Remarks on the H\"older-continuity of solutions to parabolic equations with conic singularities

TL;DR

This paper extends the understanding of H"older-continuity for solutions to linear parabolic equations, including those with conic singularities on K"ahler manifolds, under certain coefficient conditions.

Contribution

It provides a line-by-line adaptation of existing methods to prove H"older-continuity in new settings involving conic singularities and specific coefficient bounds.

Findings

H"older-continuity established for solutions with conic singularities

Applicable to parabolic equations on K"ahler manifolds with singular metrics

Conditions on coefficients ensure regularity of solutions

Abstract

This is a note on \cite{LSU} and \cite{FS}. Using their work line by line, we prove the H\"older-continuity of solutions to linear parabolic equations of mixed type, assuming the coefficient of $\frac{\partial}{\partial t}$ has time-derivative bounded from above. On a K\"ahler manifold, this H\"older estimate works when the metrics possess conic singularities along a normal crossing divisor.