Surface integrals and harmonic functions
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Using the notion of inferior mean due to M. Heins, we establish two inequalities for such a mean relative to a positive harmonic function defined on the open unit ball or halfspace in Rn+1 . 1. Introduction In connection with EP spaces, M. Heins proved the following PL-Lemma (unpublished). Lemma 1.1 (PL-Lemma). If u is a positive function on the annulus {R < |z| < 1} with a subharmonic logarithm, and γ are rectifiable Jordan curves in {r < |z| < 1} separating 0 from ∞, then
lim inf r →1 γ
γ
u(z)|dz| = lim
r →1 |z|=r
u(z)|dz|.
(1.1)
Wu showed in [4] that for a positive harmonic function in the unit disc, one has in most cases inequality, while equality occurs for functions whose boundary measures are absolutely continuous. She also showed that there exists a nonzero lower bound of the lim inf for this class of functions in the disc. The bound is achieved for functions whose boundary measures, for example, are purely singular. We generalize these results to higher dimensions. Let Ω be the open unit ball or upper half-space in Rn+1 and let S denote its boundary. Let u be a positive harmonic function on Ω, which, by Riesz’s theorem, is given by a Borel measure µ with the total measure µ on S. Definition 1.2. Let Γ be a piecewise C 1 -smooth hypersurface in Aδ = {q ∈ Ω : d(q,S) < δ } separating the two boundaries of Aδ . The inferior mean of u is defined by
IM(u) = lim inf
δ →0 Γ⊂Aδ Γ
u(q)dΓ.
(1.2)
Let ωn be the volume of the unit sphere in Rn+1 , and let Mn = ωn+1 /πωn . In this paper, we establish the following theorem. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:4 (2005) 443–448 DOI: 10.1155/JIA.2005.443
444
Surface integrals and harmonic functions
Theorem 1.3. For any positive harmonic function u on Ω with boundary measure µ, there exists the following inequality: IM(u) ≤ µ.
(1.3)
Equality occurs for those u whose boundary measures µ are absolutely continuous, when the inferior mean is attained along boundaries of Aδ not equal to S as δ → 0. Theorem 1.4. For any positive harmonic function u on Ω, with boundary measure µ, there exists the following inequality: IM(u) ≥ Mn µ.
(1.4)
Equality occurs for u with point-mass boundary measures µ concentrated at p0 , when IM(u) is attained along the boundary of the set
˜ = q ∈ Ω : d(q,S) < σ 2 \ q ∈ Ω : q − p0 < σ Ω
as σ → 0.
(1.5)
The proofs rely on Sard’s theorem (see [3]), and inequality (2.5) obtained below. 2. A surface measure lemma Given spherical angles φi ∈ [0,π], i < n, φn ∈ [0,2π], we include φ0 = π/2 and φn+1 = 0. For a point q ∈ Rn+1 , the relation between its Cartesian (x1 ,...,xn+1 ) and spherical (r,φ1 ,...,φn ) coordinates is given by x j = X j cosφ j ,
Xj = r
j −1
sinφi , r = |q|.
(2.1)
i=0
From [2, Section 676], we know that on a sphere r = const, the Jacobian of this relation satisfies
In =
n D x1 ,...,xn+1 = Xn In−1 = · · · = Xk . D r,φ1 ,...,φn k=1
(2.2)
If Sn is the unit sphere in Rn+1 and dSn is its volume element, then the vol
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