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

Valent, Tullio. "On the notion of potential for mappings between linear spaces. A generalized version of the Poincaré lemma." Bollettino dell'Unione Matematica Italiana 6-B.2 (2003): 381-392. <http://eudml.org/doc/195881>.

@article{Valent2003,
abstract = {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.},
author = {Valent, Tullio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {381-392},
publisher = {Unione Matematica Italiana},
title = {On the notion of potential for mappings between linear spaces. A generalized version of the Poincaré lemma},
url = {http://eudml.org/doc/195881},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Valent, Tullio
TI - On the notion of potential for mappings between linear spaces. A generalized version of the Poincaré lemma
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/6//
PB - Unione Matematica Italiana
VL - 6-B
IS - 2
SP - 381
EP - 392
AB - 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.
LA - eng
UR - http://eudml.org/doc/195881
ER -