  A new formulation of boundary value problems in gradient elasticity is
presented in this work. The main outcome is the construction of partial
differential systems of second order, which are typically equivalent with the
well known fourth order equation of gradient elasticity. Two alternative
methodologies are developed and presented in the present work. The first
approach is based on the framework of the pseudo-differential calculus and
exploits the special characteristics of this approach to deal with the non
local behavior of gradient elasticity. The second implementation is purely
differential and is based on the augmentation of the independent variables of
the problem. Under this concept, the constitutive equations of gradient
elasticity become part of the differential system itself and the whole
framework is reminiscent of the transformation of the wave equation to a first
order differential system. Crucial issues like existence, uniqueness and
stability of the corresponding initial boundary value problems are also
encountered via the new concepts.
