3,039 research outputs found
Fine properties of functions of bounded deformation - an approach via linear PDES
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 L1-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
On the Trace Operator for Functions of Bounded -Variation
In this paper, we consider the space of
functions of bounded -variation. For a given first order linear
homogeneous differential operator with constant coefficients , this
is the space of --functions such that the
distributional differential expression is a finite (vectorial)
Radon measure. We show that for Lipschitz domains ,
-functions have an -trace
if and only if is -elliptic (or, equivalently, if the
kernel of is finite dimensional). The existence of an
-trace was previously only known for the special cases
that coincides either with the full or the symmetric gradient of
the function (and hence covered the special cases or
). As a main novelty, we do not use the fundamental theorem of
calculus to construct the trace operator (an approach which is only available
in the - and -setting) but rather compare projections
onto the nullspace as we approach the boundary. As a sample application, we
study the Dirichlet problem for quasiconvex variational functionals with linear
growth depending on
Characterization of Generalized Young Measures Generated by Symmetric Gradients
This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer\ue2\u80\u93Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The \ue2\u80\u9clocal\ue2\u80\u9d proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti\ue2\u80\u99s rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences
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