Small parameters in the theory of isometric imbeddings of two-dimensional Riemannian manifolds in Euclidean spaces

Poznyak, È. G.; Shikin, E. V.

doi:10.1090/trans2/176/06

Cited by 11 publications

(14 citation statements)

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“…Substituting the asymptotic boundary layer profile (23) in the leading order expressions for δw S /t 2 and δw B , we get δw S /t 2 = (h 11 + νh 22 ) 2 24(1 − ν) e −2λX 1 cos λX 1 − sin λX 1 2 δw B = (h 11 + νh 22 ) 2 24(1 − ν) e −2λX 1 cos λX 1 + sin λX 1 × cos λX 1 + sin λX 1 − 2 e λX 1 .…”

Section: Boundary Layers In Very Thin Platesmentioning

confidence: 99%

Buckling transition and boundary layer in non-Euclidean plates

2009

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Non-Euclidean plates are thin elastic bodies having no stress-free configuration, hence exhibiting residual stresses in the absence of external constraints. These bodies are endowed with a threedimensional reference metric, which may not necessarily be immersible in physical space. Here, based on a recently developed theory for such bodies, we characterize the transition from flat to buckled equilibrium configurations at a critical value of the plate thickness. Depending of the reference metric, the buckling transition may be either continuous or discontinuous. In the infinitely thin plate limit, under the assumption that a limiting configuration exists, we show that the limit is a configuration that minimizes the bending content, amongst all configurations with zero stretching content (isometric immersions of the mid-surface). For small but finite plate thickness we show the formation of a boundary layer, whose size scales with the square root of the plate thickness, and whose shape is determined by a balance between stretching and bending energies.

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Section: Boundary Layers In Very Thin Platesmentioning

confidence: 99%

Buckling transition and boundary layer in non-Euclidean plates

2009

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“…The classical surface theory indicates that for the given metric, the isometric embedding or immersion can be realized if the first fundamental form and the second fundamental form satisfy the Gauss-Codazzi equations (cf. [5,35,36,44]). There have been many results for the isometric embedding of surfaces with positive Gauss curvature, which can be studied by solving an elliptic problem of the Darboux equation or the Gauss-Codazzi equations; see [27] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Christoforou in [11] obtained the small BV solution to the Gauss-Codazzi system with the same catenoid type metric as in [8]. See [18,19,20,28,31,44,46,50] for other related results on surface embeddings. For the higher dimensional isometric embeddings we refer the readers to [2,3,4,9,10,22,25,38,39,40,43] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Isometric Immersions of Surfaces with Two Classes of Metrics and Negative Gauss Curvature

Cao¹,

Huang

Wang

2015

Arch Rational Mech Anal

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The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with large data in L ∞ are obtained through the vanishing viscosity method and the compensated compactness framework. The L ∞ uniform estimate and H −1 compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type. The global weak solutions in L ∞ to the Gauss-Codazzi equations yield the C 1,1 isometric immersions of surfaces with the given metrics.

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“…Yau [39]; also see [20,33,35]). Important results have been achieved for the embedding of surfaces with positive Gauss curvature which can be formulated as an elliptic boundary value problem (cf.…”

Section: Introductionmentioning

confidence: 99%

Isometric Immersions and Compensated Compactness

Slemrod

Wang

2009

Commun. Math. Phys.

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Abstract. A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M 2 which can be realized as isometric immersions into R 3 . This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptichyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in R 3 . The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in R 3 . As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a C 1,1 isometric immersion of the two-dimensional manifold in R 3 satisfying our prescribed initial conditions. To achieve this, we introduce a vanishing viscosity method depending on the features of initial value problems for isometric immersions and present a technique to make the apriori estimates including the L ∞ control and H −1 -compactness for the viscous approximate solutions. This yields the weak convergence of the vanishing viscosity approximate solutions and the weak continuity of the Gauss-Codazzi system for the approximate solutions, hence the existence of an isometric immersion of the manifold into R 3 satisfying our initial conditions.

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Small parameters in the theory of isometric imbeddings of two-dimensional Riemannian manifolds in Euclidean spaces

Cited by 11 publications

References 0 publications

Buckling transition and boundary layer in non-Euclidean plates

Buckling transition and boundary layer in non-Euclidean plates

Isometric Immersions of Surfaces with Two Classes of Metrics and Negative Gauss Curvature

Isometric Immersions and Compensated Compactness

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