Injectivity and adjoint extremals
type: Item
instance of: scholarly article
title: Injectivity and adjoint extremals
zbMATH Open document ID: 0934.49001
author: Ulrich Clarenz
published in: Calculus of Variations and Partial Differential Equations
publication date: 2000-04-25T00:00:00Z
review text: The author studies the injectivity of local diffeomorphisms \(f:\Omega\to \mathbb{R}^n\) where \(\Omega\) is an open subset of \(\mathbb{R}^n\). His motivation stems from applications in the calculus of variations where the question of the global invertibility arises in connection with the transformations of Haar, Hölder, and Legendre. In the first part of the paper some general injectivity results are proved, one of them extending a theorem of \textit{B. Smyth} and \textit{F. Xavier} [J. Differ. Equations 130, No. 2, 406-414 (1996; Zbl 0870.58036)] about the invertibility of local diffeomorphisms \(f:\mathbb{R}^n\to \mathbb{R}^n\) to the case \(f:\Omega\to \mathbb{R}^n\). The main part of the paper deals with the special transformations which are derived from a variational integrand for scalar functions of \(n\) variables \(F= F(x,z,p)\), \(x\in\mathbb{R}^n\), \(z\in \mathbb{R}\), \(p\in\mathbb{R}^n\) and which are of the form \((x,z,p)\mapsto (x,z,y)\) with \(y= F_p\) (Legendre), \(y= p/F\) (Hölder), or \(y= F_p/(pF_p- F)\) (Haar). A typical result of the author states, e.g., that the Legendre transformation is globally injective provided that the matrix \(F_{pp}\) is nonsingular, the Weierstrass excess function is always nonnegative, and the domain of the \(p\)-variable is starshaped with respect to the origin for each fixed \((x,z)\). A further result of the paper generalizes a theorem of R. Krust for the area integrand \(F(p)= \sqrt{1+|p|^2}\) to the case of a general \(F(p)\). This theorem says that the (suitably defined) adjoint surface of a simply connected extremal surface which is a graph is again a graph.
reviewed by: F. Tomi
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 1299956
zbMATH Keywords: adjoint extremals
zbMATH Keywords: Haar transformation
zbMATH Keywords: Hölder transformation
zbMATH Keywords: injectivity of local diffeomorphisms
zbMATH Keywords: Legendre transformation
zbMATH Keywords: Weierstrass excess function
zbMATH Keywords: extremal surface
MaRDI profile type: Publication
full work available at URL: s005260050127
OpenAlex ID: W2020532459
Wikidata QID: Q123121814
DOI: 10.1007/S005260050127
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description: scientific article; zbMATH DE number 1299956