EQUATIONS OF ELASTIC HYPERSURFACES

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154 SHELLS Let ϕ be a non-trivial solution of (5.37). Then, due to (5.38), the function u := V Γ ϕ is a solution to the homogeneous equation ∆ C u(t) = 0 in both open complemented surfaces t ∈ C and in t ∈ C c . Moreover, u −+ (s) = V −1 ϕ(s) = 0 on the boundary Γ. Therefore, u(t) is a non-trivial solution to the homogeneous Dirichlet BVPs (5.16) and a similar one for the complemented surface C c with the vanishing data f(t) ≡ 0 and g(s) ≡ 0. Due to Theorem 5.7 u(t) = V Γ ϕ 0 (t) ≡ 0 in C and in C c . Then due to (5.31) and the proof is completed. Theorem 5.11 The operator is self adjoint and positive definite for a positive constant M > 0. ϕ(s) = (D νΓ V Γ ϕ) − (s) − (D νΓ V Γ ϕ) + (s) ≡ 0 −V −1 : H −1/2 (Γ) → H 1/2 (Γ) (5.39) −(V −1 w, w) Γ ≥ M‖w ∣ ∣ H −1/2 (Γ)‖ 2 (5.40) Let X θ p(C ) be either the space H θ p(C ) or W θ p(C ). Then the operator is invertible for arbitrary θ ∈ R and 1 < p < ∞. V −1 : X θ p(Γ) −→ X θ+1 p (Γ) , (5.41) Proof: Let us insert ϕ = ψ := V Γ w, w ∈ H −1/2 (S ) into the first Green formula (5.19a) for the surface S and into a similar one (∆ C (t, D)ϕ, ψ) C c + (∇ C ϕ, ∇ C ψ) C c = (D νΓ ϕ − , ψ − ) Γ (5.42) for the complemented surface; we obtain: (∆ C V Γ w, V Γ w) C + (∇ C V Γ w, ∇ C V Γ w) C = + − ( ( D νΓ V Γ w ) − + , ( V Γ w ) − + ) Γ . (5.43) Inserting in the obtained equality ∆ C V Γ ϕ(x) = 0, t ∈ C ∪ C c (see (5.38)) and applying the Plemelji formulae (5.31) we find ( + − 1 ) ∫ 2 w + W∗ 0w, V −1 w = − + ∣ ∇C − + V w(y)∣ ∣ 2 dy , (5.44) C −+ Γ where, for conciseness, we use C + := C and C − := C c Taking the difference of two formulae in (5.44) we get: ∫ − (V −1 w, w) Γ = − (w, V −1 w) Γ = ∣ ∇S V S w(y) ∣ 2 dy ≥ 0 , S = C − ∪ C + .(5.45) S Due to Lemma 5.10 V −1 w = 0 implies w = 0, i.e., the operator has a trivial kernel Ker V −1 = {0}. On the other hand, V −1 is elliptic (cf. (5.32)) and since Γ is compact, C
5. BOUNDARY VALUE PROBLEMS: EXAMPLES 155 the operator is Fredholm (cf. e.g., [Ag1]). The spectral set of −V −1 is non-negative (cf. (5.45)). Thus, we are under the scope of Corollary 2.61.II and −V −1 in (5.39) is strictly positive, self adjoint and invertible. From Theorem 4.25 follows the invertibility of V −1 in (5.41) for all −∞ < θ < ∞ and 1 < p < ∞. There remains to only check the positive definiteness (5.40). Rewrite (5.45) in the form − (V −1 w, w) Γ = ∥ ∥ ∇S V S w ∣ ∣ L2 (S ) ∥ ∥ 2 = ∥ ∥ VS w ∣ ∣ H 1 (S ) ∥ ∥ 2 − ∥ ∥ VS w ∣ ∣ L2 (S ) ∥ ∥ 2 ≥ ∥ ∥ VS w ∣ ∣ H 1 (S ) ∥ ∥ 2 − C 1 ∥ ∥VS w ∣ ∣ H 3/4 (S ) ∥ ∥ 2 ≥C 2 ∥ ∥V−1 w ∣ ∣ H 1/2 (Γ) ∥ ∥ 2 − C 2 ∥ ∥V−1 w ∣ ∣ H 1/4 (Γ) ∥ ∥ 2 ≥C 3 ∥ ∥w ∣ ∣H −1/2 (Γ) ∥ ∥ 2 − C 3 ∥ ∥w ∣ ∣H −3/4 (Γ) ∥ ∥ 2 (5.46) since the trace (V S w) −+ = V −1 w and we have applied the theorems on continuous embedding H 3/4 (S ) ⊂ L 2 (S ) (cf, Theorem 4.17) and Theorem on traces ∥ ∣ ψ − + H s−1/2 (Γ) ∥ ∥ ∣ ≤ C4∥ψ∣H s (C −+ ) ∥ (cf, Theorem 4.18). At he last step we have applied the double inequality C 5 ∥ ∥w ∣ ∣H θ−1 (Γ) ∥ ∥ ≤ ∥ ∥V−1 w ∣ ∣ H θ (Γ) ∥ ∥ ≤ C6 ∥ ∥w ∣ ∣H θ−1 (Γ) ∥ ∥ which is nothing but the invertibility of V −1 in (5.41) for p = 2. The embedding E −1 −3/4 1 : H−1 (Γ) → H −3/4 (Γ) is compact (cf. Theorem 4.17) and inequality (5.46) is interpreted as the coerciveness − (V −1 w, w) Γ ≥ C 3 ∥ ∥w ∣ ∣H −1/2 (Γ) ∥ ∥ 2 − ∥ ∥ T w ∣ ∣H −1 (Γ) ∥ ∥ 2 (5.47) where T := C 3 E −1 −3/4 is compact. Since V −1 has the trivial kernel Ker V −1 = {0}, due to Theorem 2.66 the coerciveness (5.47) implies the positive definiteness (5.40). Theorem 5.12 The operator −W +1 : H 1/2 (Γ) \ { const } → H −1/2 (Γ) (5.48) is self adjoint and its kernel is 1-dimensional, consists of constant functions only Ker W +1 = { const } . This operator is non-negative −(W +1 w, w) Γ ≥ M‖w ∣ ∣ H 1/2 (Γ)‖ 2 ∀ w ∈ H 1/2 (Γ) \ { const } (5.49) for some M > 0. The operator (cf. Corollary 5.8 for the spaces) W −1 : X θ p(Γ) \ { const } −→ X θ−1 p (Γ) , (5.50) is invertible for arbitrary θ ∈ R and 1 < p < ∞.
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EQUATIONS OF ELASTIC HYPERSURFACES

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