On parabolic equations of Kolmogorov-Fokker-Planck type

Abstract: In this thesis solutions to Kolmogorov-Fokker-Planck type equations are studied. It consists of a comprehensive summary and four scientific articles.In the first article a potential theory for certain strongly degenerate parabolic operators in unbounded domains of Lipschitz type is developed. Several fundamental properties such as energy estimates and comparison principles are proven, and in particular the main results are solvability of the continuous Dirichlet problem, a Hölder estimate at the boundary, a Carleson estimate, boundary Harnack inequalities, and a doubling property for the parabolic measure.In the second article the results about the parabolic measure studied in the first article are refined considerably under some further assumptions. The main result states that the parabolic measure is absolutely continuous with respect to the surface measure, and the associated Radon-Nikodym derivative defines a weight in the Muckenhoupt class A∞.In the third article an elliptic, a parabolic, and a Kolmogorov type operator are studied. A structural theorem, which allows results about the parabolic and Kolmogorov type operators to be concluded from the corresponding results about the elliptic operator, is proven. In particular, results about the Lp Dirichlet problem for the Kolmogorov type operator may be derived from the corresponding results for the elliptic operator, using boundary estimates developed in the first article. The established results are then applied on a homogenization problem for the Kolmogorov type operator.In the fourth article the existence and uniqueness, in bounded and unbounded Lipschitz type cylinders, of weak solutions to Cauchy-Dirichlet problems for strongly degenerate parabolic operators of Kolmogorov-Fokker-Planck type is established.