  Ultrafunctions are a particular class of functions defined on a hyperreal
field $\mathbb{R}^{\ast}\supset\mathbb{R}$. They have been introduced and
studied in some previous works. In this paper we introduce a particular space
of ultrafunctions which has special properties, especially in term of
localization of functions together with their derivatives. An appropriate
notion of integral is then introduced which allows to extend in a consistent
way the integration by parts formula, the Gauss theorem and the notion of
perimeter. This new space we introduce, seems suitable for applications to
PDE's and Calculus of Variations. This fact will be illustrated by a simple,
but meaningful example.
