Criteri per l'esistenza della soluzione in problemi di Calcolo delle variazioni

Tonelli, Leonida

doi:10.1007/bf02420224

Cited by 5 publications

(8 citation statements)

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“…One can also rescue the situation with the additional requirement that the shapes of the triangles stay bounded (so that their normals approach that of the smooth surface), but we will not explore this here. (See also [Ton21]. In this volume, the companion article [Sul08] treats curvatures of smooth and discrete surfaces, and [War08] considers convergence issues.…”

Section: Sketch Of Proofmentioning

confidence: 99%

Curves of Finite Total Curvature

Sullivan

2008

Discrete Differential Geometry

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Abstract. We consider the class of curves of finite total curvature, as introduced by Milnor. This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us connections between discrete and differential geometry. To explore these ideas, we consider theorems of Fáry/Milnor, Schur, Chakerian and Wienholtz.Keywords. Curves, finite total curvature, Fáry/Milnor theorem, Schur's comparison theorem, distortion.Here we introduce the ideas of discrete differential geometry in the simplest possible setting: the geometry and curvature of curves, and the way these notions relate for polygonal and smooth curves. The viewpoint has been partly inspired by work in geometric knot theory, which studies geometric properties of space curves in relation to their knot type, and looks for optimal shapes for given knots.After reviewing Jordan's definition of the length of a curve, we consider Milnor's analogous definition [Mil50] of total curvature. In this unified treatment, polygonal and smooth curves are both contained in the larger class of FTC (finite total curvature) curves. We explore the connection between FTC curves and BV functions. Then we examine the theorems of Fáry/Milnor, Schur and Chakerian in terms of FTC curves. We consider relations between total curvature and Gromov's distortion, and then we sketch a proof of a result by Wienholtz in integral geometry. We end by looking at ways to define curvature density for polygonal curves.A companion article [DS08] examines more carefully the topology of FTC curves, showing that any two sufficiently nearby FTC graphs are isotopic. The article [Sul08], also in this volume, looks at curvatures of smooth and discrete surfaces; the discretizations are chosen to preserve various integral curvature relations.Our whole approach in this survey should be compared to that of Alexandrov and Reshetnyak [AR89], who develop much of their theory for curves having one-sided tangents everywhere, a class somewhat more general than FTC.

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Section: Sketch Of Proofmentioning

confidence: 99%

Curves of Finite Total Curvature

Sullivan

2008

Discrete Differential Geometry

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“…By Jensen's inequality, convex functions have this property and if the target space of U has dimension one, the set of quasiconvex functions, reduces itself to the set of convex functions [51]. However, if both the domain and the target space have dimension greater than one, there exist quasiconvex functions which are not convex as shown for instance by (1.7).…”

Section: 2mentioning

confidence: 97%

Univalentσ-harmonic mappings: applications to composites

Alessandrini

Nesi

2002

ESAIM: COCV

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Abstract. This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all the two dimensional G-closure problems in conductivity.

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“…Then, Eq. (14a) admits h-transported solutions ρ (1) = αδy |γ , ρ (2) = β δx |γ (α, β ∈ R) respectively along γ (1) and along γ (2) . Both arcs are therefore abnormal.…”

Section: Examplesmentioning

confidence: 99%

“…Both arcs are therefore abnormal. Nevertheless γ is normal, since no pair ρ (1) , ρ (2) matches into a continuous non-zero virtual 1-form along γ.…”

Section: Examplesmentioning

confidence: 99%

Deformation of piecewise differentiable curves in constrained variational calculus

Massa

Pagani

2017

Differential Geometry and its Applications

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A survey of the geometric tools involved in the study of constrained variational calculus is presented. The central issue is the characterization of the admissible deformations of piecewise differentiable sections of a fibre bundle V n+1 → R, in the presence of arbitrary non-holonomic constraints. Asynchronous displacements of the corners are explicitly considered. The coordinate-independent representation of the variational equation and the associated concepts of infinitesimal control and absolute time derivative are reviewed. In the resulting algebraic environment, every admissible section is assigned a corresponding abnormality index , identified with the co-rank of a suitable linear map. Sections with vanishing index are called normal. A section is called ordinary if every solution of the variational equation vanishing at the endpoints is tangent to some finite deformation with fixed endpoints. The interplay between abnormality index and ordinariness-in particular the fact that every normal evolution is automatically an ordinary one-is discussed.

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Criteri per l'esistenza della soluzione in problemi di Calcolo delle variazioni

Cited by 5 publications

References 1 publication

Curves of Finite Total Curvature

Curves of Finite Total Curvature

Univalentσ-harmonic mappings: applications to composites

Deformation of piecewise differentiable curves in constrained variational calculus

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