Convexity Properties of Harmonic Functions on Parameterized Families of Hypersurfaces
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Convexity Properties of Harmonic Functions on Parameterized Families of Hypersurfaces Stine Marie Berge1 Received: 30 July 2019 © Mathematica Josephina, Inc. 2019
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 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. Keywords Harmonic functions · Almgren’s frequency function · Convexity Properties of Harmonic functions Mathematics Subject Classification 53B20 · 35J05 · 31B05
1 Introduction Since the paper by Almgren [2], the frequency function has been intensively used to study harmonic functions in Rn and, more generally, solutions to second-order elliptic equations. For a harmonic function h on Rn we let H (t) denote the L 2 -norm of h over the sphere of radius t. In [1], and later in [2], it was shown that the function H is geometrically convex, i.e., H r α s 1−α ≤ H (r )α H (s)1−α , 0 ≤ α ≤ 1, r , s > 0. (1.1)
B 1
Stine Marie Berge [email protected] Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
123
S. M. Berge
Inequality (1.1) is equivalent to the statement that the frequency function t H (t) , t >0 H (t)
N (t) =
is increasing. The notion of frequency function was generalized to solutions of elliptic operators of divergence form by Garafalo and Lin [8] and was shown to be almost increasing for t < t0 . They further used the result to show that the squares of solutions of the elliptic equations are Muckenhoupt weights on the ball B R with radius R > 0. In the paper of Mangoubi [13], a more explicit convexity result on Riemannian manifolds was obtained by using comparison geometry. Using this result and extending eigenfunctions to harmonic functions, Mangoubi gave a new proof that a solution u to div (grad u) = −k 2 u satisfies max |u| ≤ C1 eC2 r k
max |u|
Br ( p)
B2r ( p)
α
1−α max |u|
Br /2 ( p)
.
(1.2)
In (1.2) the positive constants C1 , C2 and 0 < α < 1 only depend on the dimension and curvature of the Riemannian manifold. Inequality (1.2) was first shown by Donnelly and Fefferman [5]. The main aim of this work is to study the L 2 -norm of harmonic functions over families of surfaces, generalizing the geometric convexity inequality (1.1). Let h be a harmonic function on a domain in a Riemannian manifold (M, g) and fix a point p ∈ M. Consider for R > 0 a
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