Last modified on 5 September 2012, at 23:22

User:Daviddaved/The square root of the minus Laplacian

The continuous Dirichlet-to-Neumann operator can be calculated explicitly for certain domains, such as a half-space, a ball and a cylinder and a shell with uniform conductivity 1. For example, for a unit ball in N-dimensions, writing the Laplace equation in spherical coordinates one gets:
$\Delta f = r^{1-N}\frac{\partial}{\partial r}\left(r^{N-1}\frac{\partial f}{\partial r}\right) + r^{-2}\Delta_{S^{N-1}}f,$

and, therefore, the Dirichlet-to-Neumann operator satisfies the following equation:

$\Lambda(\Lambda-(N-2)Id)+\Delta_{S^{N-1}} = 0$.
In two-dimensions the equation takes a particularly simple form:
$\Lambda^2=-\Delta_{S^{1}}.$

The study of material of this chapter is largely motivated by the question of Professor of Mathematics in the University of Washington Gunther Uhlmann: "Is there a discrete analog of the equation?"

Exercise (*): Prove that for the three-dimensional unit ball the Dirichlet-to-Neumann operator satisfies the following quadratic equation,

$\Lambda^2 - \Lambda + \Delta_{S^{2}} = 0.$

Exercise (*): Prove that for the Dirichlet-to-Neumann operator of a half-space of RN with uniform conductivity 1,

$\Lambda^2 = -\Delta_{R^{N-1}}.$