Singular parabolic equations of second order on manifolds with singularities

TL;DR

This paper develops an $L_p$-theory for second-order elliptic operators on manifolds with singularities, addressing degenerate or singular behaviors near singular ends, crucial for elliptic and parabolic equations on complex manifolds.

Contribution

It introduces a new $L_p$-theory framework for elliptic operators on manifolds with singularities, extending analysis to degenerate and singular cases.

Findings

Established $L_p$-theory for elliptic operators on singular manifolds

Addressed degenerate and singular behaviors near manifold ends

Applicable to non-compact and incomplete manifolds

Abstract

The main aim of this article is to establish an $L_p$-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.