User:Daviddaved/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:

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

.
In two-dimensions the equation takes a particularly simple form:

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,

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