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
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.