Characterization of the dual space of BV

Characterize the dual space of the space of functions of bounded variation, BV(\mathbb{R}^N), to support the development of variational methods for the nonsmooth 1-Laplace equation.

Background

The paper studies normalized solutions of a 1-Laplace equation in BV(\mathbb{R}^N), where the associated energy functional is locally Lipschitz but generally not Gâteaux differentiable. Standard variational arguments therefore require derivatives or generalized gradients in an appropriate dual space. The authors identify the lack of a clear description of the dual of BV(\mathbb{R}^N) as a significant obstacle, noting that this difficulty complicates the direct application of variational methods. They address it indirectly by extending the functional to the space X=L^1(\mathbb{R}^N)\cap L^{1^*}(\mathbb{R}^N), whose dual can be represented by L_{\infty,N}(\mathbb{R}^N), but the general characterization of the BV dual remains unresolved.