In this memoir we consider the Dirichlet problem for parabolic operators in a
half space with singular drift terms. In chapter I we begin the study of a parabolic
PDE modeled on the pullback of the heat equation in certain time varying domains
considered by Lewis - Murray and Hofmann - Lewis. In chapter II we obtain mutual
absolute continuity of parabolic measure and Lebesgue measure on the boundary
of this halfspace and also that the Lq(Rn) Dirichlet problem for these PDE's has a
solution when q is large enough. In chapter III we prove an analogue of a theorem of
Fefferman, Kenig, and Pipher for certain parabolic PDE's with singular drift terms.
Each of the chapters that comprise this memoir has its own numbering system and
list of references.
1991 Mathematics Subject Classification. Primary 42B20, 35K05.
keywords and phrases, parabolic measure, parabolic operators, drift terms, abso-
lute continuity, Dirichlet problem
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