Singular parabolic equations of second order on manifolds with singularities

Abstract

Abstract. The main aim of this article is to establish an Lp-theory for elliptic operators on manifolds with singularities. The particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the singular ends of the manifolds. Such a theory is of importance for the study of elliptic and parabolic equations on non-compact, or even incomplete manifolds, with or without boundary.

The Yamabe flow on incomplete manifolds

“…In this subsection, a class of singular manifolds, called singular manifolds satisfying property H λ , is introduced. This concept has proven itself useful for the theory of second order differential equations on singular manifolds in [54].…”

“…In the rest of this subsection, several properties of weighted function spaces are presented without proof. Their proofs can be found in [54].…”

“…However, the characterization of the domain of A given in [54] is erroneous. We will fix this error and state the necessary changes herein.…”

“…The theoretic basis of this paper is Section 4. Based on the results from [54] and a modification of T.X. Duong's method in [21], we can prove that a class of second order singular elliptic operators defined in (4.1) is R-sectorial.…”

Existence and maximalLp-regularity of solutions for the porous medium equation on manifolds with conical singularities

“…In [38] it was shown existence, uniqueness and maximal L q -regularity for the short time solutions, where in [39] this result was improved to long time existence and smoothness. Moreover, concerning the case of singular manifolds in the sense of H. Amann [1], in [46] it was shown existence, uniqueness and maximal continuous regularity for the short time solutions and in [47] global existence of L 1 -mild solutions; see also [48] and [49] for similar problems on such spaces. For the case of the hyperbolic space, or more generally for Riemannian manifolds with nonpositive sectional curvature, we refer to [15], [16], [19] and [53].…”