Variational Functionals for the Characterization of BV and Sobolev Spaces

This paper examines the characterization of bounded variation (BV) and Sobolev functions by using some non-local functionals. We analyze their pointwise convergence and establish a connection with the Sobolev norm and the total variation measure. By investigating a wider class of non local functionals, we provide a deeper understanding of how these approximations capture the local properties of BV and Sobolev spaces, thereby reinforcing their applicability in the Euclidean setting.