It is known that the $L^{ 2}$-norms of a harmonic function over spheres satisfies 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.
Categories: math.AP math.DG