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Journal of Convex Analysis 28 (2021), No. 2, 457--470
Copyright Heldermann Verlag 2021

An Extension Result for Generalised Special Functions of Bounded Deformation

Filippo Cagnetti
Department of Mathematics, University of Sussex, Brighton, United Kingdom

Antonin Chambolle
CMAP -- CNRS, Ecole Polytechnique, Palaiseau, France

Matteo Perugini
Institut für numerische und angewandte Mathematik, Westf.-Wilhelms-Universität, Münster, Germany

Lucia Scardia
Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom


We show an extension result for generalised special functions of bounded deformation ($GSBD^p$, for every $p>1$) and any dimension $n\geq 2$. The proof is based on a recent result of F.\,Cagnetti, A.\,Chambolle, and L.\,Scardia, showing that a function $u$ 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 of $u$.

Keywords: Free-discontinuity problems, functions of bounded deformation, Griffith's energy.

MSC: 49Q20, 70G75, 74R10.

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