  In this survey we collect some recent results obtained by the authors and
collaborators concerning the fine structure of functions of bounded deformation
(BD). These maps are $\mathrm{L}^1$-functions with the property that the
symmetric part of their distributional derivative is representable as a bounded
(matrix-valued) Radon measure. It has been known for a long time that for a
(matrix-valued) Radon measure the property of being a symmetrized gradient can
be characterized by an under-determined second-order PDE system, the
Saint-Venant compatibility conditions. This observation gives rise to a new
approach to the fine properties of BD-maps via the theory of PDEs for measures,
which complements and partially replaces classical arguments. Starting from
elementary observations, here we elucidate the ellipticity arguments underlying
this recent progress and give an overview of the state of the art. We also
present some open problems.
