Analysis of a nonconvex problem related to signal selective smoothing

We study some properties of a nonconvex variational problem. The infimum of the functional that has to be minimized fails to be attained. Instead, minimizing sequences develop gradient oscillations which allow them to decrease the value of the functional. We show an existence result for a perturbed nonconvex version of the problem, and we study the qualitative properties of the corresponding minimizer. The pattern of the gradient oscillations for the original non perturbed problem is analyzed numerically.