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On the Regularity of Harmonic Functions and Spherical Harmonic Maps Defined on Lattices

✍ Scribed by Lawrence E. Thomas

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 140 KB
Volume: 262
Category: Article
ISSN: 0022-247X
DOI: 10.1006/jmaa.2001.7575

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✦ Synopsis

A growth lemma for certain discrete symmetric Laplacians defined on a lattice Z d δ = δZ d ⊂ R d with spacing δ is proved. The lemma implies a De Giorgi theorem, that the harmonic functions for these Laplacians are equi-Hölder continuous, δ → 0. These results are then applied to establish regularity properties for the harmonic maps defined on Z d δ and taking values in an n-dimensional sphere S n , uniform in δ. Questions of the convergence δ → 0 and the Dirichlet problem for these discrete harmonic maps are also addressed.

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