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

Abstract

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