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Publication

Optimal transport of closed differential forms for convex costs

Bernard Dacorogna, Olivier Kneuss
2015
Journal Articles

Abstract

Let c : A(k-1) -> R+ be convex and Omega subset of R-n be a bounded domain. Let f(0) and f(1) be two closed k-forms on Omega satisfying appropriate boundary conditions. We discuss the minimization of integral(Omega) c (A) dx over a subset of (k - 1)-forms A on Omega such that dA + f(1) - f(0) = 0, and its connection with a transport of symplectic forms. Section 3 mainly serves as a step toward Section 4, which is richer, as it connects to variational problems with multiple minimizers. (C) 2015 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Official source

https://infoscience.epfl.ch/entities/publication/7d341337-6913-4017-8912-3f81e83824c0

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Bernard Dacorogna, Olivier Kneuss
2015
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/7d341337-6913-4017-8912-3f81e83824c0

About this result

Ontological neighbourhood

Mathematics

Geometry: Differential geometry

Related concepts (32)

Differential form

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) dx is an example of a 1-form, and can be integrated over an interval [a, b] contained in the domain of f: Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form that can be integrated over a surface S: The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms.

Closed and exact differential forms

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the of d, and a closed form is in the kernel of d. For an exact form α, α = dβ for some differential form β of degree one less than that of α. The form β is called a "potential form" or "primitive" for α.

Connection form

In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object.

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