Weak solutions of linear degenerate parabolic equations and an application in image processing

In this paper, linear degenerate parabolic diffusion equations of second order with discontinuous coefficients are studied with respect to existence and uniqueness of weak solutions. We consider the full degenerate case where the diffusion is given by a tensor field which is only positive semi-definite and essentially bounded in the whole domain. Existence of solutions in Hilbert spaces incorporating the diffusion tensor is proven and uniqueness in a certain sense is established. Moreover, we examine replacements for the missing compactness by the Lions-Aubin lemma, proving that the set of solutions associated with bounded data and bounded semi-definite coefficients is weakly relatively compact in a space of weakly continuous functions. Finally, an application to the image-processing problem of edge-preserving denoising is presented. A method based on the considered equations is introduced and numerical examples are given.