A variational weak weighted derivative: Sobolev spaces and degenerate elliptic equations∗

A new class of weak weighted derivatives and its associated Sobolev spaces is introduced and studied. The proposed notion uses a variational formulation in its definition which generalizes the usual weighting of the classical weak derivative. Such a construction naturally leads to Sobolev spaces containing classes of discontinuous functions. Weak closedness with respect to both varying functions and weights are obtained as well as density results and the validity of certain calculus rules in the respective spaces. Moreover, the local properties of functions whose weak weighted derivative exists are examined. Connections to the classical partial weak differentiation are established. In order to demonstrate the applicability, a quasi-linear degenerate elliptic equation modeling an edge-preserving denoising procedure in image processing is analyzed. It turns out that the existence of potentially discontinuous weak solutions for such problems can be ensured, utilizing the closedness and density results for the weighted Sobolev spaces.