Departement Mathematik

Analysis seminar

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For Zoom URL please contact Laura Keller

Herbstsemester 2017

Datum / Zeit Referent:in Titel Ort
17. Oktober 2017
15:15-16:15 		Dr. Filippo Cagnetti
University of Sussex
Details

Analysis Seminar

Titel Stochastic Homogenisation of Free-Discontinuity Problems
Referent:in, Affiliation Dr. Filippo Cagnetti, University of Sussex
Datum, Zeit 17. Oktober 2017, 15:15-16:15
Ort HG G 43
Abstract Free-discontinuity problems were introduced by Ennio De Giorgi and Luigi Ambrosio in 1988. These are variational problems where the energy to be minimised involves both volume and surface terms. The expression "Free-Discontinuity" refers to the fact that the set where the surface energy is concentrated is not a priori fixed, and can be described as the discontinuity set of a function. We will discuss the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised functional, whose volume andsurface integrands are characterised by asymptotic formulas involving minimisation problems on larger and larger cubes with special boundary conditions. In the proof we combine a recent deterministic Gamma-convergence result for free-discontinuity functionals with the Subadditive Ergodic Theorem by Akcoglu and Krengel. This is a joint work in collaboration with Gianni Dal Maso (SISSA), Lucia Scardia (University of Bath), and Caterina Zeppieri (University of Münster).
Stochastic Homogenisation of Free-Discontinuity Problemsread_more
HG G 43
7. November 2017
15:15-16:15 		Alexis Michelat
ETH Zurich, Switzerland
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Analysis Seminar

Titel The classification of branched Willmore two-spheres in the three-sphere and the four-sphere
Referent:in, Affiliation Alexis Michelat, ETH Zurich, Switzerland
Datum, Zeit 7. November 2017, 15:15-16:15
Ort HG G 43
Abstract In 1984, Robert Bryant, inspired by classical results of Calabi and Chern on the classification of minimal two-spheres in spheres, proved that smooth Willmore immersions from the two-sphere into the three-sphere are completely classified by a special family of minimal surfaces : they are inverse stereographic projections of complete minimal surfaces of finite total curvature and embedded planar ends from Euclidean three-space. An analogous result also holds for immersions of the two-sphere into the four-sphere thanks of a theorem of Sebástian Montiel. However, when we blow-up Willmore surfaces, the limiting immersions may have isolated branched points, in which case, the previous classifications do not hold a priori. Thanks of an optimal regularity theorem at branched points holding in any codimension, we extend these two classifications to branched immersions of the two-sphere. As a consequence, we deduce, in particular, that the width of Willmore min-max procedures in the three-sphere and the four-sphere, such as the sphere eversion, is an integer multiple of 4π. This is joint work with Tristan Rivière.
The classification of branched Willmore two-spheres in the three-sphere and the four-sphereread_more
HG G 43
14. November 2017
15:15-16:15 		Dr. Karen Yeressian Negarchi
KTH Royal Institute of Technology
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Analysis Seminar

Titel Obstacle problem with a degenerate force term
Referent:in, Affiliation Dr. Karen Yeressian Negarchi, KTH Royal Institute of Technology
Datum, Zeit 14. November 2017, 15:15-16:15
Ort HG G 43
Abstract The obstacle problem is an important example of a free boundary problem. It describes the equilibrium of an elastic membrane constrained to stay above an obstacle. In a region the membrane touches the obstacle and the boundary of this region (inside the domain) is the free boundary, which is an interesting geometric feature of the solution. Many people have contributed to the study of this problem. Frehse 1972, proved the optimal regularity of the solution. Caffarelli 1977 and 1998 studied the regularity and structure of the free boundary close to regular and singular points. Recently Figalli and Serra 2017 have obtained fine results about the structure of singular points. In the free boundary regularity results a key assumption has been that the obstacle is strictly a superharmonic function. In this talk I will present results in two papers which consider the case where we drop this assumption. Mainly we consider in a two dimensional domain an obstacle which is cubic flat on the top, thus the minus Laplacian of the obstacle is vanishing on a straight line and grows linearly away from this line. The free boundary exhibits interesting regular and irregular behaviour.
Obstacle problem with a degenerate force termread_more
HG G 43
21. November 2017
15:15-16:15 		Alessandro Pigati
ETH Zurich, Switzerland
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Analysis Seminar

Titel The regularity of parametrized stationary varifolds in two dimensions
Referent:in, Affiliation Alessandro Pigati, ETH Zurich, Switzerland
Datum, Zeit 21. November 2017, 15:15-16:15
Ort HG G 43
The regularity of parametrized stationary varifolds in two dimensions
HG G 43
28. November 2017
15:15-16:15 		Dr. Ruijun Wu
MPI Leipzig
Details

Analysis Seminar

Titel Regularity of solutions of a nonlinear sigma model with gravitino
Referent:in, Affiliation Dr. Ruijun Wu, MPI Leipzig
Datum, Zeit 28. November 2017, 15:15-16:15
Ort HG G 43
Abstract I will talk about a nonlinear sigma model with gravitino, which is a variant of the supersymmetric nonlinear sigma model. A brief description of the model will be given. After introducing the weak solutions of the Euler-Lagrange systems, we will improve the regularity of them, where the assumption of the gravitino is relevant. The weak solutions are smooth if the gravitino itself is smooth, and are continuous even if the gravitino is coarse. The special structure of the elliptic system is the key for the improvement.
Regularity of solutions of a nonlinear sigma model with gravitinoread_more
HG G 43
5. Dezember 2017
15:15-16:15 		Prof. Dr. Joaquim Serra
ETH Zurich, Switzerland
Details

Analysis Seminar

Titel On the fine structure of the free boundary for the classical obstacle problem
Referent:in, Affiliation Prof. Dr. Joaquim Serra, ETH Zurich, Switzerland
Datum, Zeit 5. Dezember 2017, 15:15-16:15
Ort HG G 43
Abstract In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. Celebrated results of Caffarelli establish that regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of $C^1$ manifolds of varying dimension. In two dimension, this $C^1$ result was improved to $C^{1,\alpha}$ by Weiss.
In the seminar I'll present recent results with A. Figalli in which we prove, in dimension 2, that singular points are locally contained in a $C^2$ curve. In higher dimensions $n\ge 3$ we show that the same result holds with $C^{1,1}$ manifolds, up to the presence of some "anomalous" points of higher codimension. In addition, we prove that the higher dimensional stratum is always contained in a $C^{1,\alpha}$, thus extending to every dimension the result of Weiss.
Finally I will discuss some examples of solutions with anomalous points which show that the bounds we can establish on the Hausdorff dimension of the anomalous points are actually optimal.
On the fine structure of the free boundary for the classical obstacle problemread_more
HG G 43
12. Dezember 2017
15:15-16:15 		Dr. Matthias Erbar
Universität Bonn
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Analysis Seminar

Titel A gradient flow approach to the Boltzmann equation
Referent:in, Affiliation Dr. Matthias Erbar, Universität Bonn
Datum, Zeit 12. Dezember 2017, 15:15-16:15
Ort HG G 43
Abstract In this talk I will present a new point of view on the spatially homogeneous Boltzmann equation viewing it as the gradient flow of theentropy. This gradient flow structure relies on a new notion of distance between probability measures that takes the collision process between particles into account and takes over the role of the Wasserstein distance. As two applications of this point of view I will present a time-discrete variational approximation scheme for the homogeneous Boltzmann equation and a new and simple proof for the convergence of Kac's random walk to the Boltzmann equation.
A gradient flow approach to the Boltzmann equationread_more
HG G 43
19. Dezember 2017
15:15-16:15 		Dr. Daniele Castorina
John Cabot University - Rome
Details

Analysis Seminar

Titel Ancient solutions of superlinear heat equations on Riemannian manifolds
Referent:in, Affiliation Dr. Daniele Castorina, John Cabot University - Rome
Datum, Zeit 19. Dezember 2017, 15:15-16:15
Ort HG G 43
Abstract We study the qualitative properties of ancient solutions of superlinear heat equations in a Riemannian manifold, with particular attention topositivity and triviality in space. This is joint work with Carlo Mantegazza (Napoli Federico II).
Ancient solutions of superlinear heat equations on Riemannian manifoldsread_more
HG G 43

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Organisatoren:innen: Alessandro Carlotto, Francesca Da Lio, Alessio Figalli, Norbert Hungerbühler, Tom Ilmanen, Thomas Kappeler, Tristan Rivière, Dietmar Salamon, Michael Struwe

Archiv: FS 24 HS 23 FS 23 HS 22 FS 22 HS 21 FS 21 HS 20 FS 20 HS 19 FS 19 HS 18 FS 18 HS 17 FS 17 HS 16 FS 16 HS 15 FS 15 HS 14 FS 14 HS 13 FS 13 HS 12 FS 12 HS 11 FS 11 HS 10 FS 10 HS 09

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