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For Zoom URL please contact Laura Keller

Autumn Semester 2020

Date / Time Speaker Title Location
22 September 2020
15:15-16:15 		Prof. Dr. Armin Schikorra
University of Pittsburgh
Event Details

Analysis Seminar

Title Scale-invariant tangent-point energies for knots and fractional harmonic maps
Speaker, Affiliation Prof. Dr. Armin Schikorra, University of Pittsburgh
Date, Time 22 September 2020, 15:15-16:15
Location Online via Zoom
Abstract I will report about the theory of minimizing and critical knots under a set of scale invariant knot energies, the so-called tangent-point energy. We obtain lower semicontinuity and weak Sobolev-convergence of minimizing sequences to critical points away from finitely many points in the domain. Extending earlier work on Moebius-, and O'Hara energies we also obtain regularity for such critical points. This is based on joint work with S. Blatt, Ph. Reiter, and N. Vorderobermeier.
Scale-invariant tangent-point energies for knots and fractional harmonic mapsread_more
Online via Zoom
29 September 2020
15:15-16:15 		Dr. Marc Pegon
Laboratoire Jacques-Louis Lions
Event Details

Analysis Seminar

Title Partial regularity for fractional harmonic maps into spheres (slides available here)
Speaker, Affiliation Dr. Marc Pegon, Laboratoire Jacques-Louis Lions
Date, Time 29 September 2020, 15:15-16:15
Location Online via Zoom
Abstract Similarly to “classical” harmonic maps, which are critical points of the Dirichlet energy, fractional harmonic maps are defined as critical points of a fractional Dirichlet energy associated with the s-power of the Laplacian, for s in (0,1). In this talk, after a brief reminder on classical harmonic maps, I will present the fractional setting and the partial regularity results we have obtained for maps valued into a sphere. In the case of half harmonic maps (s=1/2), I will also recall the connection with minimal surfaces with free boundary, which allowed us to improve known regularity results for energy minimizing maps into spheres.
Assets Slides file_download
Partial regularity for fractional harmonic maps into spheres (slides available here)read_more
Online via Zoom
20 October 2020
15:15-16:15 		Giada Franz
ETH Zurich, Switzerland
Event Details

Analysis Seminar

Title Free boundary minimal surfaces with connected boundary in the unit ball
Speaker, Affiliation Giada Franz, ETH Zurich, Switzerland
Date, Time 20 October 2020, 15:15-16:15
Location Online via Zoom
Abstract A free boundary minimal surface (FBMS) in the three-dimensional Euclidean unit ball is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ball (i.e. the unit sphere). It is natural to ask if there are FBMS in the unit ball of any given genus g and number of boundary components b. Several different examples have already been discovered, but the answer was so far still unknown already in the simple case g=b=1.
In this talk, we will present the construction of a family of FBMS with connected boundary and any given genus. This answers affirmatively to the aforementioned open question. This is joint work with Alessandro Carlotto and Mario Schulz.
Free boundary minimal surfaces with connected boundary in the unit ballread_more
Online via Zoom
27 October 2020
15:15-16:15 		Prof. Dr. Rosario Mingione
Università degli Studi di Parma
Event Details

Analysis Seminar

Title Nonautonomous functionals and regularity of minima
Speaker, Affiliation Prof. Dr. Rosario Mingione, Università degli Studi di Parma
Date, Time 27 October 2020, 15:15-16:15
Location Online via Zoom
Abstract I will first give a short survey of regularity results for minimizers of nonuniformly elliptic variational integrals. I will concentrate on the case of nonautonomous functionals, reporting on some recent joint results with Cristiana De Filippis.
Nonautonomous functionals and regularity of minimaread_more
Online via Zoom
10 November 2020
15:15-16:15 		Dr. Yi Zhang
ETH Zurich, Switzerland
Event Details

Analysis Seminar

Title Sobolev extension domains and quasiconformal mappings
Speaker, Affiliation Dr. Yi Zhang, ETH Zurich, Switzerland
Date, Time 10 November 2020, 15:15-16:15
Location Online via Zoom
Abstract We study bounded simply connected domains which admit operators extending $W^{1,\,p}$-Sobolev functions in the domains to the whole plane. When $p=2$, this class of domains has natural relations with quasiconformal mappings. We show how the results get extended for general $1\le p\le \infty$.
Sobolev extension domains and quasiconformal mappingsread_more
Online via Zoom
17 November 2020
15:15-16:15 		Dr. Elio Marconi
EPF Lausanne
Event Details

Analysis Seminar

Title On the structure of weak solutions of Burgers' equation
Speaker, Affiliation Dr. Elio Marconi, EPF Lausanne
Date, Time 17 November 2020, 15:15-16:15
Location Online via Zoom
Abstract We consider bounded weak solutions of Burgers' equation \begin{equation} u_t + \left(\frac{u^2}{2}\right)_x =0, \qquad u:(0,T)\times \mathbb{R} \to \mathbb{R} \end{equation} such that for every smooth and convex $\eta:\mathbb{R}\to \mathbb{R}$ and corresponding flux $q:\mathbb{R}\to \mathbb{R}$ defined by $q'(v)=v\eta'(v)$, the distribution $\mu_\eta:=\eta(u)_t + q(u)_x$ is a finite (signed) Radon measure.
This assumption is less restrictive than the more usual entropy condition which requires that $\mu_\eta\le 0$ and it is motivated by the study of variational problems related to micromagnetics among others. These solutions have not bounded variation in general, nevertheless they share with BV functions most of their fine properties: we will focus in particular on the rectifiability of the measure $\mu_\eta$, which detects the set of shock waves of $u$.
The analysis is based on the notion of Lagrangian representation for this class of weak solutions, which extends the classical method of characteristics to this nonsmooth setting.
On the structure of weak solutions of Burgers' equationread_more
Online via Zoom
24 November 2020
15:15-16:15 		Dr. Alessandro Audrito
ETH Zurich, Switzerland
Event Details

Analysis Seminar

Title The Neumann problem for some nonlocal elliptic operators: Regularity up to the boundary
Speaker, Affiliation Dr. Alessandro Audrito, ETH Zurich, Switzerland
Date, Time 24 November 2020, 15:15-16:15
Location Online via Zoom
Abstract In this seminar, we will review some known results concerning the regularity of solutions to some local and nonlocal elliptic equations, and then we will focus on recent developments about boundary regularity of solutions to a class of "Neumann" nonlocal elliptic equations. The new results have been obtained in a joint work with J.- C. Felipe-Navarro (UPC), X. Ros-Oton (UB).
The Neumann problem for some nonlocal elliptic operators: Regularity up to the boundaryread_more
Online via Zoom
1 December 2020
15:15-16:15 		Dr. Ali Hyder
ETH Zürich
Event Details

Analysis Seminar

Title Normal conformal metrics on R^4 with unbounded Q-curvature
Speaker, Affiliation Dr. Ali Hyder , ETH Zürich
Date, Time 1 December 2020, 15:15-16:15
Location Online via Zoom
Abstract It is known that for a smooth non-constant non-positive function $f$ with $\max f=0$ and for $\lambda>0$ small there exists a conformal metric on the 4-dimensional torus with Q-curvature $f+\lambda$. The corresponding metrics exhibit a bubbling phenomena as $\lambda$ decreses to zero. If all the maximum points of $f$ are non-degenerate, then after a suitable rescaling around a blow-up point $p$, the bubbling metrics converge to either a spherical metric or to a normal solution of $\Delta^2 u=(1+Hess(f)(p)[x,x])e^{4u}$ in $R^4$. I will talk about existence and non-existence of solutions to the above equation.
Normal conformal metrics on R^4 with unbounded Q-curvatureread_more
Online via Zoom
8 December 2020
15:15-16:15 		Prof. Dr. Felix Otto
MPI-MIS Leipzig
Event Details

Analysis Seminar

Title The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows
Speaker, Affiliation Prof. Dr. Felix Otto, MPI-MIS Leipzig
Date, Time 8 December 2020, 15:15-16:15
Location Online via Zoom
Abstract Flow of interfaces by mean curvature, in its multi-phase version, was first formulated in the context of grain growth in polycrystalline materials. The computationally efficient and very popular thresholding scheme for mean curvature flow by Osher et. al. can be naturally extended to such a multi-phase situation, even for surface tensions that depend on the lattice mismatch between the adjacent grains. This extension relies on the gradient flow structure of mean curvature flow, and the interpretation of the thresholding scheme as a corresponding ``minimizing movements'' scheme, that is, a sequence of variational problems naturally attached to the implicit time discretization of a gradient flow. This interpretation also allows for a (conditional) convergence proof based on De Giorgi's ideas for gradient flows in metric spaces. The approach is similar to the convergence proof for the minimizing movement scheme by Almgren, Taylor and Wang, as given by Luckhaus et. al. This is joint work with S. Esedoglu and T. Laux.
The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flowsread_more
Online via Zoom
15 December 2020
15:15-16:15 		Dr. Christoph Kehle
ETH Zurich, Switzerland
Event Details

Analysis Seminar

Title Diophantine Approximation as Cosmic Censor for AdS Black Holes
Speaker, Affiliation Dr. Christoph Kehle, ETH Zurich, Switzerland
Date, Time 15 December 2020, 15:15-16:15
Location Online via Zoom
Abstract The statement that general relativity is deterministic finds its mathematical formulation in the celebrated ‘Strong Cosmic Censorship Conjecture’ due to Roger Penrose. I will present my recent results on the linear analog of this conjecture in the case of negative cosmological constant and in the context of black holes. It turns out that this is intimately tied to Diophantine properties of a suitable ratio of mass and angular momentum of the black hole and that the validity of the conjecture depends in an unexpected way on the notion of genericity imposed.
Diophantine Approximation as Cosmic Censor for AdS Black Holesread_more
Online via Zoom

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