Analysis seminar

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Spring Semester 2017

Note: The highlighted event marks the next occurring event and events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Date / Time Speaker Title Location
31 January 2017
15:15-16:15
Prof. Dr. David Kinderlehrer
Carnegie Mellon University
The gradient flow of microstructure  HG G 43 
Abstract: A central problem of microstructure is to develop technologies capable of producing an arrangement, or ordering, of the material, in terms of mesoscopic parameters like geometry and crystallography, appropriate for a given application. Is there such an order in the first place? We describe the emergence of the grain boundary character distribution (GBCD), a statistic that details texture evolution, and illustrate why it should be considered a material property. The theory relies on mass transport and entropy methods, and as a consequence we seek to identify it as a gradient flow in the sense of Ambrosio, Gigli, and Savaré. In this way, the empirical texture statistic is revealed as a solution of a Fokker-Planck type equation whose limit behavior is a Boltzmann distribution, which, is in fact, observed in simulation. The development exposes the question of how to understand the circumstances under which a harvested empirical statistic is a property of the underlying process. (joint work with P. Bardsley, K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, X.-Y. Lu and S. Ta’asan)
7 March 2017
15:15-16:15
Prof. Dr. Rosario Mingione
Università degli studi di Parma
Non-standard growth conditions, double phase functionals and regularity  HG G 43 
Abstract: Regularity problems for non-uniformly elliptic operators are delicate, since they cannot be classified in a single family, but they rather show very different features and problematic aspects. I will give a brief summary of some results in this direction.
14 March 2017
15:15-16:15
Prof. Dr. Jan Metzger
Universität Potsdam
On the uniqueness of small surfaces minimizing the Willmore functional subject to a small area constraint  (CANCELLED) HG G 43 
Abstract: We consider the Willmore functional for surfaces immersed in a compact Riemannian manifold M and study minimizers subject to a small area constratint. We show that if the scalar curvature of M has a non-degenerate maximum then for small enough area these minimizers are unique. This is joint work with Tobias Lamm and Felix Schulze.
21 March 2017
15:15-16:15
Dr. Xavier Ros-Oton
University of Texas at Austin
Regularity of free boundaries in obstacle problems  HG G 43 
Abstract: We present a brief overview of the regularity theory for free boundaries in different obstacle problems. We describe how a monotonicity formula of Almgren plays a central role in the study of the regularity of the free boundary in some of these problems. Finally, we explain new strategies which we have recently developed to deal with cases in which monotonicity formulas are not available.
28 March 2017
15:15-16:15
Prof. Dr. Melanie Rupflin
University of Oxford
Title T.B.A. HG G 43 
* 5 April 2017
14:00-15:00
Prof. Dr. Max Fathi
Université de Toulouse
Gradient flows and scaling limits for interacting particle systems  HG G 19.1 
Abstract: In this talk, I will explain how the gradient flow framework for Markov chains on finite spaces that was introduced in 2011 by Maas and Mielke can be combined with the Sandier-Serfaty theorem on convergence of gradient flows to study scaling limits for interacting particle systems on lattices. The talk will be focused on the case of the simple exclusion process. Joint work with Marielle Simon (INRIA Lille).
25 April 2017
15:15-16:15
Prof. Dr. Matteo Bonforte
Universidad Autónoma de Madrid
Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains  HG G 43 
Abstract: We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L} F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$\,, with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ we can use a quite general class of linear operators that includes the three most common versions of the fractional Laplacian $(-\Delta)^s$, $01$\,. We will shortly present some recent results about existence, uniqueness and a priori estimates for a quite large class of very weak solutions, that we call weak dual solutions. We will devote special attention to the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our techniques cover also the local case s = 1 and provide new results even in this setting. A surprising instance of this problem is the possible presence of nonmatching powers for the boundary behavior: for instance, when $\mathcal{L}=(-\Delta)^s$ is a spectral power of the Dirichlet Laplacian inside a smooth domain, we can prove that, whenever $2s \ge 1 - 1/m$, solutions behave as $dist^{1/m}$ near the boundary; on the other hand, when $2s < 1 - 1/m$, different solutions may exhibit different boundary behaviors even for large times. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the elliptic case. The above results are contained on a series of recent papers in collaboration with A. Figalli, X. Ros-Oton, Y. Sire and J. L. Vazquez.
2 May 2017
15:15-16:15
Dr. Stefano Spirito
Università degli Studi dell'Aquila
Lagrangian Solutions of 2D Euler equations  HG  G 43 
Abstract: Lagrangian Solutions of two dimensional Euler equations are, roughly speaking, solutions for which the vorticity is transported by the flow of the velocity. In this talk I will firstly give an overview regarding the existence of Lagrangian solutions and secondly I will discuss a recent result with Gianluca Crippa, Camilla Nobili, and Christian Seis concerning the convergence of Navier-Stokes solutions to Lagrangian solutions of 2D Euler in the vanishing viscosity limit. A crucial point in the proof of this result is a new uniqueness theorem, interesting in its own right, for solutions of the linear continuity equation with non smooth vector field (Singular integral of an L^1 function)
9 May 2017
15:15-16:15
Dr. Armin Schikorra
Universität Freiburg
On free boundary problems for conformally invariant variational functions  HG G 43 
Abstract: I will present a regularity result at the free boundary for critical points of a large class of conformally invariant variational functionals. The main argument is that the Euler-Lagrange equation can be interpreted as a coupled system, one of local nature and one of nonlocal nature, and that both systems (and their coupling) exhibit an antisymmetric structure which leads to regularity estimates.
16 May 2017
15:15-16:15
Prof. Dr. Carlo Sinestrari
Università di Roma "Tor Vergata"
Convex ancient solutions of curvature flows  HG G 43 
Abstract: We consider compact convex hypersurfaces evolving by mean curvature flow which are ancient, that is, defined for all negative times. Solutions with these properties occur as the limit of rescalings near a singularity of a general mean convex solution of the flow. The easiest example is a shrinking sphere, but there are other known examples having an oval shape which becomes more and more eccentric for negative times. In this talk we consider various sufficient conditions which ensure that our solution is a shrinking sphere. Examples are: a uniform pinching on the principal curvatures, a growth rate assumption on the diameter, a bound on the ratio of outer and inner radius, a bound on the isoperimetric ratio. These results are in collaboration with G. Huisken. The characterization of the shrinking sphere via the uniform pinching property has been also obtained in more general contexts, such as higher codimensional mean curvature flow, or other extrinsic curvature flows (results in collaboration with S. Risa). Other related results have been obtained by R. Haslhofer and O. Hershkovits and by M. Langford and S. Lynch.
23 May 2017
15:15-16:15
Prof. Dr. Walter Craig
McMaster University
Vortex filament dynamics  HG G 43 
Abstract: The evolution of vortex filaments in three dimensions is a central problem in mathematical hydrodynamics, appearing in questions on solutions of the Euler equations as well as in the fine structure of vortex filamentation in a superfluid. It is also a setting in the analysis of partial differential equations with a compelling formulation as a Hamiltonian dynamical systems in an infinite dimensional phase space. I will give an analysis of a system of model equations for the dynamics of near-parallel vortex filaments in a three dimensional fluid. These equations can be formulated as Hamiltonian PDEs, and the talk will describe some aspects of a phase space analysis of solutions, including the construction of periodic and quasi-periodic orbits via a version of KAM theory for PDEs, and a topological principle to count multiplicity of solutions. This is ongoing joint work with L. Corsi (Georgia Tech), C.
* 30 May 2017
14:15-15:15
Prof. Dr. Tobias Lamm
KIT Karlsruhe
Limits of alpha-harmonic maps HG G 43 
* 30 May 2017
15:45-16:45
Prof. Dr. Nicos Kapouleas
Brown University
Gluing constructions for minimal surfaces and related questions HG G 43 

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