G 43
Abstract Direct methods in the calculus of variations, which is concerned with finding a minimizer u : Ω ⊂ Rn → RN for minization problems involving the integral functional f(∇u) is by now classical. In the scalar case Ω (N = 1), given other reasonable hypotheses, convexity of f is necessary and sufficient, whereas the vectorial case (N ≥ 2) revolves around find- ing good class of nonconvex integrands for which we still can ensure the existence of minimizer(s). By viewing the function u in the scalar case as a 0-form, one can envision a different vectorial extension of the scalar problem, which is to study the minization problems for f(du), where u is a differential k- Ω form on Ω. Furthermore, by considering several differential forms, i.e by studying the functionals f(du1,...,dum), it is possible to view both Ω extensions as special cases of a unified framework. In this talk, we shall see that such a unification can be achieved and it both clarifies and raises a number of interesting points. On one hand, known facts like the so-called ‘divergence structure’ and cancellations of the determinants are seen as central to the framework and they are put in their most general and natural setting - the exterior algebra. On the other hand, the general case introduces a lack of coercivity issue due to gauge invariance. Part of the work was done in collaboration with Saugata Bandyopadhyay and Bernard Dacorogna.
Calculus of Variations: A Differential Form perspectiveread_more
HG G 43
17 April 2018
15:15-16:15
Prof. Dr. Donatella Donatelli
University of L'Aquila
Event Details

Analysis Seminar

Title On a singular limit in astrophysics
Speaker, Affiliation Prof. Dr. Donatella Donatelli, University of L'Aquila
Date, Time 17 April 2018, 15:15-16:15
Location HG G 43
Abstract A broad range of interesting phenomena in science and engineering occur under various scaling regimes, among them it is particularly relevant the low Mach number regime, in which the fluid velocity is much less than the speed of sound. In this talk I will focus on one example arising from astrophysics, where the modeling equations are given by the coupling of the compressible Navier Stokes equations with equations that take into account of the chemical reactions and heat effects. One feature of these flows is that they take place under a low Mach number and high Reynolds number regime and so they are affected by the presence of high oscillating acoustic waves. In order to understand this type of dynamic one has to derive a model for low speed flows (low Mach number) in a hydrostatically balanced, radially stratified background that removes acoustic waves and allows for the development of finite amplitude temperature and density variation. Here, we analyze a simplified model arising in astrophysics and we identify the asymptotic limit in the regime of low Mach, low Froude and high Reynolds number. The system is driven by a long range gravitational potential. We show convergence to an anelastic system for ill-prepared initial data. The proof is based on frequency localized Strichartz estimates for the acoustic equation based on the recent work of Metcalfe and Tataru.
On a singular limit in astrophysicsread_more
HG G 43
8 May 2018
15:15-16:15
Prof. Dr. Filippo Santambrogio
Université Paris Sud
Event Details

Analysis Seminar

Title Modeling and regularity issues in Mean Field Games
Speaker, Affiliation Prof. Dr. Filippo Santambrogio, Université Paris Sud
Date, Time 8 May 2018, 15:15-16:15
Location HG G 43
Abstract In the talk, I will first present a typical Mean Field Game problem, as in the theory introduced by Lasry and Lions in 2006, concentrating on the case where the game has a variational structure (i.e., the equilibrium can be found by minimizing a global energy) and is purely deterministic (no diffusion, no stochastic control). I will explain why regularity questions are natural and useful for rigorously proving that minimizers are equilibria, making the connection with the incompressible Euler equation in the Brenier's variational formalism. Then, I will sketch two regularity results: the first is obtained via convex duality and gives H^1 estimates, the second via optimal transport techniques and provides L^\infty estimates. This last result will be compared to a similar one by Lions, based on elliptic PDE techniques. The content of the talk comes from joint works with A. Mészáros, P. Cardaliaguet, A. Prosinski and H. Lavenant.
Modeling and regularity issues in Mean Field Gamesread_more
HG G 43
15 May 2018
15:15-16:15
Prof. Dr. Armin Schikorra
Universität Basel
Event Details

Analysis Seminar

Title Hölder-Analysis of the Topology of the Heisenberg group
Speaker, Affiliation Prof. Dr. Armin Schikorra, Universität Basel
Date, Time 15 May 2018, 15:15-16:15
Location HG G 43
Abstract The Heisenberg groups are examples of sub-Riemannian manifolds homeomorphic, but not diffeomorphic to the Euclidean space. Their metric is derived from curves which are only allowed to move in so-called horizontal directions. When one considers approximation or extension problems for Sobolev maps into Riemannian manifolds it is known that topological properties of the target manifold play a role. However, due to the homeomorphism, the topology of the Heisenberg group is the same as the Euclidean space. A notion of Hölder topology is needed. I will report on some progress (with Hajlasz) on some topological features of the Heisenberg group, in particular on an embedding question due to Gromov.
Hölder-Analysis of the Topology of the Heisenberg groupread_more
HG G 43
22 May 2018
15:15-16:15
Prof. Dr. David Jerison
Massachusetts Institute of Technology
Event Details

Analysis Seminar

Title Compactness and singular limits of free boundary problems in the plane
Speaker, Affiliation Prof. Dr. David Jerison, Massachusetts Institute of Technology
Date, Time 22 May 2018, 15:15-16:15
Location HG G 43
Abstract We will describe the classical one-phase free boundary problem of Alt and Caffarelli and our joint work with N. Kamburov characterizing one-phase solutions to the Euler-Lagrange equation when that phase is a simply-connected region in the plane. If the curva- ture is large, then the solution resembles the double hairpin solution discovered by Helein, Hauswirth and Pacard. We prove this in a strong sense in the spirit of theorems of Colding and Minicozzi concerning minimal surfaces in this simpler free-boundary context. We will also explain the direct connection between free boundaries and minimal surfaces discovered by M. Traizet.
Compactness and singular limits of free boundary problems in the planeread_more
HG G 43

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Organizers: Alessandro Carlotto, Francesca Da Lio, Alessio Figalli, Norbert Hungerbühler, Thomas Kappeler, Tristan Rivière, Dietmar Salamon, Michael Struwe

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