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Extension results for harmonic functions which vanish on cylindrical surfaces

Extension results for harmonic functions which vanish on cylindrical surfaces The Schwarz reflection principle applies to a harmonic function which continuously vanishes on a relatively open subset of a planar or spherical boundary surface. It yields a harmonic extension to a predefined larger domain and provides a simple formula for this extension. Although such a point-to-point reflection law is unavailable for other types of surface in higher dimensions, it is natural to investigate whether similar harmonic extension results still hold. This article describes recent progress on such results for the particular case of cylindrical surfaces, and concludes with several open questions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Extension results for harmonic functions which vanish on cylindrical surfaces

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-018-0213-0
Publisher site
See Article on Publisher Site

Abstract

The Schwarz reflection principle applies to a harmonic function which continuously vanishes on a relatively open subset of a planar or spherical boundary surface. It yields a harmonic extension to a predefined larger domain and provides a simple formula for this extension. Although such a point-to-point reflection law is unavailable for other types of surface in higher dimensions, it is natural to investigate whether similar harmonic extension results still hold. This article describes recent progress on such results for the particular case of cylindrical surfaces, and concludes with several open questions.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Feb 10, 2018

References