On an Extension of the Space of Bounded Deformations

Abstract

As for functions of bounded variations ($BV$), the space of functions of bounded deformations ($BD$) can be equipped with different topologies. We propose an extension of the traditional space of $BD$ functions equipped with the weak-$\star$ topology. It is shown that in this fine extension, both compactness and weak-$\star$ continuity of the trace hold under weak assumptions. This is relevant for applications, where the specimen is typically exposed to forces on the boundary. We give an application to a functional with a linearly growing energy density.