Exotic function spaces and (their use in the theory of) integration by compensation: Lect.2
Tuesday, 27.10.09, 16:15-17:15, Raum 404, Eckerstr. 1
A) Lorentz spaces\n- definitions (with and without interpolation)\n- elementary properties (Hölder's, Young's inequalities, ...)\n- embeddings (comparisons to Lebesgue and Sobolev spaces)\n- use for regularity (Wente estimates, integration by compensation,\n"div-curl" type results)\n\nB) Hardy spaces, BMO, and Morrey-Campanato spaces\n- definitions\n- elementary properties\n- embeddings\n- use for regularity\n\nC) Applications for one or two specific problems\n- details to be clarified\n
Surfaces of Willmore type in Riemannian manifolds
Wednesday, 28.10.09, 16:15-17:15, Hörsaal II, Albertstr. 23b
In this talk I present recent results on surfaces of\nWillmore type in three dimensional Riemannian manifolds. These are\nsurfaces that are critical for the Willmore energy subject to an area\nconstraint. I present an analysis of spherical surfaces of Willmore\ntype with positive mean curvature in geodesic balls of small\nradius. As a result we obtain that such surfaces are well approximated\nby geodesic spheres. This enables us to derive necessary conditions\nfor the existence of such surfaces related to the scalar curvature of\nthe ambient manifold.\n
Exotic function spaces and (their use in the theory of) integration by compensation: Lect3
Tuesday, 3.11.09, 16:15-17:15, Raum 404, Eckerstr. 1
A) Lorentz spaces\n- definitions (with and without interpolation)\n- elementary properties (Hölder's, Young's inequalities, ...)\n- embeddings (comparisons to Lebesgue and Sobolev spaces)\n- use for regularity (Wente estimates, integration by compensation,\n"div-curl" type results)\n\nB) Hardy spaces, BMO, and Morrey-Campanato spaces\n- definitions\n- elementary properties\n- embeddings\n- use for regularity\n\nC) Applications for one or two specific problems\n- details to be clarified
On 2-scalar curvature
Tuesday, 10.11.09, 14:00-15:00, Hörsaal Virologie, Hermann-Herder-Straße 11
In this talk, we will first recall the definition\nof the k-scalar curvature and review the existence of the\ncorresponding Yamabe type problem for this curvature. Then\nwe discuss the analysis about the 2-scalar curvature and\npresent a 3-dimensional sphere theorem as an application.
Gauß curvature flows of entire graphs
Tuesday, 10.11.09, 16:00-17:00, Hörsaal Virologie, Hermann-Herder-Straße 11
We study entire graphs in Euclidean space that evolve with normal velocity equal to a power of the Gauß curvature. Mild restrictions on the initial data ensure that smooth solutions exist for all positive times. For initial data close to cones, we obtain stability results. This is joint work with John Urbas.\n
Numerical solutions of Euler equations by Discontinuous Galerkin finite element method.
Tuesday, 17.11.09, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Polyharmonische Abbildungen
Tuesday, 17.11.09, 16:15-17:15, Raum 404, Eckerstr. 1
Polyharmonische Abbildungen zwischen Riemannschen Mannigfaltigkeiten\n sind die kritischen Punkte von Energiefunktionalen h"oherer Ordnung,\n die das Dirichlet-Integral verallgemeinern. Im Vortrag wird dabei nur\n die ``kritische Dimension'' des Definitionsbereichs diskutiert. Hier\n geht es um die Existenz schwacher L"osungen mit der\n W"armeleitungsmethode sowie um die Regularit"at.
Harmonic maps and Dirac-harmonic maps from degenerating surfaces
Tuesday, 24.11.09, 16:15-17:15, Raum 404, Eckerstr. 1
We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy. We find conditions that are both necessary and sufficient for the compactness in \(W^{1,2}\) and \(C^{0}\) modulo bubbles of sequences of such maps and then establish a generalized energy identity. Similar methods can be applied to study certain compactness of sequences of Dirac-harmonic maps from degenerating spin surfaces.
Harmonic maps and Dirac-harmonic maps from degenerating surfaces
Tuesday, 24.11.09, 16:15-17:15, Raum 404, Eckerstr. 1
We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy. We find conditions that are both necessary and sufficient for the compactness in \(W^{1,2}\) and \(C^{0}\) modulo bubbles of sequences of such maps and then establish a generalized energy identity. Similar methods can be applied to study certain compactness of sequences of Dirac-harmonic maps from degenerating spin surfaces.\n\n
Calderón & Zygmund estimates for parabolic obstacle problems
Wednesday, 25.11.09, 16:15-17:15, Hörsaal II, Albertstr. 23b
Obstacle problems are a classical topic in the regularity theory of partial differential equations: they arise in a quantity of situations when modeling physical phenomena, and they are also used as an effective approximation tool allowing to face several fundamental issues in non-linear potential theory. A typical approach consists of recovering the regularity of solutions starting from that of the given obstacle.\nIn this talk we shall establish the natural Calderón & Zygmund theory for solutions of parabolic obstacle problems, proving that the gradient of solutions is as integrable as that of the assigned obstacles. Thereby, the considered obstacle function is allowed to be discontinuous. The involved operators are in divergence form and allowed to be degenerate, and the simplest model we have in mind is the parabolic p-Laplace operator.
Foliations of asymptotically flat manifolds by surfaces of Willmore type
Wednesday, 25.11.09, 17:30-18:30, Hörsaal II, Albertstr. 23b
In this talk we show the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Hawking mass. This is joint work with T. Lamm and J. Metzger.
Harmonic maps and Dirac-harmonic maps from degenerating surfaces
Tuesday, 1.12.09, 16:15-17:15, Raum 404, Eckerstr. 1
We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy. We find conditions that are both necessary and sufficient for the compactness in \(W^{1,2}\) and \(C^{0}\) modulo bubbles of sequences of such maps and then establish a generalized energy identity. Similar methods can be applied to study certain compactness of sequences of Dirac-harmonic maps from degenerating spin surfaces.\n\n
Exotic function spaces and (their use in the theory of) integration by compensation : Lect 4.
Tuesday, 1.12.09, 16:15-17:15, Raum 404, Eckerstr. 1
T.b.a
Tuesday, 8.12.09, 16:15-17:15, Raum 404, Eckerstr. 1
Two recent results in anisotropic PDE
Tuesday, 15.12.09, 16:15-17:15, Raum 404, Eckerstr. 1
New Harnack inequalities for Ricci flow
Wednesday, 16.12.09, 16:15-17:15, Hörsaal II, Albertstr. 23b
Exotic function spaces and (their use in the theory of) integration by compensation : Lect 5
Tuesday, 19.1.10, 16:15-17:15, Raum 404, Eckerstr. 1
Regularity of optimal transportation maps on compact locally nearly spherical manifolds
Wednesday, 20.1.10, 16:15-17:15, Hörsaal II, Albertstr. 23b
Given a couple of smooth positive measures of same total mass on a compact connected Riemannian manifold M, we look for a smooth optimal transportation map G, pushing one measure to the other at a least total squared\ndistance cost, directly by using the continuity method to produce a classical solution of the elliptic equation of Monge-Ampere type satisfied by the\npotential function u, such that G=exp(grad u). This approach boils down to proving an a priori upper bound on the Hessian of u. In this talk, based\non the recent local C^2 estimate of MaTrudinger-Wang, we treat the case\nof manifolds with curvature sufficiently close to 1 in C^2 norm.
T.b.a
Tuesday, 26.1.10, 16:15-17:15, Raum 404, Eckerstr. 1
T.b.a
Tuesday, 2.2.10, 16:15-17:15, Raum 404, Eckerstr. 1
Immersionen mit gleichmäßiger lokaler Lipschitz Darstellung
Tuesday, 2.2.10, 16:15-17:15, Raum 404, Eckerstr. 1
"Almost positivity" in the fourth order clamped plate equation
Wednesday, 3.2.10, 16:15-17:15, Hörsaal II, Albertstr. 23b
A classical example for a fourth order problem in mechanics\nis the linear clamped plate boundary value problem. "Linear questions" may be considered as well understood. This changes completely as soon as one poses\nthe simplest "nonlinear question": What can be said\nabout positivity preserving? Does a clamped plate bend upwards\nwhen being pushed upwards? It is known\nthat the answer is "no" in general. However, there are \npositivity issues as e.g. "almost positivity" to be discussed.\n\n\n\nThe lecture is based on joint work with F. Robert (Nice) \nand G. Sweers (Cologne).
Complete Ricci-flat open Kähler manifolds
Wednesday, 10.2.10, 16:15-17:15, Hörsaal II, Albertstr. 23b
Existence results and asymptotic properties of complete Ricci-flat Kähler metrics on open manifolds are presented.\nOpen manifolds are complements of a divisor D in a compact complex manifold. The level of smoothness of D plays a decisive role not only for the results but also for the techniques available to tackle the problem.
Geometric quantization on manifolds with boundary
Friday, 19.2.10, 16:15-17:15, Hörsaal II, Albertstr. 23b
We will describe a recent joint work with Xiaonan Ma where we prove a kind of geometric quantization formula for symplectic manifolds with boundary. The boundaryless case is the famous Guillemin-Sternberg conjecture first proved by Meinrenken and Meinrenken-Sjamaar. Our new result resolves a conjecture of Vergne on the geometric quantization formula on noncompact manifolds.\n\n