Modul: MAT675 PDE and Mathematical Physics

## On global solutions of the obstacle problem

Prof. Dr. Georg Weiss talk

Speaker invited by: Prof. Dr. Xavier Ros-Oton

**Date:** 17.10.19 **Time:** 18.10 - 19.10 **Room:** Y27H35/36

While compact coincidence sets have been completely characterised in the classical obstacle problem, concerning the characterisation of unbounded coincidence sets much less is known, and a characterisation is related to more than 20 year old conjectures in potential theory. Only in the two dimensional case a complete classification of coincidence sets has been carried out by M. Sakai, determining the respective Riemann mapping. The conjecture is that apart from the compact case only half spaces as well as so-called elliptical paraboloids are possible. Different from other problems in which the solution shares the self-similar structure of its level set and that self-similarity may in certain cases be proved, the solution does not have a self-similar structure in our case. Transform methods like the Hodograph transform do not seem to help either. In our talk we present a positive result in higher dimensions.