Dirichlet-to-Neumann operator

Summary

The Dirichlet-to-Neumann operator of a domain sends boundary values to normal derivatives. Given a function $f$ on the boundary, it solves the Dirichlet problem for Laplace's equation with data $f$. Then it returns the normal derivative of the solution on the boundary; conductivity equations work the same way. It is a basic example of a pseudo-differential operator on the boundary. It carries the boundary measurements of electrical impedance tomography, in the Calderón problem, and its eigenvalues are the Steklov eigenvalues.