Mikko Salo
University of Helsinki, Finland
Date: May 1, 2006
We consider the problem of determining the coefficients of a Schrodinger equation from boundary measurements. This is a well known model case for many inverse problems for elliptic equations. Recently, Kenig, Sjostrand, and Uhlmann were able to interpret many of the earlier results on such problems in a more general context. Their treatment was based on Carleman estimates, special solutions to Schrodinger equations, and analytic microlocal analysis. In another article with Dos Santos Ferreira, these authors extended the results to the magnetic Schrodinger equation with limited boundary measurements.
In the case of nonsmooth coefficients, it turns out
that the special solutions produced by Carleman estimates may not
have enough regularity to allow for the solution of the inverse
problem. We will give an improved construction of special solutions
which combines Carleman estimates with (an extension of) a
pseudodifferential conjugation technique due to Nakamura and
Uhlmann), so that the solutions will have sufficient regularity. As a
consequence, we extend the results of Dos Santos Ferreira et al. to
Holder continuous coefficients. This is joint work with Kim Knudsen
from Aalborg University.