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Autumn Semester 2014

Date / Time Speaker Title Location
30 September 2014
15:15-16:15 		Robin Neumayer
UT Austin
Event Details

Analysis Seminar

Title A stability result for the anisotropic isoperimetric inequality
Speaker, Affiliation Robin Neumayer, UT Austin
Date, Time 30 September 2014, 15:15-16:15
Location HG G 43
Abstract For a set E that almost minimizes perimeter among sets of the same volume, quantitative isoperimetric inequalities measure how "close" this set is to the unique perimeter minimizer. A recent paper of Fusco and Julin gives quantitative control of the oscillation of the boundary of such a set. In this talk, we will generalize this result for the anisotropic case, where perimeter is weighted with respect to some fixed convex set K.
A stability result for the anisotropic isoperimetric inequalityread_more
HG G 43
7 October 2014
15:15-16:15 		Frabrice Planchon
Université de Nice
Event Details

Analysis Seminar

Title Beyond dispersion for the wave equation inside a convex domain
Speaker, Affiliation Frabrice Planchon , Université de Nice
Date, Time 7 October 2014, 15:15-16:15
Location HG G 43
Abstract Usually, Strichartz estimates follow almost trivially from dispersion, using duality and interpolation. For the wave equation inside a strictly convex domain, however, the resulting theorem is not sharp and we will present arguments which in some sense average over the space-time regions where swallowtail singularities (where the worse loss occur) appear and recover Strichartz estimates which would be induced by cusp-like losses. This is joint work with O. Ivanovici and G. Lebeau
Beyond dispersion for the wave equation inside a convex domainread_more
HG G 43
14 October 2014
15:15-16:15 		Javier Morales
UT Austin
Event Details

Analysis Seminar

Title Gradient flows in metric spaces and transportation costs between positive measures
Speaker, Affiliation Javier Morales, UT Austin
Date, Time 14 October 2014, 15:15-16:15
Location HG G 43
Abstract The theory of gradient flows in Hilbert spaces allows us to prove existence of solutions to parabolic equations and it provides contraction and energy estimates. The minimizing movement scheme from Ennio De Giorgi allows the extension of this theory to metric spaces. This scheme was first applied by R. Jordan, D.Kinderlehrer and F. Otto to the space of probability measures and it yielded contraction estimates and existence for new families of evolutionary equations. I will outline these ideas and then discuss new applications based on transportation costs between positive measures that allow the use of the minimizing movement scheme to build solutions to parabolic and reaction diffusion equations with boundary conditions.
Gradient flows in metric spaces and transportation costs between positive measuresread_more
HG G 43
4 November 2014
15:15-16:15 		Po-Ning Chen
Columbia University
Event Details

Analysis Seminar

Title Quasi-local mass in general relativity.
Speaker, Affiliation Po-Ning Chen, Columbia University
Date, Time 4 November 2014, 15:15-16:15
Location HG G 43
Abstract While the total energy of an isolated system in general relativity is well-studied, the concept of energy in general relativity remains a challenging problem because of the lack of a quasi-local description. In this talk, we survey several definition of quasi-local mass including the Hawking mass, the Brown-York mass, and their applications. We then describe a new proposal of quasi-local mass/energy and discuss its application and properties.
Quasi-local mass in general relativity.read_more
HG G 43
11 November 2014
14:00-15:00 		Simon Blatt
Universität Karlsruhe
Event Details

Analysis Seminar

Title The Supercritical Lane - Emden equation and its gradient flow
Speaker, Affiliation Simon Blatt, Universität Karlsruhe
Date, Time 11 November 2014, 14:00-15:00
Location HG G 43
Abstract In this talk we will consider the Lane-Emden equation $\Delta u + |u|^{p-2}u = 0$ and its gradient flow for supercritical exponents $p > 2^\ast = \frac {2n}{n-2}$ on an $n$-dimensional domain, , $n\geq 3$. Among many other things, these equations are used to model the gravitational equilibrium of polytropic stars. We will derive Morrey bounds, show the existence of tangent maps, and discuss partial regularity at the first singular time.
The Supercritical Lane - Emden equation and its gradient flowread_more
HG G 43
11 November 2014
15:15-16:15 		Dana Mendelson
MIT
Event Details

Analysis Seminar

Title Local nonsqueezing for the cubic nonlinear Klein-Gordon equation on $\bT^3$
Speaker, Affiliation Dana Mendelson, MIT
Date, Time 11 November 2014, 15:15-16:15
Location HG G 43
Abstract Consider the periodic defocusing cubic nonlinear Klein-Gordon equation in three dimensions in the symplectic phase space $H^{\frac{1}{2}}(\bT^3) \times H^{-\frac{1}{2}}(\bT^3)$. In this talk, we will present a local nonsqueezing result for this equation. In this space, the global wellposedness of this equation is still open and there is no uniform control on the local time of existence of solutions. We use an almost sure global wellposedness result for this equation to define a set of full measure with respect to a suitable randomization of the initial data on which the flow is globally defined. The proof of nonsqueezing then relies on Gromov's nonsqueezing theorem and an approximation result for the flow, which uses probabilistic estimates for the nonlinear component of the flow map and deterministic critical stability theory.
Local nonsqueezing for the cubic nonlinear Klein-Gordon equation on $\bT^3$ read_more
HG G 43
25 November 2014
14:00-15:00 		Prof. Dr. Glen Wheeler
University of Wollongong
Event Details

Analysis Seminar

Title Unstable Willmore Surfaces
Speaker, Affiliation Prof. Dr. Glen Wheeler, University of Wollongong
Date, Time 25 November 2014, 14:00-15:00
Location HG G 43
Abstract In this talk I will describe some recent work with Anna Dall'Acqua (Ulm) and Klaus Deckelnick (OvGU Magdeburg) that established the existence of Willmore surfaces with boundary that are unstable. The natural approach of using Palais-Smale and Mountain Pass doesn't (really) work. I'll explain why this is the case. We overcame this problem by using a completely different (and new) approach. I will finish by describing some open questions arising from the work.
Unstable Willmore Surfacesread_more
HG G 43
25 November 2014
15:15-16:15 		Maria Colombo
Scuola Normale of Pisa
Event Details

Analysis Seminar

Title Regularity for double phase variational problems
Speaker, Affiliation Maria Colombo, Scuola Normale of Pisa
Date, Time 25 November 2014, 15:15-16:15
Location HG G 43
Abstract In the eighties, Zhikov introduced some integral functionals to model non-homogeneous composites made by different basic materials. A model type is given by a variational integral whose integrand switches between two different types of elliptic phases according to a continuous coefficient, which depends on the position. If the elliptic phases are far from each other in a sharply quantified way, minimizers can be discontinuous. If this is not the case, we prove sharp regularity for minimizers, namely the Holder continuity of the gradient of the deformation. This is joint work with Paolo Baroni and Giuseppe Mingione.
Regularity for double phase variational problemsread_more
HG G 43
2 December 2014
14:15-15:15 		Dr. Fabiana Leoni
University "La Sapienza" of Rome
Event Details

Analysis Seminar

Title Local estimates and global non-existence results for fully nonlinear differential inequalities
Speaker, Affiliation Dr. Fabiana Leoni, University "La Sapienza" of Rome
Date, Time 2 December 2014, 14:15-15:15
Location HG G 43
Abstract We present some new a priori estimates for viscosity solutions of second order, fully nonlinear elliptic inequalities in possibly unbounded domains. We consider both degenerate elliptic inequalities with "absorbing" lower order terms satisfying generalized Keller-Osserman type conditions and uniformly elliptic inequalities with reaction power-like zero order terms. When the inequalities are posed in the whole space or in cone-like domains we give necessary and sufficient conditions for the existence of sub and supersolutions.
Local estimates and global non-existence results for fully nonlinear differential inequalitiesread_more
HG G 43
2 December 2014
15:30-16:30 		Mikaela Iacobelli
University "La Sapienza" of Rome
Event Details

Analysis Seminar

Title A gradient flow approach to quantization of measures
Speaker, Affiliation Mikaela Iacobelli, University "La Sapienza" of Rome
Date, Time 2 December 2014, 15:30-16:30
Location HG G 43
Abstract The problem of quantization of a $d$-dimension probability distribution by discrete probabilities with a given number of points can be stated as follows: Given a probability density $\rho$, approximate it in the Wasserstein metric by a convex combination of a finite number $N$ of Dirac masses. In a recent paper we studied a gradient flow approach to this problem in one dimension. By embedding the problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles tends to infinity. Under some suitable regularity assumptions on the density, we prove uniform stability and quantitative convergence result for the discrete and continuous dynamics.
A gradient flow approach to quantization of measuresread_more
HG G 43
9 December 2014
15:15-15:15 		Dr. Vedran Sohinger
ETH Zürich
Event Details

Analysis Seminar

Title The Gross-Pitaevskii hierarchy on periodic domains
Speaker, Affiliation Dr. Vedran Sohinger, ETH Zürich
Date, Time 9 December 2014, 15:15-15:15
Location HG G 43
Abstract : The Gross-Pitaevskii hierarchy is a system of infinitely many linear PDEs which occurs in the derivation of the nonlinear Schrodinger equation from the dynamics of many-body quantum system. We will study this problem in the periodic setting. Even though the hierarchy is linear, it is non-closed, in the sense that the equation for the k^th density matrix depends on the (k+1)^st density matrix. This structure poses its challenges in the study of the problem, in particular in the understanding of uniqueness of solutions. Moreover, by randomizing in the collision operator, it is possible to use probabilistic techniques to study related hierarchies at low regularities. I will summarize some recent results obtained partly in joint work with Philip Gressman and Gigliola Staffilani, as well as with Sebastian Herr.
The Gross-Pitaevskii hierarchy on periodic domainsread_more
HG G 43
16 December 2014
15:15-16:15 		Christopher Nerz
Universität Tübingen
Event Details

Analysis Seminar

Title Asymptotic flatness as a geometric property.
Speaker, Affiliation Christopher Nerz, Universität Tübingen
Date, Time 16 December 2014, 15:15-16:15
Location HG G 43
Abstract In mathematical general relativity, one often considers 'isolated' gravitational systems. This means that on every time-slice $\{t=const\}$, (almost) the whole mass is contained within a bounded domain. These time-slices are mathematically modeled by asymptotically flat Riemannian manifolds. Asymptotic flatness on the other hand is defined by the existence of a coordinate system satisfying suitable assumptions, i.e. this definition depends on a choice of coordinates. In this talk, we characterize asymptotic flatness by existence of a suitable foliation by hypersurfaces of constant mean curvature, first constructed by Huisken-Yau. In particular, we conclude that asymptotic flatness is a geometric property, i.e. it is coordinate independent, as to be expected by the physical motivation.
Asymptotic flatness as a geometric property.read_more
HG G 43

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Organizers: Francesca Da Lio, Tom Ilmanen, Thomas Kappeler, Tristan Rivière, Michael Struwe

Archive: SS 24 AS 23 SS 23 AS 22 SS 22 AS 21 SS 21 AS 20 SS 20 AS 19 SS 19 AS 18 SS 18 AS 17 SS 17 AS 16 SS 16 AS 15 SS 15 AS 14 SS 14 AS 13 SS 13 AS 12 SS 12 AS 11 SS 11 AS 10 SS 10 AS 09

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