EQUATIONS OF ELASTIC HYPERSURFACES

5. BOUNDARY VALUE PROBLEMS: EXAMPLES 153
Proof: See Theorem 4.36, (4.180) for the mapping properties and (4.183a), (4.183b) for the
Plemelji formulae.
The concluding assertion about ellipticity also is standard and we quote [CW1, Du2, Fi1,
KGBB1, Le1, Ma1] etc. for the proofs in case of domains in R n .
Lemma 5.9 (The direct potential method). The boundary pseudodifferential equation
(V −1 ψ)(s) = (N C (s, D)f)(s) − 1 2 g + (W 0(s, D)g)(s)
= (D νΓ N C (s, D)f)(s) + (M ∗ −(s, D)g)(s) , (5.33)
M − (s, D)u(s):=− 1 2 u(s) + W 0(s, D)u(s) , s ∈ Γ , (5.34)
where ψ(s) := (D νΓ ϕ) + (s) is the unknown function, is equivalent to the Dirichlet BVP (5.16)
via the representation formula (5.26), in which ϕ + = g and (D νΓ ϕ) + = ψ is the solution of
(5.33).
The boundary pseudodifferential equation
(W +1 (s, D)ψ)(s) = −(D νΓ N C (s, D)f)(s) + 1 2 h + (W∗ 0(s, D)h)(s)
= −(D νΓ N C (s, D)f)(s) + (M ∗ +(s, D)h)(s) , (5.35)
M + (s, D)u(s):= 1 2 u(s) + W 0(s, D)u(s) , s ∈ Γ , (5.36)
where ψ(s) := ϕ + (s) is the unknown function, is equivalent to the Neumann BVP (5.17) via
the representation formula (5.26), in which (D νΓ ϕ) + = h and ϕ + = ψ is the solution of
(5.35).
Proof (cf. Theorem 4.42): By inserting the known data ϕ + = g (for the Dirichlet BVP)
or (D νΓ ϕ) + = h (for the Neumann BVP) into the representation formulae (5.26) and by
applying the appropriate Plemelji formulae (5.31) and get the equivalent boundary pseudodifferential
equations (5.33) and (5.35), respectively.
The next Lemma is only auxiliary one. The result is sharpened in Theorem 5.11.
Lemma 5.10 The homogeneous equation
V −1 (s, D)ϕ(s) = 0 , ϕ ∈ H θ p(Γ) , s ∈ Γ (5.37)
has only a trivial solution ϕ = 0 for arbitrary s > 0 and 1 < p < ∞.
Proof (cf. Lemma 4.44: From (5.27) and (5.25) follows that
∮
∮
∆ C V Γ ψ(t)= ∆ C K ∆ (t, t − s)ψ(s) ds = δ(t − s)ψ(s) ds = 0 ∀ t ∉ Γ, (5.38)
Γ
Γ
since δ(t − s) = 0 for all t ∉ Γ and all s ∈ Γ.