EQUATIONS OF ELASTIC HYPERSURFACES

4. BOUNDARY VALUE PROBLEMS: GENERAL RESULTS 133
is bounded. Consider the adjoint operator with respect to the sesquilinear form
µ−1
∑
(Aϕ, ψ) = (Aϕ, ψ) C − ((B j ϕ) + , (C j ψ) + ) Γ = (ϕ, A ∗ ψ) , (4.143)
j=0
(cf. I Green formula (4.24)) which maps
j=0
ϕ, ψ ∈ C ∞ (C , C N ) ,
µ−1
∏
A ∗ : H µ p
(C ) × W 1/p+m j−µ
′ p
(Γ) −→ H −µ
′ p
(C ) . (4.144)
′
Definition 4.33 We say that the BVP (4.1) is fredholm (equivalently: the mapping (4.142) is
Fredholm) if:
i. the homogeneous BVP with f = g 0 = · · · = g µ−1 only has a finite number of linearly
independent solutions in H µ p(C );
ii. the adjoint homogeneous equation U ∗ Ψ = 0 only has a finite number of linearly independent
solutions Ψ k := (ψ k , ψ0, k . . . , ψµ−1) k ⊤ , k = 1, . . . , l, in the space H µ p
(C ) ×
′
µ−1 ∏
W 1/p+m j−µ
p
(Γ);
′
j=0
iii. has a solution ϕ ∈ H µ p(C ) for those data f, g 0 , . . . , g µ−1 which are orthogonal
µ−1
∑
(f, ψ k ) C + (g j , ψj k ) Γ = 0 , k = 1, . . . , l , (4.145)
j=0
to all solutions { Ψ k
} l
k=1 to the adjoint homogeneous equation U∗ Ψ = 0.
It is well known that the mapping (4.142) is Fredholm if and only if the adjoint mapping
(4.144) is Fredholm.
Theorem 4.34 Let the ”basic” operator A(t, D) in (4.2) be properly elliptic of order m =
2µ and {B j } µ−1
j=0 be the Dirichlet system of boundary operators.
The BVP (4.1) is fredholm (equivalently: the mapping (4.142) is Fredholm) if and only
if the system of boundary operators {B j (s, D} µ−1
j=0 covers the operator A(s, D) (see Lemma
4.31 and Remark 4.32).
The BVP (4.1) has a solution ϕ ∈ W µ p(C ) only for those data { }
f, g 0 , . . . , g µ−1 ∈
p (C ) × µ−1 ∏
W µ−m j−1/p
p (Γ) which satisfy the orthogonality conditions (4.145).
H −µ
j=0
Proof: We will only describe principal steps of the proof, leaving the details to a qualified
reader (also see [CDS1] for the 2-dimensional case and BVPs with mixed boundary conditions).
We shall consider an equivalent formulation -the operator A in (4.142).