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

Abstract

AbstractIn this paper we prove a regularity and rigidity result for displacements in $$GSBD^p$$ G S B D p , for every $$p>1$$ p > 1 and any dimension $$n\ge 2$$ n ≥ 2 . We show that a displacement in $$GSBD^p$$ G S B D p with a small jump set coincides with a $$W^{1,p}$$ W 1 , p function, 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 $$GSBD^p$$ G S B D p .