Green's functions for parabolic systems of second order in time-varying domains

TL;DR

This paper develops Green's functions for second order parabolic systems in complex, non-smooth, time-varying domains, providing pointwise estimates under certain regularity assumptions on solutions.

Contribution

It constructs Green's functions in non-smooth, time-varying domains with new boundary regularity conditions and derives global estimates assuming solution regularity.

Findings

Constructed Green's functions in non-smooth, time-varying domains.

Derived global pointwise estimates for Green's functions.

Applicable to complex perturbations of real equations.

Abstract

We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and 1/2-H\"older continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior H\"older continuity estimate. We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local H\"older continuity estimate. In particular, our results apply to complex perturbations of a single real equation.