Korn and Poincaré-Korn inequalities for functions with a small jump set

AbstractIn this paper we prove a regularity and rigidity result for displacements in$$GSBD^p$$GSBDp, for every$$p>1$$p>1and any dimension$$n\ge 2$$n≥2. We show that a displacement in$$GSBD^p$$GSBDpwith a small jump set coincides with a$$W^{1,p}$$W1,pfunction, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in$$GSBD^p$$GSBDp.