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Fine properties of functions with bounded deformation. (English) Zbl 0890.49019
The paper deals with the space of vector fields with bounded deformation: $$\text{BD}(\Omega) = \{u \in L^1(\Omega;R^n)$$: $$E_{ij}u$$ is a Radon measure with bounded total variation, $$i,j = 1,...,n \}$$, where $$E_{ij}u = {1 \over 2} (D_iu^j + D_ju^i)$$, $$Eu = \{E_{ij}u\}$$, which was introduced in late 70-ties in connection with mathematical theory of plasticity. After recalling some fine properties of functions with bounded variation (elements of $$\text{BV}(\Omega;R^n)$$ spaces) the authors prove that many of these properties can be suitably extended to $$\text{BD}(\Omega)$$ spaces. Especially they analyse the set of Lebesgue points and the set where such functions have one-sided approximate limits. They prove a theorem (analogous as in the case of $$BV$$-functions) stating that the symmetric distributional derivative $$Eu$$ can be decomposed (with respect to the Lebesgue measure) into three parts: $$Eu = E^au + E^su = {\mathcal E}u{\mathcal L}^n + E^ju + E^cu$$ (i.e., absolutely continuous, jump and Cantor part, respectively). One of the main results of the paper is the extension to BD-functions of the so-called structure theorem, known for BV-functions and showing that the above parts of the derivative can be recovered from the corresponding ones of the one-dimensional sections. The space $$\text{SBD}(\Omega)$$ of special functions with bounded deformation (for which $$E^cu = 0$$) is defined and characterized in a similar manner as SBV-functions. Finally, it is proved that any function $$u \in \text{BD}(\Omega)$$ is approximately differentiable in $${\mathcal L}^n$$-almost every point in $$\Omega$$, and the proof is complemented by some remarks, one of them stating that in the case where $$E^cu = 0$$, $${\mathcal E}u \in L^p(\Omega)$$ for some $$p>1$$, this proof can be used to obtain the Korn’s inequality.

##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 26B05 Continuity and differentiation questions 46E40 Spaces of vector- and operator-valued functions
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