Green's function for second order parabolic systems with Neumann boundary condition

Abstract

We study the Neumann Green's function for second order parabolic systems in divergence form with time-dependent measurable coefficients in a cylindrical domain $\mathcal{Q}=\Omega\times (-\infty,\infty)$, where $\Omega\subset \mathbb{R}^n$ is an open connected set such that a multiplicative Sobolev embedding inequality holds there. Such a domain includes, for example, a bounded Sobolev extension domain, a special Lipschitz domain, and an unbounded domain with compact Lipschitz boundary. We construct the Neumann Green's function in $\mathcal{Q}$ under the assumption that weak solutions of the systems satisfy an interior H\"older continuity estimate. We also establish global Gaussian bounds for Neumann Green's function under an additional assumption that weak solutions with zero Neumann data satisfy a local boundedness estimate. In the scalar case, such a local boundedness estimate is a consequence of De Giorgi-Moser-Nash theory holds for equations with bounded measurable coefficients in Sobolev extension domains, while in the vectorial case, one may need to impose further regularity assumptions on the coefficients of the system as well as on the domain to obtain such an estimate. We present a unified approach valid for both the scalar and vectorial cases and discuss some applications of our results including the construction of Neumann functions for second order elliptic systems with measurable coefficients in two dimensional domains.