Convex integration theory without integration

Theillière, Mélanie

doi:10.1007/s00209-021-02785-9

Cited by 3 publications

(4 citation statements)

References 29 publications

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“…To build the sequence (S k ) k , we propose in [6] to replace the usual formula of the Convex Integration Theory by another one, called Corrugation Process:…”

Section: Corrugation Processmentioning

confidence: 99%

“…Consequently, this formula performs the deformations required to build the sequence (S k ) k provided γ has values in a well chosen region. For more details and for a proof of this proposition see [6]. We give below a coordinate free expression of the Corrugation Process: Definition 5.…”

Section: Proposition 4 ([6]) the Mapmentioning

confidence: 99%

“…However the construction of a holonomic solution often requires to repeat the Corrugation Process in several directions ∂ j and consequently needs to re-build at each step the loop family γ on a different subsolution at each time. In [6], we propose to simplify this approach by constructing a bigger loop family γ γ that could be used indifferently regardless of the subsolution. This simplification leads to introduce the notion of surrounding loop family and then the notion of Kuiper relation.…”

Section: Proposition 4 ([6]) the Mapmentioning

confidence: 99%

“…In [6] the reader will find a proof of the proposition as well as examples of Kuiper relations. In the next section, we prove that the relation ofisometric maps is Kuiper in codimension one.…”

Section: Proposition 4 ([6]) the Mapmentioning

confidence: 99%

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Corrugation Process and $ε$-isometric maps

Theillière

2020

Preprint

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Convex Integration is a theory developed in the '70s by M. Gromov. This theory allows to solve families of differential problems satisfying some convex assumptions. From a subsolution, the theory iteratively builds a solution by applying a series of convex integrations. In a previous paper [6], we proposed to replace the usual convex integration formula by a new one called Corrugation Process. This new formula is of particular interest when the differential problem under consideration has the property of being of Kuiper. In this paper, we consider the differential problem of -isometric maps and we prove that it is Kuiper in codimension 1. As an application, we construct -isometric maps from a short map having a conical singularity.1.2. Differential relations. We now introduce the formalism of Gromov's Convex Integration Theory [2]. This theory can be seen as a wide generalization of Nash's approach. It provides a powerful tool to solve a large family of differential constraints. We denote by

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“…To build the sequence (S k ) k , we propose in [6] to replace the usual formula of the Convex Integration Theory by another one, called Corrugation Process:…”

Section: Corrugation Processmentioning

confidence: 99%

Section: Proposition 4 ([6]) the Mapmentioning

confidence: 99%

Section: Proposition 4 ([6]) the Mapmentioning

confidence: 99%

Section: Proposition 4 ([6]) the Mapmentioning

confidence: 99%

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Corrugation Process and $ε$-isometric maps

Theillière

2020

Preprint

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From the Nash–Kuiper theorem of isometric embeddings to the Euler equations for steady fluid motions: Analogues, examples, and extensions

Slemrod

2023

Journal of Mathematical Physics

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Direct linkages between regular or irregular isometric embeddings of surfaces and steady compressible or incompressible fluid dynamics are investigated in this paper. For a surface ( M, g) isometrically embedded in [Formula: see text], we construct a mapping that sends the second fundamental form of the embedding to the density, velocity, and pressure of steady fluid flows on ( M, g). From a Partial Differential Equations perspective, this mapping sends solutions to the Gauss–Codazzi equations to the steady Euler equations. Several families of special solutions of physical or geometrical significance are studied in detail, including the Chaplygin gas on standard and flat tori as well as the irregular isometric embeddings of the flat torus. We also discuss tentative extensions to multiple dimensions.

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