EQUATIONS OF ELASTIC HYPERSURFACES

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4 SHELLS The form is correctly defined because the differential operator ∂ V is tangential and can be applied to a function f defined on the surface S only. Then, for a given f, there exists a vector field ∇ S f ∈ TS such that ω f (V ) := 〈∇ S f, V 〉 for all V ∈ V (X) , (1.3) which is, classical differential geometry, the definition of surface Gradient of a function f ∈ C 1 (S ) and maps ∇ S : C ∞ (S ) → TS . (1.4) The surface divergence Div S : V (S ) → C ∞ (S ) (1.5) of a smooth tangential vector field is, by the definition, the dual operator with the opposite sign 〈Div S V, f〉 := −〈V, ∇ S f〉 , ∀ V ∈ TS , ∀ f ∈ C 1 (S ) . (1.6) Expressing these operators in intrinsic parameters of the surface S (tangential vector fields, Metric tensor etc.) one obtains (cf. [Ca1, Ci4, Ku1, Ta2]): Div S U := [ det G S ] −1/2 n∑ j=1 { [det ] } 1/2U ∑ j ∂ j GS = n ∇ S f = ∑ j,k (gjk ∂ j f) ∂ k . j=1 U j ;j = Tr ∇ S U , (1.7) where U j ;k denotes the componentwise covariant derivative, while ∇ S U is the tensor field defined by the covariant derivative (∇ S U)V := ∂U S V (see § 2.7 and [Ta2, Ch. 2, § 3]). Their composition is the Laplace-Beltrami operator ∆ S f := Div S ∇ S f = [ ] −1/2 ∑n−1 det G S which is self-adjoint due to (1.5): j,k=1 ∂ j {g jk[ ] } 1/2∂k det G S f , f ∈ C 2 (S ) , (1.8) ∆ ∗ S = ( ∇ S Div S ) ∗ = ( DivS ) ∗ ( ∇S ) ∗ = ∇S Div S = ∆ S . (1.9) The intrinsic parameters enable generalization to arbitrary manifolds, not necessarily immersed in the Euclidean space R n . On the other hand it is sometimes more convenient to record these operators in cartesian coordinates. For example, it is easy to ascertain that the surface gradient is nothing but the collection of the weakly tangential Günter’s (covariant) derivatives ∇ S = D S := (D 1 , . . . , D n ) ⊤ , D j := ∂ j − ν j (X )∂ ν = ∂ dj , (1.10) where ∂ ν := ∑ n j=1 ν j∂ j denotes the normal derivative (cf. [Gu1], [KGBB1], [Du1]). Here, for each 1 ≤ j ≤ n, the first-order differential operator D j = ∂ dj is the directional derivative along the tangential vector d j := π S e j . The operator (the matrix) π S : R n → TS , π S (t) = I − ν(t)ν ⊤ (t) = [ δ jk − ν j (t)ν k (t) ] n×n , t ∈ S (1.11)
1. INTRODUCTION 5 defines the canonical orthogonal projection π 2 S = π S onto the tangential space to S at the point t ∈ S : (ν, π S v) = ∑ j ν j v j − ∑ j,k ν 2 j ν k v k = 0 for all v = (v 1 , . . . , v n ) ⊤ ∈ R n . and, as usual, e j = (δ jk ) 1≤k≤n ∈ R n , with δ jk denoting the Kronecker symbol. Moreover, the surface divergence coincides with the operator Div S U = n∑ D j U j , for U = j=1 n∑ U j ∂ j ∈ TS (1.12) and the Laplace-Beltrami operator with (see also [MM1, pp. 2ff and p. 8.]) ∆ S ϕ := Div S ∇ S ϕ = j=1 n∑ Dj 2 ϕ , ϕ ∈ C 2 (S ) . (1.13) j=1 Relatively simple form of recorded operators enables simplified treatment of corresponding boundary value problems, which require proofs of Korn’s inequalities or similar. Besides Laplace-Beltrami and Láme operators are develop representation of the Hodge- Laplacian, Bochner-Laplacian and Stoke’s, as well as their associated boundary value problems, on a hypersurface S in R n . Writing them in the cartesian coordinates in R n and demonstrate the investigation of classical boundary value problems for them. Alternatively, the Laplace-Beltrami operator (1.13) is the natural operator associated with the Euler-Lagrange equations for the variational integral E [u] = − 1 ∫ ‖Du‖ 2 dS. (1.14) 2 A similar approach, based on the principle that, at equilibrium, the displacement minimizes the potential energy (a Koiter’s model), leads to the following expression for the Lamé operator L S on S : S L S U = µπ S Div S ∇ S U + (λ + µ) ∇ S Div S U + µ H 0 S W S U , (1.15) (cf. (1.11) for the projection π S ) for arbitrary (tangential) vector fields U on S , which are tangential to S . Above, λ, µ ∈ R are the Lamé moduli, whereas H 0 S = −Div S ν := − n∑ D j ν j , W S = − [ D j ν k ]n×n . (1.16) j=1 Note, that HS 0 := (n − 1)−1 HS 0 and W S represent, respectively, the mean curvature (cf. (2.204)) and the Weingarten mapping (cf. (2.233)) of S . This identification ensures that the boundary-value problem ⎧ ⎪⎨ ⎪⎩ U = (U 1 , ..., U n ) ⊤ ∈ H s+1/2 (S ), R n ) , U · ν = 0 in S , µ π S ( DivS ∇ S U ) + (λ + µ) ∇ S Div S U + µ H 0 S W S U = 0 in S , U ∣ ∣ Γ = f ∈ H s (∂S , R n ) , f · ν = f · ν Γ = 0 on Γ := ∂S (1.17)
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2. OUTLINE OF DIFFERENTIAL GEOMETRY
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3. CALCULUS OF DIFFERENTIAL OPERATO
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4. BOUNDARY VALUE PROBLEMS: GENERAL
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5. BOUNDARY VALUE PROBLEMS: EXAMPLE
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BIBLIOGRAPHY 181 [CD1] O.Chkadua, R
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BIBLIOGRAPHY 183 [Hr1] L. Hörmande
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BIBLIOGRAPHY 185 [Si1] I. Simonenko
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EQUATIONS OF ELASTIC HYPERSURFACES

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