Mapping properties of the heat operator on edge manifolds

TL;DR

This paper investigates the behavior of the heat operator on differential forms within edge manifolds, extending classical results to spaces with singularities and establishing foundational properties for solving related parabolic equations.

Contribution

It generalizes the mapping properties of the heat operator to edge manifolds, including those with singularities, and introduces algebraic boundary conditions at edges.

Findings

Established mapping properties of the heat operator on edge manifolds.

Extended classical results to spaces with edge singularities.

Proved short-time existence of solutions for certain semilinear parabolic equations.

Abstract

We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extending the classical results from smooth manifolds and conical spaces. The estimates, together with strong continuity of the heat operator, yield short-time existence of solutions to certain semilinear parabolic equations. Our discussion reviews and generalizes earlier work by Jeffres and Loya.