Embeddings for the space $L{D}_{\gamma }^p$ on sets of finite perimeter

Given an open set with finite perimeter $\mathrm{\Omega }\subset {\mathbb{R}}^n$, we consider the space $L{D}_{\gamma }^p(\mathrm{\Omega })$, $1\le p<\mathrm{\infty }$, of functions with $p$th-integrable deformation tensor on $\mathrm{\Omega }$ and with $p$ th-integrable trace value on the essential boundary of $\mathrm{\Omega }$. We establish the continuous embedding $L{D}_{\gamma }^p(\mathrm{\Omega })\subset L^{pN\mathrm{/}(N-1)}(\mathrm{\Omega })$. The space $L{D}_{\gamma }^p(\mathrm{\Omega })$ and this embedding arise naturally in studying the motion of rigid bodies in a viscous, incompressible fluid.