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

TL;DR

This paper introduces and analyzes the embedding properties of the space $LD_eta^{p}$ on sets with finite perimeter, relevant for fluid-structure interaction problems involving rigid bodies in viscous fluids.

Contribution

It establishes a continuous embedding of $LD_eta^{p}( ext{Omega})$ into an $L^{pN/(N-1)}$ space, connecting deformation and boundary trace integrability.

Findings

Proves the continuous embedding $LD_eta^{p}( ext{Omega}) o L^{pN/(N-1)}( ext{Omega})$.

Provides a functional framework for studying rigid body motion in viscous fluids.

Links geometric measure theory with fluid mechanics applications.

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.