2019/07/18 by Berge, Stine Marie
#31B05 #35J05 #53B20 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1907.08085
It is known that the L2-norms of a harmonic function over spheres satisfies some convexity inequality strongly linked to the Almgren's frequency function. We examine the L2-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 L2-norms of harmonic functions over these hypersurfaces. To illustrate our result, we consider ellipses with constant eccentricity and growing tori in 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.