Fine properties of functions with bounded deformation.

*(English)*Zbl 0890.49019The 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.

Reviewer: Z.Denkowski (Kraków)

##### MSC:

49Q20 | Variational problems in a geometric measure-theoretic setting |

26B05 | Continuity and differentiation questions |

46E40 | Spaces of vector- and operator-valued functions |