Title
Free boundary methods in inverse problems
Abstract
We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? If this happens, the obstacle would be invisible for this choice of incident wave. This phenomenon is closely related to inverse problems with a single measurement.
There are several results on corner scattering showing that obstacles with corners typically scatter every incident wave. It was recently observed that the theory of free boundary problems in PDE can be used to shed light on this issue. This suggests a dichotomy for boundary points of any invisible obstacle: either the boundary is regular, or the complement of the obstacle has to be very thin near the point.
The analysis boils down to understanding solutions of the elliptic equation $\Delta u = f \chi_D$, where $\chi_D$ is the characteristic function of the obstacle $D$ and both $u$ and $f$ may have varying sign. The optimal Sobolev regularity of solutions, a blowup analysis as in geometric measure theory, and a classification of blowup solutions are essential parts of the argument. This talk is based on joint work with Henrik Shahgholian (KTH).
