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

Abstract

Given an open set with finite perimeter $\Omega \subset {\mathbb{R}}^n$, we consider the space $L{D}_{\gamma }^p(\Omega )$, $1\le p<\infty$, 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 $L{D}_{\gamma }^p(\Omega )\subset L^{pN/(N-1)}(\Omega )$. The space $L{D}_{\gamma }^p(\Omega )$ and this embedding arise naturally in studying the motion of rigid bodies in a viscous, incompressible fluid.

20 pages, 3 figures