The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem |
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Authors: | Daniel Grieser |
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Institution: | 1. Institut für Mathematik, Carl von Ossietzky Universit?t Oldenburg, D-26111, Oldenburg, Germany
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Abstract: | We show how ‘test’ vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian
manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that
a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger’s constant may be obtained
precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger’s constant in
terms of the inradius of a plane domain.
Received: 13 June 2005 |
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Keywords: | Primary 35P15 Secondary 51M16 49N15 |
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