# User:DVD206/The square root of the minus Laplacian

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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:
${\displaystyle \Delta f=r^{1-N}{\frac {\partial }{\partial r}}\left(r^{N-1}{\frac {\partial f}{\partial r}}\right)+r^{-2}\Delta _{S^{N-1}}f,}$

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

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

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,

${\displaystyle \Lambda ^{2}-\Lambda +\Delta _{S^{2}}=0.}$

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

${\displaystyle \Lambda ^{2}=-\Delta _{R^{N-1}}.}$