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arXiv · 0806.0653

The Square of the Dirichlet-to-Neumann map equals minus Laplacian

David V. Ingerman
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Abstract

The Dirichlet-to-Neumann maps connect boundary values of harmonic functions. It is an amazing fact that the square of the non-local Dirichlet-to-Neumann map for the uniform conductivity 1 on the unit disc equals minus the local(!) Laplace operator on the boundary circle. To establish a new connection between discrete and continuous Dirichlet-to-Neumann maps and for the approximations I construct a finite and an infinite graphs which Dirichlet-to-Neumann map have the same property: \Lambda^2(1) = - d^2/d \theta^2. The construction gives a new continued fraction identity. It is interesting to consider the geometric and probabilistic (trajectories of the random walk) consequences of this localizing identity unifying discrete and continuous equations for potentials.

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David V. Ingerman. 2008-06-03. The Square of the Dirichlet-to-Neumann map equals minus Laplacian. https://arxiv.org/abs/0806.0653

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