Abstract:

PDE specialists normally strive to construct solutions of PDEs and study properties of the solutions. In many applications one is interested in the reverse question - given the values (from experiments) of the solutions of the PDE on the boundary of the region, determine the coefficients of the PDE. We will introduce three inverse problems and the associated theoretical results and unsolved problems:

• determining the shape of a drum from its eigenvalues - the famous “Can you hear the shape of a drum”;
• determining the shape of a vocal tract from pressure measurements near the mouth;
• determining the electrical conductivity of an inhomogeneous body from current and voltage measurements on the boundary.

If there is time we will describe the techniques used to solve the conductivity problem.