EQUATIONS OF ELASTIC HYPERSURFACES

34 SHELLS
ii. The integration by parts
∫
∮
∂ j f(y)g(y) dy =
Ω
S
∫
ν j (τ)f(τ)g(τ) dS − f(y)∂ j g(y) dy (2.121)
Ω
holds for arbitrary f, g ∈ W 2 2(S ).
Proof: Formula (2.119) is a direct consequences of (2.120):
∫
Div F (y) dy = ∑ ∫
∂ j f j (y)dy = ∑ ∮
∮
ν j (τ), f j (τ) dS =
Ω j Ω j S
S
〈ν(τ), F (τ)〉 dS .
Since f, g ∈ W 2 2(S ) implies fg ∈ W 2 1(S ), we can apply (2.120) to the Leibnitz
equality ∂ j [ψ(y)ϕ(y)] = ϕ(y)∂ j ψ(y) + ψ(y)∂ j ϕ(y) and get easily (2.121).
Let us consider the normal derivative
∂ ν ϕ := ν · ∇ϕ =
n∑
ν j ∂ j ϕ , ϕ ∈ C 1 (Ω) . (2.122)
j=1
Corollary 2.33 (Green’s formula). Let Ω ⊂ R n be a domain with Lipschitz boundary.
For the Laplace operator
∆ := ∂ 2 1 + · · · + ∂ 2 n (2.123)
and functions ϕ, ψ ∈ W 1 2(Ω) the following I and II Green formulae are valid:
∫
∫
Ω
Ω
∮
(∆ψ)(y)ϕ(y)dy =
∫
(∆ψ)(y)ϕ(y)dy =
∂Ω
Ω
(∂ ν ψ)(τ)ϕ(τ) dS −
n∑
∫
j=1
Ω
(∂ j ψ)(y)(∂ j ϕ)(y)dy (2.124)
ψ(y)(∆ϕ)(y)dy
∮
[
+ (∂ν ψ)(τ)ϕ(τ) + ψ(τ)(∂ ν ϕ)(τ) ] dS (2.125)
∂Ω
Proof: Let, for time being, ϕ, ψ ∈ C 2 (Ω). By applying (2.121) we prove I Green formulae
in (2.124).
By writing a similar formula
∫
(ψ)(y)∆ϕ(y)dy
Ω
∮
=
∂Ω
(∂ ν ψ)(τ)ϕ(τ) dS S −
n∑
∫
j=1
Ω
(∂ j ψ)(y)(∂ j ϕ)(y)dy (2.126)
and taking the difference with (2.124), we prove II Green formulae in (2.125).