The Analysis and Geometry of Hardy's Inequality by Alexander A. Balinsky W. Desmond Evans & Roger T. Lewis

Author:Alexander A. Balinsky, W. Desmond Evans & Roger T. Lewis
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(3.8.5)

For V defined in (3.8.3), consider the terms appearing in in which is the inward normal on ,

and

We therefore have that

A similar calculation shows that

and we conclude from (3.8.3) that

Then, using this fact in (3.8.5) and applying (3.8.2) (with replaced by α), we obtain

(3.8.6)

Observe that in this case δ 1 =  | x | −ρ, δ 2 = R − | x | ,

and ∇δ 1 = −∇δ 2 = x∕ | x | , implying that

On choosing α = 2 and allowing in (3.8.6), we establish the first inequality (3.8.4). The last inequality follows since

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