By Philippe G. Ciarlet

curvilinear coordinates. This therapy contains specifically an instantaneous facts of the three-d Korn inequality in curvilinear coordinates. The fourth and final bankruptcy, which seriously is determined by bankruptcy 2, starts by means of a close description of the nonlinear and linear equations proposed by way of W.T. Koiter for modeling skinny elastic shells. those equations are “two-dimensional”, within the feel that they're expressed by way of curvilinear coordinates used for de?ning the center floor of the shell. The life, specialty, and regularity of suggestions to the linear Koiter equations is then verified, thank you this time to a primary “Korn inequality on a floor” and to an “in?nit- imal inflexible displacement lemma on a surface”. This bankruptcy additionally features a short advent to different two-dimensional shell equations. apparently, notions that pertain to di?erential geometry consistent with se,suchas covariant derivatives of tensor ?elds, also are brought in Chapters three and four, the place they seem so much obviously within the derivation of the elemental boundary worth difficulties of three-d elasticity and shell conception. sometimes, parts of the cloth coated listed below are tailored from - cerpts from my publication “Mathematical Elasticity, quantity III: conception of Shells”, released in 2000by North-Holland, Amsterdam; during this recognize, i'm indebted to Arjen Sevenster for his sort permission to depend upon such excerpts. Oth- clever, the majority of this paintings was once considerably supported through can provide from the study can provide Council of Hong Kong specified Administrative sector, China [Project No. 9040869, CityU 100803 and venture No. 9040966, CityU 100604].

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E. in Ω, together imply that ∇Φ(x) ∈ O3+ for almost all x ∈ U . By (i), there thus exist c ∈ E3 and Q ∈ O3+ such that Φ(x) = Θ(x) = c + Q ox for almost all x = Θ(x) ∈ U , or equivalently, such that Ξ(x) := ∇Θ(x)∇Θ(x)−1 = Q for almost all x ∈ U. Since the point x0 ∈ Ω is arbitrary, this relation shows that Ξ ∈ L1loc (Ω). By a classical result from distribution theory (cf. 6]), Sect. 8] Uniqueness of immersions with the same metric tensor 37 we conclude from the assumed connectedness of Ω that Ξ(x) = Q for almost all x ∈ Ω, and consequently that Θ(x) = c + QΘ(x) for almost all x ∈ Ω.

Let Ω be a connected and simply-connected open subset of n n ) ∈ C 2 (Ω; S3> ), n ≥ 0, be matrix ﬁelds satisfying Rqijk = 0 in R3 . Let Cn = (gij Ω, n ≥ 0, such that lim n→∞ Cn − I 2,K = 0 for all K Ω. Then there exist immersions Θn ∈ C 3 (Ω; E3 ) satisfying (∇Θn )T ∇Θn = Cn in Ω, n ≥ 0, such that lim n→∞ Θn − id 3,K = 0 for all K Ω where id denotes the identity mapping of the set Ω, the space R3 being identiﬁed here with E3 (in other words, id(x) = x for all x ∈ Ω). Three-dimensional diﬀerential geometry 40 [Ch.

7-1), of which Liouville’s theorem is the special case corresponding to Θ(x) = x for all x ∈ Ω. (3) The result established in part (i) of the above proof asserts that, given a connected open subset Ω of R3 , if a mapping Θ ∈ H 1 (Ω; E3 ) is such that ∇Θ(x) ∈ O3+ for almost all x ∈ Ω, then there exist c ∈ E3 and Q ∈ O3+ such that Θ(x) = c + Qox for almost all x ∈ Ω. This result thus constitutes a generalization of Liouville’s theorem. 7-3), an assumption about the sign of det ∇Θ becomes necessary.