Oberseminar Analysis Summer 2023
Organizers: S. Conti, H. Koch, S. Müller, B. Niethammer, A. Rüland, M. Rumpf, C. Thiele, J.J.L. Velázquez, R. Schubert
- Thursday, April 13 at 14:15 in Lipschitz Saal
Likhit Ganedi (Aachen) Phase Separation in Heterogeneous FluidsVariants of the famous Modica-Mortola functional can be used to model phase separation in heterogeneous fluids by supposing a free energy of the fluid mixture to have periodic heterogeneities of the scale δ and the phase separation to happen on the scale of ε. Γ-limits depend on the rate of convergence between δ and ε and we will discuss recent results in various regimes.
- Thursday, April 27 at 14:15 in SR 0.011
Idriss Mazari (Paris Dauphine) Quantitative inequalities in optimal control theory and convergence of thresholding schemesWe will give an overview of recent progress in the study of quantitative inequalities for optimal control problems. In particular, we will show how they can be used to obtain convergence results for thresholding schemes, which are of great importance in the simulation of optimal control problems. This is a joint work with A. Chambolle and Y. Privat.
- Thursday, May 11 at 14:15 in SR 0.011
Xian Liao (Karlsruhe) On the two-dimensional stratified regularity problemThe vortex patch problem for two-dimensional Euler equations is a well-known stratified regularity problem in fluid mechanics. We will recall Chemin's idea and Bertozzi-Constantin's observation in solving this problem. Some related regularity problems for elliptic equations will also be discussed.
- Thursday, May 25 at 16:00 in SR 1.008
Max Wardetzky (Göttingen) Efficient Simulations of Random Walks on Riemannian ManifoldsAccording to a version of Donsker's theorem, geodesic random walks on Riemannian manifolds converge to the respective Brownian motion. From a computational perspective, however, evaluating geodesics can be costly. We therefore introduce approximate geodesic random walks based on the concept of retractions. We show that these approximate walks converge to the correct Brownian motion in the Skorokhod topology as long as the geodesic equation is approximated up to second order. As a result we obtain an efficient algorithm for sampling Brownian motion on compact Riemannian manifolds. We will also discuss some open questions concerning possible extensions of our method to the setting of sub-Riemannian manifolds. This is joint work with Michael Herrmann, Simon Schwarz and Anja Sturm.
- Thursday, June 22 at 16:15 in SR 1.008
Roberta Marziani (Dortmund) A non-parametric Plateau problem with partial free boundaryWe consider a Plateau problem in codimension 1 in the non-parametric setting. A Dirichlet boundary datum is given only on a part of the boundary of a convex domain in the plane. Where the Dirichlet datum is not prescribed, we allow the solution to have a free contact with the plane domain. We show existence of a solution, and prove some regularity for the corresponding area-minimizing surface. Finally we compare the solutions we find with classical solutions provided by Meeks and Yau, and show that they are equivalent at least in the case that the Dirichlet boundary datum is assigned in at most 2 connected components of the boundary of the domain. This result was obtained in collaboration with Giovanni Bellettini and Riccardo Scala.
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