By Philippe G. Ciarlet

curvilinear coordinates. This remedy comprises specifically an instantaneous evidence of the third-dimensional Korn inequality in curvilinear coordinates. The fourth and final bankruptcy, which seriously depends on bankruptcy 2, starts off through an in depth description of the nonlinear and linear equations proposed through W.T. Koiter for modeling skinny elastic shells. those equations are “two-dimensional”, within the experience 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 strategies to the linear Koiter equations is then validated, 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 contains a short creation to different two-dimensional shell equations. curiously, notions that pertain to di?erential geometry in line with se,suchas covariant derivatives of tensor ?elds, also are brought in Chapters three and four, the place they seem so much clearly within the derivation of the elemental boundary worth difficulties of 3-dimensional elasticity and shell idea. sometimes, parts of the fabric coated listed below are tailored from - cerpts from my publication “Mathematical Elasticity, quantity III: idea of Shells”, released in 2000by North-Holland, Amsterdam; during this admire, i'm indebted to Arjen Sevenster for his type permission to depend on such excerpts. Oth- clever, the majority of this paintings was once considerably supported through delivers from the study provides Council of Hong Kong distinct Administrative area, China [Project No. 9040869, CityU 100803 and venture No. 9040966, CityU 100604].

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**Example text**

R∈O n We assert that the matrices Q deﬁned in this fashion satisfy limn→∞ Qn An = I. For otherwise, there would exist a subsequence (Qp )p≥0 of the sequence (Qn )n≥0 and δ > 0 such that |Qp Ap − I| = inf 3 |RAp − I| ≥ δ for all p ≥ 0. R∈O Since lim |Ap | = lim p→∞ ρ((Ap )T Ap ) = p→∞ ρ(I) = 1, the sequence (Ap )p≥0 is bounded. Therefore there exists a further subsequence (Aq )q≥0 that converges to a matrix S, which is orthogonal since ST S = lim (Aq )T Aq = I. q→∞ But then lim ST Aq = ST S = I, q→∞ which contradicts inf R∈O3 |RAq − I| ≥ δ for all q ≥ 0.

N→∞ Since the set O3 is compact, there exist matrices Qn ∈ O3 , n ≥ 0, such that |Qn An − I| = inf 3 |RAn − I|. R∈O n We assert that the matrices Q deﬁned in this fashion satisfy limn→∞ Qn An = I. For otherwise, there would exist a subsequence (Qp )p≥0 of the sequence (Qn )n≥0 and δ > 0 such that |Qp Ap − I| = inf 3 |RAp − I| ≥ δ for all p ≥ 0. R∈O Since lim |Ap | = lim p→∞ ρ((Ap )T Ap ) = p→∞ ρ(I) = 1, the sequence (Ap )p≥0 is bounded. Therefore there exists a further subsequence (Aq )q≥0 that converges to a matrix S, which is orthogonal since ST S = lim (Aq )T Aq = I.

Hence Cof ∇Θ(x) = (det ∇Θ(x))∇Θ(x)−T = ∇Θ(x)−T for almost all x ∈ Ω, on the one hand. Since, on the other hand, div Cof ∇Θ = 0 in (D (B))3 Three-dimensional diﬀerential geometry 36 [Ch. 1]), we conclude that ∆Θ = div Cof ∇Θ = 0 in (D (B))3 . Hence Θ = (Θj ) ∈ (C ∞ (Ω))3 . For such mappings, the identity ∆(∂i Θj ∂i Θj ) = 2∂i Θj ∂i (∆Θj ) + 2∂ik Θj ∂ik Θj , together with the relations ∆Θj = 0 and ∂i Θj ∂i Θj = 3 in Ω, shows that ∂ik Θj = 0 in Ω. The assumed connectedness of Ω then implies that there exist a vector c ∈ E3 and a matrix Q ∈ O3+ (by assumption, ∇Θ(x) ∈ O3+ for almost all x ∈ Ω) such that Θ(x) = c + Q ox for almost all x ∈ Ω.