Strictly continuous extension of functionals with linear growth to the space BV

Abstract

The main result of this paper is a proof of the continuity of a family of integral functionals defined on the space of functions of bounded variation with respect to a topology under which smooth functions are dense. These functionals occur often in the Calculus of Variations as the extension of integral problems defined over weakly differentiable functions with linear growth, and the result in this paper sheds light on the question of what the 'correct' extension is in this context. The result is proved via a combination of Reshetnyak's Continuity Theorem and a map assigning a lifting $\mu[u]\in\mathbf{M}(\Omega\times\mathbb{R}^m;\mathbb{R}^{m\times d})$ to each $u\in BV(\Omega;\mathbb{R}^{m})$ and is valid for a large class of integrands satisfying $|f(x,y,A)|\leq C(1+|y|^{d/(d-1)}+|A|)$. In the case where $f$ exhibits $d/(d-1)$ growth in the $y$ variable, an embedding result from the theory of concentration-compactness is needed.