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Topological Properties of Neumann Domains
A Laplacian eigenfunction on a two-dimensional manifold dictates some natural
partitions of the manifold; the most apparent one being the well studied nodal
domain partition. An alternative partition is revealed by considering a set of
distinguished gradient flow lines of the eigenfunction - those which are
connected to saddle points. These give rise to Neumann domains. We establish
complementary definitions for Neumann domains and Neumann lines and use basic
Morse homology to prove their fundamental topological properties. We study the
eigenfunction restrictions to these domains. Their zero set, critical points
and spectral properties allow to discuss some aspects of counting the number of
Neumann domains and estimating their geometry
Numerical computation of the conformal map onto lemniscatic domains
We present a numerical method for the computation of the conformal map from
unbounded multiply-connected domains onto lemniscatic domains. For -times
connected domains the method requires solving boundary integral
equations with the Neumann kernel. This can be done in
operations, where is the number of nodes in the discretization of each
boundary component of the multiply connected domain. As demonstrated by
numerical examples, the method works for domains with close-to-touching
boundaries, non-convex boundaries, piecewise smooth boundaries, and for domains
of high connectivity.Comment: Minor revision; simplified Example 6.1, and changed Example 6.2 to a
set without symmetr
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