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Convexity Properties of Harmonic Functions on Parameterized Families of Hypersurfaces

  • Stine Marie BergeEmail author
Article
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Abstract

It is known that the \(L^{2}\)-norms of a harmonic function over spheres satisfy some convexity inequality strongly linked to the Almgren’s frequency function. We examine the \(L^{2}\)-norms of harmonic functions over a wide class of evolving hypersurfaces. More precisely, we consider compact level sets of smooth regular functions and obtain a differential inequality for the \(L^{2}\)-norms of harmonic functions over these hypersurfaces. To illustrate our result, we consider ellipses with constant eccentricity and growing tori in \({\mathbf {R}}^3.\) Moreover, we give a new proof of the convexity result for harmonic functions on a Riemannian manifold when integrating over spheres. The inequality we obtain for the case of positively curved Riemannian manifolds with non-constant curvature is slightly better than the one previously known.

Keywords

Harmonic functions Almgren’s frequency function Convexity Properties of Harmonic functions 

Mathematics Subject Classification

53B20 35J05 31B05 

Notes

Acknowledgements

The author would like to thank Eugenia Malinnikova for her guidance and Dan Mangoubi for his insightful suggestions. The author was partially supported by the BFS/TFS project Pure Mathematics in Norway.

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Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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