Boundary layer potentials with some applications from imaging
The use of integral equations to solve boundary value problems has a long history, beginning with the study of gravitation and electrostatics in the late 18th and early 19th centuries. It was shown that solutions of the Neumann and Dirichlet problems for Laplace's equation can be represented by boundary layer potentials, which are certain integrals taken over the boundary of the domain of interest. Similar techniques can be used to study the solutions of other basic PDEs from mathematical physics. While these techniques are considered outdated for the treatment of PDE, they are still important today in many applications. We will start with an accessible introduction to boundary layer integrals and then discuss their applications to some medical imaging techniques, such as magnetoacoustic imaging and photoacoustic tomography.