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Convexity Properties of Harmonic Functions on Parameterized Families of Hypersurfaces

Berge, Stine Marie
Journal article, Peer reviewed
Accepted version
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URI
http://hdl.handle.net/11250/2638585
Date
2019
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  • Institutt for matematiske fag [1390]
  • Publikasjoner fra CRIStin - NTNU [19694]
Original version
Journal of Geometric Analysis. 2019, 1-27.   10.1007/s12220-019-00307-y
Abstract
It is known that the L2L2-norms of a harmonic function over spheres satisfy some convexity inequality strongly linked to the Almgren’s frequency function. We examine the L2L2-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 L2L2-norms of harmonic functions over these hypersurfaces. To illustrate our result, we consider ellipses with constant eccentricity and growing tori in R3.R3. 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.
Publisher
Springer Verlag
Journal
Journal of Geometric Analysis

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