During this course I will give two results which help allow to identify coefficients in a partial differential operator from knowledge of solutions at the boundary. One point of interest in the proofs presented is an effort to give close to optimal regularity assumptions on the coefficients of the operators but which still allow us to recover the coefficients. The first result shows that we can recover the boundary values of coefficients of a second order, divergence form elliptic operator from knowledge of the map that takes the boundary values of solutions to the normal derivative at the boundary. This is work of the speaker which built on earlier work of Kohn and Vogelius, Sylvester and Uhlmann, and Alessandrini. The second part of the course considers the recovery of a potential $q$ in an operator $(-\Delta )^m +q$ where the principal of the operator is the polyharmonic operator. We show that we can recover $q$ from a Dirichlet to Neumann type map or (almost) equivalently from the Cauchy data for solutions of this operator. Our results include cases where $q$ is a distribution of negative order and we will indicate some questions which remain open. This work is joint with L. Gauthier and D. Faraco. It builds on earlier results of Krupchyk, Lassas and Uhlmann for polyharmonic operators and makes use of an important technique of Haberman and Tataru.

During this course I will give two results which help allow to identify coefficients in a partial differential operator from knowledge of solutions at the boundary. One point of interest in the proofs presented is an effort to give close to optimal regularity assumptions on the coefficients of the operators but which still allow us to recover the coefficients. The first result shows that we can recover the boundary values of coefficients of a second order, divergence form elliptic operator from knowledge of the map that takes the boundary values of solutions to the normal derivative at the boundary. This is work of the speaker which built on earlier work of Kohn and Vogelius, Sylvester and Uhlmann, and Alessandrini. The second part of the course considers the recovery of a potential $q$ in an operator $(-\Delta )^m +q$ where the principal of the operator is the polyharmonic operator. We show that we can recover $q$ from a Dirichlet to Neumann type map or (almost) equivalently from the Cauchy data for solutions of this operator. Our results include cases where $q$ is a distribution of negative order and we will indicate some questions which remain open. This work is joint with L. Gauthier and D. Faraco. It builds on earlier results of Krupchyk, Lassas and Uhlmann for polyharmonic operators and makes use of an important technique of Haberman and Tataru.

During this course I will give two results which help allow to identify coefficients in a partial differential operator from knowledge of solutions at the boundary. One point of interest in the proofs presented is an effort to give close to optimal regularity assumptions on the coefficients of the operators but which still allow us to recover the coefficients. The first result shows that we can recover the boundary values of coefficients of a second order, divergence form elliptic operator from knowledge of the map that takes the boundary values of solutions to the normal derivative at the boundary. This is work of the speaker which built on earlier work of Kohn and Vogelius, Sylvester and Uhlmann, and Alessandrini. The second part of the course considers the recovery of a potential $q$ in an operator $(-\Delta )^m +q$ where the principal of the operator is the polyharmonic operator. We show that we can recover $q$ from a Dirichlet to Neumann type map or (almost) equivalently from the Cauchy data for solutions of this operator. Our results include cases where $q$ is a distribution of negative order and we will indicate some questions which remain open. This work is joint with L. Gauthier and D. Faraco. It builds on earlier results of Krupchyk, Lassas and Uhlmann for polyharmonic operators and makes use of an important technique of Haberman and Tataru.