For more information about this meeting, contact Victor Nistor, Jinchao Xu, Stephanie Zerby, Xiantao Li, Yuxi Zheng, Kris Jenssen, Hope Shaffer.

Title: | Regularity results for Schroedinger operators |

Seminar: | Computational and Applied Mathematics Colloquium |

Speaker: | Bernd Ammann, Regensburg University, Germany |

Abstract: |

The talk will present regularity statements for elliptic operators with certain types of singularities. Regularity results are often an important ingredient in the design of numerical methods. The main application is to Schroedinger operators.
In fact, we will consider two types of singularities: at first we study
operators similar to the Schroedinger operator, i.e. elliptic differential operators with a potential comparable to $r^{-s}$ where $r$ is the distance to a
submanifold and where $s$ is positive, but smaller than the order of the operator. In this frist case, we consider the points with infinite potential as singularities. Secondly we study elliptic operators on bounded domains, with non-smooth but piecewise smooth boundary. In this case we consider the non-smooth boundary points as singularities.
In both cases, one cannot expect the usual regularity results in the
standard Sobolev spaces close to the singularity.
In the talk we use blow-up methods to derive suitable Sobolev spaces which are well-adapted the geometry of the singularities. In these modified Sobolev spaces, one obtains regularity results, generalizing the standard regularity statements to operators on domains with non-smooth but piecewise smooth boundary and to Schr\"odinger type operators. The blow-up method is formalized in the language of Lie manifolds, a class of non-compact complete manifolds. This class of manifolds contains b-manifolds in the sense of Melrose, asymptotically hyperbolic manifolds and many more. We then describe a systematic way to conformally blow-up a Lie-manifold along a given submanifold, and the result will be a Lie manifold with "new" points at infinity.
The results I will present are joint work with Victor Nistor, Catarina Carvalho, Robert Lauter, and Alexandru Ionescu. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 02 / 10 / 2012 |

Time: | 03:35pm - 04:25pm |