In this paper we prove a regularity and rigidity result for displacements in \(GSBD^p\), for every \(p>1\) and any dimension \(n\ge 2\). 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 \(GSBD^p\).

在本文中，我们证明了\(GSBD^p\) 中位移的规律性和刚性结果，对于每个\(p>1\)和任何维度\(n\ge 2\)。我们表明\(GSBD^p\)中的位移与一个小跳跃集与\(W^{1,p}\)函数重合，直到一个小集，其周长和体积由跳。这是 Conti、Focardi 和 Iurlano 的结果推广到更高维度。这样做的结果是，这样的位移满足，最多一个小集合，庞加莱-科恩和科恩不等式。作为一个应用，我们推导出一个近似结果，这意味着\(GSBD^p\) 中存在位移的近似梯度。