  This paper discusses properties of periodic functions, focusing on (systems
of) partial differential equations with periodicity boundary conditions, called
"cellular problems". These cellular problems arise naturally from the
asymptotic study of PDEs with rapidly oscillating coefficients; this study is
called "homogenization theory". We believe the present paper may shed a new
light on well-known concepts, for instance by showing hidden links between
Green's formula, the div-curl lemma and Donati's representation theorem. We
state and prove three extensions of Donati's Theorem adapted to the periodic
framework which, beyond their own importance, are essential for understanding
the variational formulations of cellular problems in strain, in stress and in
displacement. Section 4 presents a self-contained study of properties of traces
of a function and their relations with periodicity properties of that function.
