On the notion of potential for mappings between linear spaces. A generalized version of the Poincaré lemma

Valent, Tullio

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 2, pp. 381-392

Résumé

An approach to the theory of linear differential forms in a radial subset of an (arbitrary) real linear space $X$ without a Banach structure is proposed. Only intrinsic (partially linear) topologies on $X$ are (implicitly) involved in the definitions and statements. Then a mapping $F \colon U\subseteq X \to Y$, with $X$, $Y$ real linear spaces and $U$ a radial subset of $X$, is considered. After showing a representation theorem of those bilinear forms $\langle \cdot,\cdot \rangle$ on $X\times Y$ for which $\langle x, y\rangle =0$$\forall x\in X$$ \Rightarrow y=0$, we observe that the assignment of such a bilinear form allows to associate (in a natural way) a linear differential form to the mapping $F$; this fact spontaneously leads us to a definition of potentialness for $F$. This definition has a special interest in the case when the mapping $F$ describes a boundary and, or, initial value problem; a simple example, originated from finite elasticity, is explained in sect. 6.