EQUATIONS OF ELASTIC HYPERSURFACES

5. BOUNDARY VALUE PROBLEMS: EXAMPLES 165
Proof: The Green formulae (5.95) is a direct consequence of the integration by parts (cf.
(2.228)) and of the Divergence Theorem, i.e. of (3.47) with P = Div C .
If U ∈ TC denotes the displacement, natural boundary conditions for L C , as easily
viewed from formulae (5.95), include prescribing the displacement (the Dirichlet type BVP)
{
(LC (t, D) U)(t) = F (t), t ∈ C ,
(5.97)
U + (s) = G(s),
s ∈ Γ
or the traction–trace of the stress (Neumann type BVP)
{
(LC (t, D) U)(t) = F (t), t ∈ C ,
(T C (ν Γ , D) U) + (s) = H(s), s ∈ Γ .
(5.98)
Further constraints on data in BVPs (5.97) and (5.98) are
F ∈ X −1 (C ) , G ∈ W 1/2 (Γ) , H ∈ W −1/2 (Γ) and look for U ∈ X 1 (C ) . (5.99)
Thee constraints are similar to those (5.20) for the Laplace-Beltrami operator.
Let us choose the function B ∈ C ∞ (R n ) to be supported in the complemented domain
supp B ⊂ C c := S \C and consider the fundamental solution K L (t, t−τ) to the perturbed
elasticity operator L S (t, D) − BI (see Theorem 5.22). Any solution of BVPs (5.97), (5.20)
and (5.98), (5.20) is represented by the formulae (cf. (5.26))
U(t) = (N C F )(t) + (W Γ U + )(t) − (V Γ (T C (ν Γ , D)U) + )(t) , t ∈ C , (5.100)
where the corresponding potential operators are defined as follows (cf. (5.27))
∮
(N C (t, D)ϕ)(t) := K L (t, t − τ)ϕ(τ) dS ,
∮
(W Γ (t, D)ϕ)(t) :=
C
[(T C (ν Γ (s), D s )K L )(t, s − t)] ⊤ ϕ(s) ds , (5.101)
Γ
∮
(V Γ (t, D)ϕ)(t) :=
K L (t, t − s)ϕ(s) ds , t ∈ C .
The direct values of potential operators
Γ
V 0 (t, D) := V Γ (t, D) , B 1 (t, D)V 0 (t, D) := −T C (ν Γ , D)V Γ (t, D) ,
V 1 (t, D) := −W Γ (t, D) , V 1 (t, D)B 1 (t, D) := T C (ν Γ , D)W Γ (t, D) ,
(5.102)
where B 0 (t, D) = I and B 1 (t, D) = −T C (ν Γ , D), we denote by
V −1 (s, D) := V 00 (t, D) , W ∗ 0(t, D) := −V 10 (t, D) ,
W 0 (t, D) := −V 01 (t, D) , W +1 (s, D) := V 11 (t, D) ,
(5.103)