For degenerate elliptic partial differential equations, it is often desirable to show that a weak solution is smooth. The first and most difficult step in this process is establishing local Hölder continuity. Sufficient conditions for establishing continuity have already been documented in [FP], [SW1], and [MRW], and their necessity in [R]. However, the complexity of the equations discussed in those works makes it difficult to understand the core structure of the arguments employed. Here, we present a harmonic-analytic method for establishing Hölder continuity of weak solutions in context of a simple linear equation div(Q?u) = f in a homogeneous space structure in order to showcase the form of the argument. Ad- ditionally, we correct an oversight in the adaptation of the John-Nirenberg inequality presented in [SW1], restricting it to a much smaller class of balls.

Liver perfusion can be modelled by Darcy's flow in multiple connected com- partments. The first part of the present thesis shows in detail the existence of a solution to the multi-compartmental model. The flow in each compartment in this model is characterized by a permeability tensor, which is obtained from the geometry of liver vasculature. It turns out that this tensor might be singular, which potentially causes solvability problems. The second part deals with this abnormality in one compartment. By using the theory of degenerate Sobolev spaces, an appropriate weak formulation is defined. Analogues of Poincar'e and traces inequalities in this degenerate setting are proved, which also imply the existence of the weak solutions. In addition, this part justifies another possibil- ity how to deal with degenerate permeability, which is regularizing the tensor by adding a small isotropic permeability to it. In the third part, the aim is to find subdomains of autonomous perfusion with respect to the source positions. This is formulated as a minimization problem and several numerical results are presented. 1