Representation of Multivariate Functions via the Potential Theory and Applications to Inequalities
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Research Article Representation of Multivariate Functions via the Potential Theory and Applications to Inequalities Florica C. Cˆırstea1 and Sever S. Dragomir2 1 2
Department of Mathematics, Australian National University, Canberra, ACT 0200, Australia School of Computer Science and Mathematics, Victoria University, P.O. Box 14428, Melbourne City, Victoria 8001, Australia
Correspondence should be addressed to Sever S. Dragomir, [email protected] Received 12 February 2007; Revised 2 August 2007; Accepted 9 November 2007 Recommended by Siegfried Carl We use the potential theory to give integral representations of functions in the Sobolev spaces W 1,p Ω, where p ≥ 1 and Ω is a smooth bounded domain in RN N ≥ 2. As a byproduct, we obtain sharp inequalities of Ostrowski type. Copyright q 2008 F. C. Cˆırstea and S. S. Dragomir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and main results Let N ≥ 2 and let ·, · denote the canonical inner product on RN × RN . If ωN stands for the area of the surface of the N − 1-dimensional sphere, then ωN 2π N/2 /ΓN/2, where Γ ∞ −tunit s−1 is the gamma function defined by Γs 0 e t dt for s > 0 see 1, Proposition 0.7. Let E denote the normalized fundamental solution of Laplace equation: ⎧ 1 ⎪ ⎪ x / 0 if N 2, ⎪ ⎨ 2π ln |x|, Ex ⎪ 1 ⎪ ⎪ , x / 0 if N ≥ 3. ⎩ 2 − NωN |x|N−2
1.1
Unless otherwise stated, we assume throughout that Ω ⊂ RN is a bounded domain with C2 boundary ∂Ω. Let ν denote the unit outward normal to ∂Ω and let dσ indicate the N−1-dimensional area element in ∂Ω. The Green-Riemann formula says that any function
2
Journal of Inequalities and Applications
f ∈ C2 Ω ∩ C1 Ω satisfying Δf ∈ CΩ can be represented in Ω as follows see 2, Section 2.4:
∂f ∂E fy xEx − y dσx
fx x − y − Ex − yΔfxdx, ∂ν ∂ν ∂Ω Ω
∀y ∈ Ω, 1.2
where ∂f/∂νx is the normal derivative of f at x ∈ ∂Ω. In particular, if f ∈ C0∞ Ω the set of functions in C∞ Ω with compact support in Ω, then 1.2 leads to the representation formula fy
Ω
Ex − yΔfxdx,
∀y ∈ Ω.
1.3
For a continuous function h on ∂Ω, the double-layer potential with moment h is defined by uh y
hx ∂Ω
∂E x − y dσx. ∂ν
1.4
Expression 1.4 may be interpreted as the potential produced by dipoles located on ∂Ω; the direction of which at any point x ∈ ∂Ω coincides with that of the exterior normal ν, while its intensity is equal to hx. The double-layer potential is well defined in RN and it satisfies the Laplace equation Δu 0 in RN \ ∂Ω see Proposition 2.8. For other properties of the double-layer potential, see Lemma 2.9 and Proposition 2.10. The double-layer potential plays an important role in solving boundary value problems of elliptic equations. The representation of the solution of the interior/exterior Dirichlet problem for Laplace’s equ
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