The aim of the present paper is
that of developing a theory for the Weierstrass-Cesari integral of the calculus of
variations ℐ =∫TF(p,q) over a variety T (not necessarily continuous nor
Sobolev’s), with respect to a nonlinear parametric integrand F(p,q). As
is well known, in the case of continuous BV varieties, Cesari framed the
integral ℐ in an abstract, general setting by means of the Burkill-Cesari
integration process on a “quasi-additive” set function. We introduce here a
suitable condition of quasi-additivity type for a couple of set functions (the
Γ-quasi additivity) which allows to adopt Cesari formulation, even in this
more general situation and represents the key-idea for the outcome of our
research.
