Injectivity and adjoint extremals
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.
scientific article; zbMATH DE number 1299956