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

created by cagnetti on 09 Apr 2020
modified on 02 Jun 2020

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Submitted Paper

Inserted: 9 apr 2020
Last Updated: 2 jun 2020

Year: 2020

Abstract:

In this paper we prove a regularity and rigidity result for displacements in $GSBD^p$, for every $p>1$ and any dimension $n\geq 2$. We show that a displacement in $GSBD^p$ with a small jump set coincides with a $W^{1,p}$ 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, Poincar\'e-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in $GSBD^p$.