Theory of parabolic differential equations on singular manifolds and its applications to geometric analysis

Abstract

The main objective of this work is to establish a Holder and Lp theory for elliptic operators on manifolds with singularities. These operators may exhibit degenerate or singular behaviors while approaching the singular ends of the manifolds. Then we apply the theory established herein to several parabolic equations arising in geometric analysis, degenerate boundary value problems, and boundary blow-up problems.