Minimal Surfaces

Dierkes, Ulrich; Hildebrandt, Stefan; Sauvigny, Friedrich

doi:10.1007/978-3-642-11698-8

Cited by 90 publications

(106 citation statements)

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“…The regularity up to the boundary for the first case is classic, and is essentially the one for the Plateau problem, for which we refer to [24,23]. We now study the second possibility.…”

Section: Morrey-type Estimates and Conclusionmentioning

confidence: 99%

A Resolution of the Poisson Problem for Elastic Plates

Lio

Palmurella

Rivière

2020

Arch Rational Mech Anal

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The Poisson problem consists in finding an immersed surface Σ ⊂ R m minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area. This problem represents a non-linear model for the equilibrium state of thin, clamped elastic plates originating from the work of S. Germain and S.D. Poisson in the early XIX century. We present a solution to this problem in the case of boundary data of class C 1,1 and when the boundary curve is simple and closed. The minimum is realised by an immersed disk, possibly with a finite number of branch points in its interior, which is of class C 1,α up to the boundary for some 0 < α < 1, and whose Gauss map extends to a map of class C 0,α up to the boundary.

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“…The regularity up to the boundary for the first case is classic, and is essentially the one for the Plateau problem, for which we refer to [24,23]. We now study the second possibility.…”

Section: Morrey-type Estimates and Conclusionmentioning

confidence: 99%

A Resolution of the Poisson Problem for Elastic Plates

Lio

Palmurella

Rivière

2020

Arch Rational Mech Anal

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“…It is now easy to verify that for Q = R 3 with the Euclidean metric δ KL , the Euler-Lagrange equations (6.8) for graph of a surface ζ : R 2 ⊃ U → R 3 , ζ(x, y) = (x, y, u(x, y)), are equivalent to the well-known single second-order differential equation for an unknown function u : R 2 ⊃ U → R, (6.11) (1 + u 2 y )u xx − 2u x u y u xy + (1 + u 2 x )u yy = 0, cf. Dierkes, Hildebrandt, and Sauvigny [7].…”

Section: Example: Variational Functional For Minimal Submanifoldsmentioning

confidence: 99%

The fundamental Lepage form in variational theory for submanifolds

Urban

Brajerčík

2018

Int. J. Geom. Methods Mod. Phys.

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A setting for global variational geometry on Grassmann fibrations is presented. The integral variational functionals for finite dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert form in the classical mechanics. Prolongations of immersions, diffeomorphisms and vector fields to the Grassmann fibrations are introduced as geometric tools for the variations of immersions. The first infinitesimal variation formula together with its consequences, the Euler-Lagrange equations for extremal submanifolds and the Noether theorem for invariant variational functionals are proved. The theory is illustrated on the variational functional for minimal submanifolds.2000 Mathematics Subject Classification. 58E30; 58A20; 58D19; 53A10.

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“…The form of Thomsen surface can be obtained from [19] and [20]. Equation (1) shows the equation for Thomsen Surface.…”

Section: Generation Of Thomsen Surface Figurementioning

confidence: 99%

Form-Finding Of Thomsen Surface Using Nonlinear Analysis Method

Ibrahim

Hadi

Yee

2018

J. Phys.: Conf. Ser.

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Abstract. Tensioned fabric structure is the structure composed fabric surface as main material in tensioned fabric structure. Tensioned fabric structure with different surface form could be realized. Their variations as possible choice form of minimal surface for tensioned fabric structure have been studied. In this analysis, the form of Thomsen Surface has been used in this study. This study is to carry out form-finding computational analysis of Thomsen Surface for u=v=0.4 and u=v=1.0. The computational form-finding analysis used in this study is nonlinear analysis method. The surface of Thomsen Surface with variable u=v=0.4 and u=v=1.0 obtained in this study will provide an alternative shapes for designers to be considered for adoption in tensionel fabric structures.

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Minimal Surfaces

Cited by 90 publications

References 0 publications

A Resolution of the Poisson Problem for Elastic Plates

A Resolution of the Poisson Problem for Elastic Plates

The fundamental Lepage form in variational theory for submanifolds

Form-Finding Of Thomsen Surface Using Nonlinear Analysis Method

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