Calculus of Variations and Geometric Measure Theory

F. Cagnetti - A. Chambolle - L. Scardia

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

created by cagnetti on 09 Apr 2020
modified on 13 May 2021


Accepted Paper

Inserted: 9 apr 2020
Last Updated: 13 may 2021

Journal: Math. Ann.
Year: 2020


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, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in $GSBD^p$.