Approximation of Flat W 2,2 Isometric Immersions by Smooth Ones

Hornung, Peter

doi:10.1007/s00205-010-0374-y

Cited by 57 publications

(102 citation statements)

References 13 publications

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“…Then . z ; z / satisfy the hypotheses of proposition 2.5 in [13]. The claim follows from conclusion (ii) of that proposition.…”

Section: Relations Between Framed Curves and Surfacesmentioning

confidence: 53%

“…In fact, the set I 0 can be open and dense. An example is given in the appendix to [13]. There is strong numerical evidence (cf.…”

Section: The Euler-lagrange Systemmentioning

confidence: 94%

“…Moreover, for all x; y 2 S n C ru , the alternative PROOF: We only sketch the proof; details can be found in [13]. There exists r 2 .0;…”

Section: Relations Between Framed Curves and Surfacesmentioning

confidence: 96%

“…Hence s˙2 C 0 .OE T; T / by Lemma 2.1. Moreover, Lemma 2.2 implies that satisfies (1.12) (i.e., is admissible in the sense of [12,13]), and 0 .a/ 0 .t/ To prove part (ii), notice that by smoothness there is c > 0 such that 1 s˙ c on OE T; T . Let z be any (arc length parametrized) smooth extension of to all of R. Using the fact that is C 1 in a neighborhood of ..˙T /;˙N.˙T // by Lemma 2.1, we conclude that there is…”

Section: Relations Between Framed Curves and Surfacesmentioning

confidence: 97%

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Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Hornung

2010

Comm Pure Appl Math

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Let S R 2 be a bounded domain with boundary of class C 1 , and let g ij D ı ij denote the flat metric on R 2 . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of @S) of all W 2;2 isometric immersions of the Riemannian manifold .S; g/ into R 3 . In this article we derive the Euler-Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C 3 away from a certain singular set † and C 1 away from a larger singular set † [ † 0 . We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near † 0 .Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates.

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“…Then . z ; z / satisfy the hypotheses of proposition 2.5 in [13]. The claim follows from conclusion (ii) of that proposition.…”

Section: Relations Between Framed Curves and Surfacesmentioning

confidence: 53%

“…In fact, the set I 0 can be open and dense. An example is given in the appendix to [13]. There is strong numerical evidence (cf.…”

Section: The Euler-lagrange Systemmentioning

confidence: 94%

“…Moreover, for all x; y 2 S n C ru , the alternative PROOF: We only sketch the proof; details can be found in [13]. There exists r 2 .0;…”

Section: Relations Between Framed Curves and Surfacesmentioning

confidence: 96%

Section: Relations Between Framed Curves and Surfacesmentioning

confidence: 97%

See 2 more Smart Citations

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Hornung

2010

Comm Pure Appl Math

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“…There has recently been a number of works devoted to this topic, see [12,13,16,20]. The results of Sect.…”

Section: The Structure Of Isometric Immersionsmentioning

confidence: 98%

Bending of thin periodic plates

Cherdantsev

Cherednichenko

2015

Calc. Var.

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We show that nonlinearly elastic plates of thickness h → 0 with an ε-periodic structure such that ε −2 h → 0 exhibit non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional elasticity: in general, their effective stored-energy density is "discontinuously anisotropic" in all directions. The proof relies on a new result concerning an additional isometric constraint that deformation fields must satisfy on the microscale. Mathematics Subject Classification

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Bilayer Plates: Model Reduction, Γ‐Convergent Finite Element Approximation, and Discrete Gradient Flow

Bartels

Bonito

Nochetto

2015

Comm Pure Appl Math

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The bending of bilayer plates is a mechanism that allows for large deformations via small externally induced lattice mismatches of the underlying materials. Its mathematical modeling, discussed herein, consists of a nonlinear fourthorder problem with a pointwise isometry constraint. A discretization based on Kirchhoff quadrilaterals is devised and its -convergence is proved. An iterative method that decreases the energy is proposed, and its convergence to stationary configurations is investigated. Its performance, as well as reduced model capabilities, are explored via several insightful numerical experiments involving large (geometrically nonlinear) deformations.

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Approximation of Flat W 2,2 Isometric Immersions by Smooth Ones

Cited by 57 publications

References 13 publications

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Bending of thin periodic plates

Bilayer Plates: Model Reduction, Γ‐Convergent Finite Element Approximation, and Discrete Gradient Flow

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