CAN THE NONLOCAL CHARACTERIZATION OF SOBOLEV SPACES BY BOURGAIN ET AL. BE USEFUL FOR SOLVING VARIATIONAL PROBLEMS?

We question whether the recent characterization of Sobolev spaces by Bourgain, Brezis, and Mironescu ( 2001) could be useful to solve variational problems on W-1,W-p(Omega). To answer this, we introduce a sequence of functionals so that the seminorm is approximated by an integral operator involving a differential quotient and a radial mollifier. Then, for the approximated formulation, we prove existence, uniqueness, and convergence of the solution to the unique solution of the initial formulation. We show that these results can also be extended in the BV-case. Interestingly, this approximation leads to a unified implementation, for Sobolev spaces (including with high p-values) and for the BV space. Finally, we show how this theoretical study can indeed lead to a numerically tractable implementation, and we give some image diffusion results as an illustration.