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Autumn Semester 2018

Date / Time Speaker Title Location
2 October 2018
15:15-16:15 		Prof. Dr. Tobias Lamm
KIT Karlsruher Institut für Technologie
Event Details

Analysis Seminar

Title Expanders of the harmonic map flow
Speaker, Affiliation Prof. Dr. Tobias Lamm, KIT Karlsruher Institut für Technologie
Date, Time 2 October 2018, 15:15-16:15
Location HG G 43
Expanders of the harmonic map flow
HG G 43
9 October 2018
15:15-16:15 		Prof. Dr. Serena Dipierro
The University of Western Australia, Perth
Event Details

Analysis Seminar

Title A free boundary problem driven by the biharmonic operator
Speaker, Affiliation Prof. Dr. Serena Dipierro, The University of Western Australia, Perth
Date, Time 9 October 2018, 15:15-16:15
Location HG G 43
Abstract In this talk we describe a minimization problem involving the biharmonic operator, and we discuss some recent results obtained in collaboration with A. Karakhanyan and E. Valdinoci about the free boundary condition, the regularity of the solutions and that of their free boundary, and a monotonicity formula.
A free boundary problem driven by the biharmonic operatorread_more
HG G 43
6 November 2018
15:15-16:15 		Prof. Dr. Luca Martinazzi
Università degli Studi di Padova
Event Details

Analysis Seminar

Title News on the Moser-Trudinger inequality
Speaker, Affiliation Prof. Dr. Luca Martinazzi, Università degli Studi di Padova
Date, Time 6 November 2018, 15:15-16:15
Location HG G 43
Abstract The existence of critical points for the Moser-Trudinger inequality for large energies has been open for a long time. We will first show how a collaboration with G. Mancini allows to recast the Moser-Trudinger inequality and the existence of its extremals (originally due to L. Carleson and A. Chang) under a new light, based on sharp energy estimate. Building upon a recent subtle work of O. Druet and P-D. Thizy, in a work in progress with O. Druet, A. Malchiodi and P-D. Thizy, we use these estimates to compute the Leray-Schauder degree of the Moser-Trudinger equation (via a suitable use of the Poincaré-Hopf theorem), hence proving that for any bounded non-simply connected domain the Moser-Trudinger inequality admits critical points of arbitrarily high energy. In a work in progress with F. De Marchis, O. Druet, A. Malchiodi and P-D. Thizy, we expect to use a variational argument to treat the case of a closed surface.
News on the Moser-Trudinger inequalityread_more
HG G 43
13 November 2018
15:15-16:15 		Prof. Dr. Enno Lenzmann
Universität Basel
Event Details

Analysis Seminar

Title A New Rearrangement Technique in Fourier Space and Symmetry Results for Higher-Order PDE
Speaker, Affiliation Prof. Dr. Enno Lenzmann, Universität Basel
Date, Time 13 November 2018, 15:15-16:15
Location HG G 43
Abstract In this talk, I discuss a new method to prove general symmetry results for variational problems in R^n involving (pseudo)-differential operators of arbitrary order. In particular, we can deal with problems, where the classical Polya-Szegö principle fails to be applicable. The method is based on symmetric-decreasing rearrangements in Fourier space. We obtain a class of sharp rearrangement inequalities. As some applications, we prove radial symmetry of optimizers for Gagliardo-Nirenberg inequalities in L^2 with arbitrary differential order and some general Adams-Moser-Trudinger type in equalities in R^n. This is joint work with Jérémy Sok.
A New Rearrangement Technique in Fourier Space and Symmetry Results for Higher-Order PDEread_more
HG G 43
20 November 2018
15:15-16:15 		Dr. Katarzyna Mazowiecka
Université catholique de Louvain
Event Details

Analysis Seminar

Title On the size of the singular set of minimizing harmonic maps
Speaker, Affiliation Dr. Katarzyna Mazowiecka, Université catholique de Louvain
Date, Time 20 November 2018, 15:15-16:15
Location HG G 43
Abstract In this talk, I will consider minimizing harmonic maps from 3-dimensional domains into the two dimensional sphere and present an extension of Almgren and Lieb’s linear law on the bound of the singular set as well as Hardt and Lin’s stability theorem for singularities. I will also discuss new higher dimensional counterparts of those theorems. This is joint work with Michal Miskiewicz and Armin Schikorra.
On the size of the singular set of minimizing harmonic mapsread_more
HG G 43
20 November 2018
16:30-17:30 		Prof. Dr. Matteo Bonforte
Universidad Autónoma de Madrid
Event Details

Analysis Seminar

Title Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains
Speaker, Affiliation Prof. Dr. Matteo Bonforte, Universidad Autónoma de Madrid
Date, Time 20 November 2018, 16:30-17:30
Location HG G 43
Abstract We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=\Delta u^m$, posed in a smooth bounded domain $\Omega\subset \RR^N$, in the exponent range $m_s=(N-2)_+/(N+2)<m<1$. It is known that bounded positive solutions $u(t,x)$ of such problem extinguish in a finite time $T=T(u_0)$ and also that they approach a separate variable solution $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$. It has been shown in 2012 that $v(x,t)=u(t,x)\,(T-t)^{-1/(1-m)}$ tends to $S(x)$ as $t\to T^-$, uniformly in the relative error norm; to our knowledge, this is the best asymptotic result valid in the whole range of parameters. Starting from this result, we investigate the fine asymptotic behaviour, which amounts to derive (sharp) rates of convergence for the relative error. The proof is based on a new entropy method and one of the fundamental ingredients of such method is a (improved) weighted Poincaré inequality, that we show to be true on a generic bounded domain. Another essential aspect of the method is the new concept of "almost orthogonality", which can be thought as a nonlinear analogous of the classical orthogonality condition needed to obtain improved Poincaré inequalities and sharp convergence rates for linear flows. This is a joint work with Alessio Figalli.
Extinction Rates for Fast Diffusion Equations on Generic Bounded Domainsread_more
HG G 43
27 November 2018
15:15-16:15 		Dr. Paul Laurain
Université Paris Diderot
Event Details

Analysis Seminar

Title Energy convexity for some conformally invariant problems and applications
Speaker, Affiliation Dr. Paul Laurain, Université Paris Diderot
Date, Time 27 November 2018, 15:15-16:15
Location HG G 43
Abstract After a short introduction to harmonic maps, I will explain why the energy convexity is hard to obtain especially when the target is positively curved. Then we will give a new proof of the original result of Colding and Minicozzy. As we will underline it, the main advantage of this proof is that it relies only on the $\eps$-regularity property and some Hardy inequality. Then we will be able to get new results for biharmonic maps and free boundary harmonic maps. Finally, we will focus on the later case, explaining how this comes as a crucial step in the existence of free-boundary minimal disc realizing the width of a manifold with boundary. This work is joined with L. Lin and P. Petrides.
Energy convexity for some conformally invariant problems and applicationsread_more
HG G 43
4 December 2018
15:15-16:15 		Prof. Dr. Gian Paolo Leonardi
Università di Modena
Event Details

Analysis Seminar

Title Approximate curvatures of a varifold
Speaker, Affiliation Prof. Dr. Gian Paolo Leonardi, Università di Modena
Date, Time 4 December 2018, 15:15-16:15
Location HG G 43
Abstract Varifolds, i.e. Radon measures on the Grassmannian bundle of unoriented tangent d-planes of a Riemannian n-manifold M, represent a variational generalization of unoriented, d-dimensional submanifolds of M. By a suitable extension of classical variation operators, we introduce a notion of approximate second fundamental form that is well-defined for a generic varifold. Rectifiability, compactness, and convergence results are proved, showing in particular the consistency and stability of approximate curvatures with respect to varifold convergence. If restricted to the case of "discrete varifolds", this theory provides a general framework for extracting key features from discrete geometric data. Some numerical tests on point clouds (evaluation of curvatures and geometric flows, also in presence of noise and singularities) will be shown. We shall finally discuss some future perspectives and open problems. This is a joint research with Blanche Buet (Univ. Paris XI - Orsay) and Simon Masnou (Univ. Lyon 1).
Approximate curvatures of a varifoldread_more
HG G 43
11 December 2018
15:15-16:15 		Francesco Palmurella
ETH Zurich, Switzerland
Event Details

Analysis Seminar

Title A Resolution of the Poisson Problem for Elastic Plates
Speaker, Affiliation Francesco Palmurella, ETH Zurich, Switzerland
Date, Time 11 December 2018, 15:15-16:15
Location HG G 43
Abstract The Poisson problem consists in finding a surface immersed in the Euclidean space minimising Germain's elastic energy (known as Willmore energy in geometry) with assigned boundary, boundary Gauss map and area; it constitutes a non-linear model for the equilibrium state of thin, clamped elastic plates. We present a solution, and discuss its partial boundary regularity, to a variationally equivalent version of this problem when the boundary curve is simple and closed, as in the most classical version of the Plateau problem. This is a joint work with F. Da Lio & T. Rivière.
A Resolution of the Poisson Problem for Elastic Platesread_more
HG G 43
18 December 2018
15:15-16:15 		Dr. Reto Buzano
School of Mathematical Sciences, Queen Mary University of London
Event Details

Analysis Seminar

Title Bubbling analysis for closed and free boundary minimal hypersurfaces
Speaker, Affiliation Dr. Reto Buzano, School of Mathematical Sciences, Queen Mary University of London
Date, Time 18 December 2018, 15:15-16:15
Location HG G 43
Abstract We will discuss a variety of weak and strong compactness results for sequences of minimal hypersurfaces (with and without boundary) in compact Riemannian manifolds of dimension between 3 and 7. In particular, exploiting a precise bubbling analysis, we obtain new strong compactness theorems in dimension 3 that extend in particular results of Choi-Schoen and Fraser-Li from ambient manifolds with positive Ricci curvature to manifolds with only positive scalar curvature under the additional assumption of an index bound. The presented results have been obtained in joint work with Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp.
Bubbling analysis for closed and free boundary minimal hypersurfacesread_more
HG G 43

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