Continuous maximal regularity on singular manifolds and its applications

Abstract

In this article, we set up the continuous maximal regularity theory for a class of linear differential operators on manifolds with singularities. These operators exhibit degenerate or singular behaviors while approaching the singular ends. Particular examples of such operators include differential operators defined on domains, which degenerate fast enough toward the boundary. Applications of the theory established herein are shown to the Yamabe flow, the porous medium equation, the parabolic $p$-Laplacian equation and the thin film equation. Some comments about the boundary blow-up problem, and waiting time phenomena for singular or degenerate parabolic equations can also be found in this paper.

References in corpus (7)

• Some Remarks on Uniformly Regular Riemannian Manifolds
• Continuous maximal regularity on uniformly regular Riemannian manifolds
• Anisotropic Function Spaces on Singular Manifolds
• A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows
• Singular parabolic equations of second order on manifolds with singularities
• The Yamabe flow on incomplete manifolds
• Existence and maximal $L^p$-regularity of solutions for the porous medium equation on manifolds with conical singularities

Cited by in corpus (4)

• Singular parabolic equations of second order on manifolds with singularities
• The Yamabe flow on incomplete manifolds
• Wellposedness of a nonlocal nonlinear diffusion equation of image processing
• Function Spaces on Uniformly Regular and Singular Riemannian Manifolds