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Of the structure of the Euler mapping. (English) Zbl 0337.33012

Let \(\mathbb R^n\), \(\mathbb R^m\) be real Euclidean spaces of dimension \(n, m\), respectively, \(L(\mathbb R^n, \mathbb R^m)\) the vector space of all linear mappings from \(\mathbb R^n\) into \(\mathbb R^m\), and \(L_s^2(\mathbb R^n, \mathbb R^n)\) the vector space of all symmetric bilinear mappings from \(\mathbb R^n\times\mathbb R^n\) into \(\mathbb R^n\). For open subsets \(U\subset\mathbb R^n\) and \(V\subset\mathbb R^n\) define \(\mathcal J^1=U\times V\times L(\mathbb R^n, \mathbb R^m)\) and \(\mathcal J^2 = U\times V\times L(\mathbb R^n, \mathbb R^n)\times L_s^2(\mathbb R^n, \mathbb R^n)\). Assume \(\mathcal J^1\) and \(\mathcal J^2\) are differentiable manifolds in natural coordinates. Let \(L\) be a Lagrange function and \(\mathcal E_\mu(L)=0\), \(\mu=1,2,\ldots,m\), the Euler equations defining the extremals associated with \(L\). Certain sufficient conditions for the identical vanishing of the left side of the Euler equations are known. However, a complete description of the Lagrange functions satisfying the Euler 1-form \(\mathcal E_\mu(L)=0\), \((\mu=1)\) is not known except when \(m=1\). An attempt is made to provide such a description.
Let \(\mathcal L(\mathcal J^1)\) and \(\Omega^1(\mathcal J^2)\) denote the vector spaces of all Lagrange functions and Euler 1-forms on \(\mathcal J^2\), respectively. An effort is made to study the kernel of the linear mapping \(\mathcal L(\mathcal J^1)\ni L\to \mathcal E(L)\in\Omega^1(\mathcal J^2)\). After establishing several lemmas, the equivalence of \(\mathcal E(L)=0\) to the existence of an \(n\)-form satisfying certain conditions is proved.
Reviewer: N. A. Warsi

MSC:

58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
58E30 Variational principles in infinite-dimensional spaces
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